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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|---|---|---|---|---|---|---|---|---|---|---|
continuous_linear_equiv.comp_cont_diff_iff
(e : F ≃L[𝕜] G) :
cont_diff 𝕜 n (e ∘ f) ↔ cont_diff 𝕜 n f | by simp only [← cont_diff_on_univ, e.comp_cont_diff_on_iff] | lemma | continuous_linear_equiv.comp_cont_diff_iff | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_on_univ"
] | Composition by continuous linear equivs on the left respects higher differentiability. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to_on.comp_continuous_linear_map
(hf : has_ftaylor_series_up_to_on n f p s) (g : G →L[𝕜] E) :
has_ftaylor_series_up_to_on n (f ∘ g)
(λ x k, (p (g x) k).comp_continuous_linear_map (λ _, g)) (g ⁻¹' s) | begin
let A : Π m : ℕ, (E [×m]→L[𝕜] F) → (G [×m]→L[𝕜] F) :=
λ m h, h.comp_continuous_linear_map (λ _, g),
have hA : ∀ m, is_bounded_linear_map 𝕜 (A m) :=
λ m, is_bounded_linear_map_continuous_multilinear_map_comp_linear g,
split,
{ assume x hx,
simp only [(hf.zero_eq (g x) hx).symm, function.comp... | lemma | has_ftaylor_series_up_to_on.comp_continuous_linear_map | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"continuous.comp_continuous_on",
"continuous_linear_map.map_zero",
"has_fderiv_at",
"has_ftaylor_series_up_to_on",
"is_bounded_linear_map",
"is_bounded_linear_map_continuous_multilinear_map_comp_linear"
] | If `f` admits a Taylor series `p` in a set `s`, and `g` is linear, then `f ∘ g` admits a Taylor
series in `g ⁻¹' s`, whose `k`-th term is given by `p k (g v₁, ..., g vₖ)` . | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at.comp_continuous_linear_map {x : G}
(g : G →L[𝕜] E) (hf : cont_diff_within_at 𝕜 n f s (g x)) :
cont_diff_within_at 𝕜 n (f ∘ g) (g ⁻¹' s) x | begin
assume m hm,
rcases hf m hm with ⟨u, hu, p, hp⟩,
refine ⟨g ⁻¹' u, _, _, hp.comp_continuous_linear_map g⟩,
apply continuous_within_at.preimage_mem_nhds_within',
{ exact g.continuous.continuous_within_at },
{ apply nhds_within_mono (g x) _ hu,
rw image_insert_eq,
exact insert_subset_insert (imag... | lemma | cont_diff_within_at.comp_continuous_linear_map | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at",
"continuous_within_at.preimage_mem_nhds_within'",
"nhds_within_mono"
] | Composition by continuous linear maps on the right preserves `C^n` functions at a point on
a domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.comp_continuous_linear_map
(hf : cont_diff_on 𝕜 n f s) (g : G →L[𝕜] E) :
cont_diff_on 𝕜 n (f ∘ g) (g ⁻¹' s) | λ x hx, (hf (g x) hx).comp_continuous_linear_map g | lemma | cont_diff_on.comp_continuous_linear_map | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on"
] | Composition by continuous linear maps on the right preserves `C^n` functions on domains. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.comp_continuous_linear_map {f : E → F} {g : G →L[𝕜] E}
(hf : cont_diff 𝕜 n f) : cont_diff 𝕜 n (f ∘ g) | cont_diff_on_univ.1 $
cont_diff_on.comp_continuous_linear_map (cont_diff_on_univ.2 hf) _ | lemma | cont_diff.comp_continuous_linear_map | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_on.comp_continuous_linear_map"
] | Composition by continuous linear maps on the right preserves `C^n` functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.iterated_fderiv_within_comp_right
{f : E → F} (g : G →L[𝕜] E) (hf : cont_diff_on 𝕜 n f s)
(hs : unique_diff_on 𝕜 s) (h's : unique_diff_on 𝕜 (g⁻¹' s)) {x : G}
(hx : g x ∈ s) {i : ℕ} (hi : (i : ℕ∞) ≤ n) :
iterated_fderiv_within 𝕜 i (f ∘ g) (g ⁻¹' s) x =
(iterated_fderiv_within 𝕜 i ... | (((hf.ftaylor_series_within hs).comp_continuous_linear_map g).eq_ftaylor_series_of_unique_diff_on
hi h's hx).symm | lemma | continuous_linear_map.iterated_fderiv_within_comp_right | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on",
"iterated_fderiv_within",
"unique_diff_on"
] | The iterated derivative within a set of the composition with a linear map on the right is
obtained by composing the iterated derivative with the linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_equiv.iterated_fderiv_within_comp_right
(g : G ≃L[𝕜] E) (f : E → F) (hs : unique_diff_on 𝕜 s) {x : G} (hx : g x ∈ s) (i : ℕ) :
iterated_fderiv_within 𝕜 i (f ∘ g) (g ⁻¹' s) x =
(iterated_fderiv_within 𝕜 i f s (g x)).comp_continuous_linear_map (λ _, g) | begin
induction i with i IH generalizing x,
{ ext1 m,
simp only [iterated_fderiv_within_zero_apply,
continuous_multilinear_map.comp_continuous_linear_map_apply] },
{ ext1 m,
simp only [continuous_multilinear_map.comp_continuous_linear_map_apply,
continuous_linear_equiv.coe_coe, iterated_fderiv... | lemma | continuous_linear_equiv.iterated_fderiv_within_comp_right | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"continuous_linear_equiv.coe_coe",
"continuous_linear_equiv.comp_fderiv_within",
"continuous_linear_equiv.comp_right_fderiv_within",
"continuous_linear_map.coe_comp'",
"continuous_multilinear_map.comp_continuous_linear_map_apply",
"continuous_multilinear_map.comp_continuous_linear_map_equivL",
"continuo... | The iterated derivative within a set of the composition with a linear equiv on the right is
obtained by composing the iterated derivative with the linear equiv. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.iterated_fderiv_comp_right
(g : G →L[𝕜] E) {f : E → F} (hf : cont_diff 𝕜 n f) (x : G) {i : ℕ} (hi : (i : ℕ∞) ≤ n) :
iterated_fderiv 𝕜 i (f ∘ g) x =
(iterated_fderiv 𝕜 i f (g x)).comp_continuous_linear_map (λ _, g) | begin
simp only [← iterated_fderiv_within_univ],
apply g.iterated_fderiv_within_comp_right hf.cont_diff_on unique_diff_on_univ unique_diff_on_univ
(mem_univ _) hi,
end | lemma | continuous_linear_map.iterated_fderiv_comp_right | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"iterated_fderiv",
"iterated_fderiv_within_univ",
"unique_diff_on_univ"
] | The iterated derivative of the composition with a linear map on the right is
obtained by composing the iterated derivative with the linear map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry_equiv.norm_iterated_fderiv_within_comp_right
(g : G ≃ₗᵢ[𝕜] E) (f : E → F) (hs : unique_diff_on 𝕜 s) {x : G} (hx : g x ∈ s) (i : ℕ) :
‖iterated_fderiv_within 𝕜 i (f ∘ g) (g ⁻¹' s) x‖ = ‖iterated_fderiv_within 𝕜 i f s (g x)‖ | begin
have : iterated_fderiv_within 𝕜 i (f ∘ g) (g ⁻¹' s) x =
(iterated_fderiv_within 𝕜 i f s (g x)).comp_continuous_linear_map (λ _, g),
from g.to_continuous_linear_equiv.iterated_fderiv_within_comp_right f hs hx i,
rw [this, continuous_multilinear_map.norm_comp_continuous_linear_isometry_equiv]
end | lemma | linear_isometry_equiv.norm_iterated_fderiv_within_comp_right | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"continuous_multilinear_map.norm_comp_continuous_linear_isometry_equiv",
"iterated_fderiv_within",
"unique_diff_on"
] | Composition with a linear isometry on the right preserves the norm of the iterated derivative
within a set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
linear_isometry_equiv.norm_iterated_fderiv_comp_right
(g : G ≃ₗᵢ[𝕜] E) (f : E → F) (x : G) (i : ℕ) :
‖iterated_fderiv 𝕜 i (f ∘ g) x‖ = ‖iterated_fderiv 𝕜 i f (g x)‖ | begin
simp only [← iterated_fderiv_within_univ],
apply g.norm_iterated_fderiv_within_comp_right f unique_diff_on_univ (mem_univ (g x)) i,
end | lemma | linear_isometry_equiv.norm_iterated_fderiv_comp_right | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"iterated_fderiv_within_univ",
"unique_diff_on_univ"
] | Composition with a linear isometry on the right preserves the norm of the iterated derivative
within a set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_equiv.cont_diff_within_at_comp_iff (e : G ≃L[𝕜] E) :
cont_diff_within_at 𝕜 n (f ∘ e) (e ⁻¹' s) (e.symm x) ↔
cont_diff_within_at 𝕜 n f s x | begin
split,
{ assume H,
simpa [← preimage_comp, (∘)] using H.comp_continuous_linear_map (e.symm : E →L[𝕜] G) },
{ assume H,
rw [← e.apply_symm_apply x, ← e.coe_coe] at H,
exact H.comp_continuous_linear_map _ },
end | lemma | continuous_linear_equiv.cont_diff_within_at_comp_iff | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at"
] | Composition by continuous linear equivs on the right respects higher differentiability at a
point in a domain. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_equiv.cont_diff_at_comp_iff (e : G ≃L[𝕜] E) :
cont_diff_at 𝕜 n (f ∘ e) (e.symm x) ↔ cont_diff_at 𝕜 n f x | begin
rw [← cont_diff_within_at_univ, ← cont_diff_within_at_univ, ← preimage_univ],
exact e.cont_diff_within_at_comp_iff
end | lemma | continuous_linear_equiv.cont_diff_at_comp_iff | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at",
"cont_diff_within_at_univ"
] | Composition by continuous linear equivs on the right respects higher differentiability at a
point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_equiv.cont_diff_on_comp_iff (e : G ≃L[𝕜] E) :
cont_diff_on 𝕜 n (f ∘ e) (e ⁻¹' s) ↔ cont_diff_on 𝕜 n f s | begin
refine ⟨λ H, _, λ H, H.comp_continuous_linear_map (e : G →L[𝕜] E)⟩,
have A : f = (f ∘ e) ∘ e.symm,
by { ext y, simp only [function.comp_app], rw e.apply_symm_apply y },
have B : e.symm ⁻¹' (e ⁻¹' s) = s,
by { rw [← preimage_comp, e.self_comp_symm], refl },
rw [A, ← B],
exact H.comp_continuous_l... | lemma | continuous_linear_equiv.cont_diff_on_comp_iff | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on"
] | Composition by continuous linear equivs on the right respects higher differentiability on
domains. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_equiv.cont_diff_comp_iff (e : G ≃L[𝕜] E) :
cont_diff 𝕜 n (f ∘ e) ↔ cont_diff 𝕜 n f | begin
rw [← cont_diff_on_univ, ← cont_diff_on_univ, ← preimage_univ],
exact e.cont_diff_on_comp_iff
end | lemma | continuous_linear_equiv.cont_diff_comp_iff | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_on_univ"
] | Composition by continuous linear equivs on the right respects higher differentiability. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to_on.prod (hf : has_ftaylor_series_up_to_on n f p s)
{g : E → G} {q : E → formal_multilinear_series 𝕜 E G} (hg : has_ftaylor_series_up_to_on n g q s) :
has_ftaylor_series_up_to_on n (λ y, (f y, g y)) (λ y k, (p y k).prod (q y k)) s | begin
set L := λ m, continuous_multilinear_map.prodL 𝕜 (λ i : fin m, E) F G,
split,
{ assume x hx, rw [← hf.zero_eq x hx, ← hg.zero_eq x hx], refl },
{ assume m hm x hx,
convert (L m).has_fderiv_at.comp_has_fderiv_within_at x
((hf.fderiv_within m hm x hx).prod (hg.fderiv_within m hm x hx)) },
{ ass... | lemma | has_ftaylor_series_up_to_on.prod | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"continuous.comp_continuous_on",
"continuous_multilinear_map.prodL",
"formal_multilinear_series",
"has_fderiv_at.comp_has_fderiv_within_at",
"has_ftaylor_series_up_to_on"
] | If two functions `f` and `g` admit Taylor series `p` and `q` in a set `s`, then the cartesian
product of `f` and `g` admits the cartesian product of `p` and `q` as a Taylor series. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at.prod {s : set E} {f : E → F} {g : E → G}
(hf : cont_diff_within_at 𝕜 n f s x) (hg : cont_diff_within_at 𝕜 n g s x) :
cont_diff_within_at 𝕜 n (λx:E, (f x, g x)) s x | begin
assume m hm,
rcases hf m hm with ⟨u, hu, p, hp⟩,
rcases hg m hm with ⟨v, hv, q, hq⟩,
exact ⟨u ∩ v, filter.inter_mem hu hv, _,
(hp.mono (inter_subset_left u v)).prod (hq.mono (inter_subset_right u v))⟩
end | lemma | cont_diff_within_at.prod | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at",
"filter.inter_mem"
] | The cartesian product of `C^n` functions at a point in a domain is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.prod {s : set E} {f : E → F} {g : E → G}
(hf : cont_diff_on 𝕜 n f s) (hg : cont_diff_on 𝕜 n g s) :
cont_diff_on 𝕜 n (λ x : E, (f x, g x)) s | λ x hx, (hf x hx).prod (hg x hx) | lemma | cont_diff_on.prod | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on"
] | The cartesian product of `C^n` functions on domains is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.prod {f : E → F} {g : E → G}
(hf : cont_diff_at 𝕜 n f x) (hg : cont_diff_at 𝕜 n g x) :
cont_diff_at 𝕜 n (λ x : E, (f x, g x)) x | cont_diff_within_at_univ.1 $ cont_diff_within_at.prod
(cont_diff_within_at_univ.2 hf)
(cont_diff_within_at_univ.2 hg) | lemma | cont_diff_at.prod | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at",
"cont_diff_within_at.prod"
] | The cartesian product of `C^n` functions at a point is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.prod {f : E → F} {g : E → G} (hf : cont_diff 𝕜 n f) (hg : cont_diff 𝕜 n g) :
cont_diff 𝕜 n (λ x : E, (f x, g x)) | cont_diff_on_univ.1 $ cont_diff_on.prod (cont_diff_on_univ.2 hf)
(cont_diff_on_univ.2 hg) | lemma | cont_diff.prod | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_on.prod"
] | The cartesian product of `C^n` functions is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.comp_same_univ
{Eu : Type u} [normed_add_comm_group Eu] [normed_space 𝕜 Eu]
{Fu : Type u} [normed_add_comm_group Fu] [normed_space 𝕜 Fu]
{Gu : Type u} [normed_add_comm_group Gu] [normed_space 𝕜 Gu]
{s : set Eu} {t : set Fu} {g : Fu → Gu} {f : Eu → Fu}
(hg : cont_diff_on 𝕜 n g t) (hf : cont_di... | begin
unfreezingI { induction n using enat.nat_induction with n IH Itop generalizing Eu Fu Gu },
{ rw cont_diff_on_zero at hf hg ⊢,
exact continuous_on.comp hg hf st },
{ rw cont_diff_on_succ_iff_has_fderiv_within_at at hg ⊢,
assume x hx,
rcases (cont_diff_on_succ_iff_has_fderiv_within_at.1 hf) x hx
... | lemma | cont_diff_on.comp_same_univ | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on",
"cont_diff_on_succ_iff_has_fderiv_within_at",
"cont_diff_on_top",
"cont_diff_on_zero",
"continuous_on.comp",
"continuous_within_at.preimage_mem_nhds_within'",
"continuous_within_at_inter'",
"enat.nat_induction",
"filter.inter_mem",
"has_fderiv_within_at",
"mem_of_mem_nhds_within"... | Auxiliary lemma proving that the composition of `C^n` functions on domains is `C^n` when all
spaces live in the same universe. Use instead `cont_diff_on.comp` which removes the universe
assumption (but is deduced from this one). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.comp
{s : set E} {t : set F} {g : F → G} {f : E → F}
(hg : cont_diff_on 𝕜 n g t) (hf : cont_diff_on 𝕜 n f s) (st : s ⊆ f ⁻¹' t) :
cont_diff_on 𝕜 n (g ∘ f) s | begin
/- we lift all the spaces to a common universe, as we have already proved the result in this
situation. -/
let Eu : Type (max uE uF uG) := ulift E,
let Fu : Type (max uE uF uG) := ulift.{(max uE uG) uF} F,
let Gu : Type (max uE uF uG) := ulift.{(max uE uF) uG} G,
-- declare the isomorphisms
have iso... | lemma | cont_diff_on.comp | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on",
"cont_diff_on.comp_same_univ",
"continuous_linear_equiv.coe_apply",
"continuous_linear_equiv.ulift",
"function.comp_apply"
] | The composition of `C^n` functions on domains is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.comp'
{s : set E} {t : set F} {g : F → G} {f : E → F}
(hg : cont_diff_on 𝕜 n g t) (hf : cont_diff_on 𝕜 n f s) :
cont_diff_on 𝕜 n (g ∘ f) (s ∩ f⁻¹' t) | hg.comp (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _) | lemma | cont_diff_on.comp' | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on"
] | The composition of `C^n` functions on domains is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.comp_cont_diff_on {s : set E} {g : F → G} {f : E → F}
(hg : cont_diff 𝕜 n g) (hf : cont_diff_on 𝕜 n f s) :
cont_diff_on 𝕜 n (g ∘ f) s | (cont_diff_on_univ.2 hg).comp hf subset_preimage_univ | lemma | cont_diff.comp_cont_diff_on | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_on"
] | The composition of a `C^n` function on a domain with a `C^n` function is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.comp {g : F → G} {f : E → F}
(hg : cont_diff 𝕜 n g) (hf : cont_diff 𝕜 n f) :
cont_diff 𝕜 n (g ∘ f) | cont_diff_on_univ.1 $ cont_diff_on.comp (cont_diff_on_univ.2 hg)
(cont_diff_on_univ.2 hf) (subset_univ _) | lemma | cont_diff.comp | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_on.comp"
] | The composition of `C^n` functions is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at.comp
{s : set E} {t : set F} {g : F → G} {f : E → F} (x : E)
(hg : cont_diff_within_at 𝕜 n g t (f x))
(hf : cont_diff_within_at 𝕜 n f s x) (st : s ⊆ f ⁻¹' t) :
cont_diff_within_at 𝕜 n (g ∘ f) s x | begin
assume m hm,
rcases hg.cont_diff_on hm with ⟨u, u_nhd, ut, hu⟩,
rcases hf.cont_diff_on hm with ⟨v, v_nhd, vs, hv⟩,
have xmem : x ∈ f ⁻¹' u ∩ v :=
⟨(mem_of_mem_nhds_within (mem_insert (f x) _) u_nhd : _),
mem_of_mem_nhds_within (mem_insert x s) v_nhd⟩,
have : f ⁻¹' u ∈ 𝓝[insert x s] x,
{ apply... | lemma | cont_diff_within_at.comp | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at",
"filter.inter_mem",
"le_rfl",
"mem_of_mem_nhds_within",
"nhds_within_mono",
"nhds_within_restrict''"
] | The composition of `C^n` functions at points in domains is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at.comp_of_mem
{s : set E} {t : set F} {g : F → G} {f : E → F} (x : E)
(hg : cont_diff_within_at 𝕜 n g t (f x))
(hf : cont_diff_within_at 𝕜 n f s x) (hs : t ∈ 𝓝[f '' s] f x) :
cont_diff_within_at 𝕜 n (g ∘ f) s x | (hg.mono_of_mem hs).comp x hf (subset_preimage_image f s) | lemma | cont_diff_within_at.comp_of_mem | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at"
] | The composition of `C^n` functions at points in domains is `C^n`,
with a weaker condition on `s` and `t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at.comp' {s : set E} {t : set F} {g : F → G}
{f : E → F} (x : E)
(hg : cont_diff_within_at 𝕜 n g t (f x)) (hf : cont_diff_within_at 𝕜 n f s x) :
cont_diff_within_at 𝕜 n (g ∘ f) (s ∩ f⁻¹' t) x | hg.comp x (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _) | lemma | cont_diff_within_at.comp' | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at"
] | The composition of `C^n` functions at points in domains is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.comp_cont_diff_within_at {n} (x : E)
(hg : cont_diff_at 𝕜 n g (f x)) (hf : cont_diff_within_at 𝕜 n f s x) :
cont_diff_within_at 𝕜 n (g ∘ f) s x | hg.comp x hf (maps_to_univ _ _) | lemma | cont_diff_at.comp_cont_diff_within_at | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at",
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.comp (x : E)
(hg : cont_diff_at 𝕜 n g (f x))
(hf : cont_diff_at 𝕜 n f x) :
cont_diff_at 𝕜 n (g ∘ f) x | hg.comp x hf subset_preimage_univ | lemma | cont_diff_at.comp | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at"
] | The composition of `C^n` functions at points is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.comp_cont_diff_within_at
{g : F → G} {f : E → F} (h : cont_diff 𝕜 n g)
(hf : cont_diff_within_at 𝕜 n f t x) :
cont_diff_within_at 𝕜 n (g ∘ f) t x | begin
have : cont_diff_within_at 𝕜 n g univ (f x) :=
h.cont_diff_at.cont_diff_within_at,
exact this.comp x hf (subset_univ _),
end | lemma | cont_diff.comp_cont_diff_within_at | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.comp_cont_diff_at {g : F → G} {f : E → F} (x : E)
(hg : cont_diff 𝕜 n g) (hf : cont_diff_at 𝕜 n f x) : cont_diff_at 𝕜 n (g ∘ f) x | hg.comp_cont_diff_within_at hf | lemma | cont_diff.comp_cont_diff_at | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_fst : cont_diff 𝕜 n (prod.fst : E × F → E) | is_bounded_linear_map.cont_diff is_bounded_linear_map.fst | lemma | cont_diff_fst | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"is_bounded_linear_map.cont_diff",
"is_bounded_linear_map.fst"
] | The first projection in a product is `C^∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.fst {f : E → F × G} (hf : cont_diff 𝕜 n f) : cont_diff 𝕜 n (λ x, (f x).1) | cont_diff_fst.comp hf | lemma | cont_diff.fst | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff"
] | Postcomposing `f` with `prod.fst` is `C^n` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.fst' {f : E → G} (hf : cont_diff 𝕜 n f) : cont_diff 𝕜 n (λ x : E × F, f x.1) | hf.comp cont_diff_fst | lemma | cont_diff.fst' | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_fst"
] | Precomposing `f` with `prod.fst` is `C^n` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_fst {s : set (E × F)} : cont_diff_on 𝕜 n (prod.fst : E × F → E) s | cont_diff.cont_diff_on cont_diff_fst | lemma | cont_diff_on_fst | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff.cont_diff_on",
"cont_diff_fst",
"cont_diff_on"
] | The first projection on a domain in a product is `C^∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.fst {f : E → F × G} {s : set E} (hf : cont_diff_on 𝕜 n f s) :
cont_diff_on 𝕜 n (λ x, (f x).1) s | cont_diff_fst.comp_cont_diff_on hf | lemma | cont_diff_on.fst | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_fst {p : E × F} : cont_diff_at 𝕜 n (prod.fst : E × F → E) p | cont_diff_fst.cont_diff_at | lemma | cont_diff_at_fst | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at"
] | The first projection at a point in a product is `C^∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.fst {f : E → F × G} {x : E} (hf : cont_diff_at 𝕜 n f x) :
cont_diff_at 𝕜 n (λ x, (f x).1) x | cont_diff_at_fst.comp x hf | lemma | cont_diff_at.fst | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at"
] | Postcomposing `f` with `prod.fst` is `C^n` at `(x, y)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.fst' {f : E → G} {x : E} {y : F} (hf : cont_diff_at 𝕜 n f x) :
cont_diff_at 𝕜 n (λ x : E × F, f x.1) (x, y) | cont_diff_at.comp (x, y) hf cont_diff_at_fst | lemma | cont_diff_at.fst' | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at",
"cont_diff_at.comp",
"cont_diff_at_fst"
] | Precomposing `f` with `prod.fst` is `C^n` at `(x, y)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.fst'' {f : E → G} {x : E × F} (hf : cont_diff_at 𝕜 n f x.1) :
cont_diff_at 𝕜 n (λ x : E × F, f x.1) x | hf.comp x cont_diff_at_fst | lemma | cont_diff_at.fst'' | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at",
"cont_diff_at_fst"
] | Precomposing `f` with `prod.fst` is `C^n` at `x : E × F` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at_fst {s : set (E × F)} {p : E × F} :
cont_diff_within_at 𝕜 n (prod.fst : E × F → E) s p | cont_diff_fst.cont_diff_within_at | lemma | cont_diff_within_at_fst | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at"
] | The first projection within a domain at a point in a product is `C^∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_snd : cont_diff 𝕜 n (prod.snd : E × F → F) | is_bounded_linear_map.cont_diff is_bounded_linear_map.snd | lemma | cont_diff_snd | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"is_bounded_linear_map.cont_diff",
"is_bounded_linear_map.snd"
] | The second projection in a product is `C^∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.snd {f : E → F × G} (hf : cont_diff 𝕜 n f) : cont_diff 𝕜 n (λ x, (f x).2) | cont_diff_snd.comp hf | lemma | cont_diff.snd | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff"
] | Postcomposing `f` with `prod.snd` is `C^n` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.snd' {f : F → G} (hf : cont_diff 𝕜 n f) : cont_diff 𝕜 n (λ x : E × F, f x.2) | hf.comp cont_diff_snd | lemma | cont_diff.snd' | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_snd"
] | Precomposing `f` with `prod.snd` is `C^n` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_snd {s : set (E × F)} : cont_diff_on 𝕜 n (prod.snd : E × F → F) s | cont_diff.cont_diff_on cont_diff_snd | lemma | cont_diff_on_snd | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff.cont_diff_on",
"cont_diff_on",
"cont_diff_snd"
] | The second projection on a domain in a product is `C^∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.snd {f : E → F × G} {s : set E} (hf : cont_diff_on 𝕜 n f s) :
cont_diff_on 𝕜 n (λ x, (f x).2) s | cont_diff_snd.comp_cont_diff_on hf | lemma | cont_diff_on.snd | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_snd {p : E × F} : cont_diff_at 𝕜 n (prod.snd : E × F → F) p | cont_diff_snd.cont_diff_at | lemma | cont_diff_at_snd | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at"
] | The second projection at a point in a product is `C^∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.snd {f : E → F × G} {x : E} (hf : cont_diff_at 𝕜 n f x) :
cont_diff_at 𝕜 n (λ x, (f x).2) x | cont_diff_at_snd.comp x hf | lemma | cont_diff_at.snd | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at"
] | Postcomposing `f` with `prod.snd` is `C^n` at `x` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.snd' {f : F → G} {x : E} {y : F} (hf : cont_diff_at 𝕜 n f y) :
cont_diff_at 𝕜 n (λ x : E × F, f x.2) (x, y) | cont_diff_at.comp (x, y) hf cont_diff_at_snd | lemma | cont_diff_at.snd' | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at",
"cont_diff_at.comp",
"cont_diff_at_snd"
] | Precomposing `f` with `prod.snd` is `C^n` at `(x, y)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.snd'' {f : F → G} {x : E × F} (hf : cont_diff_at 𝕜 n f x.2) :
cont_diff_at 𝕜 n (λ x : E × F, f x.2) x | hf.comp x cont_diff_at_snd | lemma | cont_diff_at.snd'' | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at",
"cont_diff_at_snd"
] | Precomposing `f` with `prod.snd` is `C^n` at `x : E × F` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at_snd {s : set (E × F)} {p : E × F} :
cont_diff_within_at 𝕜 n (prod.snd : E × F → F) s p | cont_diff_snd.cont_diff_within_at | lemma | cont_diff_within_at_snd | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at"
] | The second projection within a domain at a point in a product is `C^∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.comp₂ {g : E₁ × E₂ → G} {f₁ : F → E₁} {f₂ : F → E₂}
(hg : cont_diff 𝕜 n g) (hf₁ : cont_diff 𝕜 n f₁) (hf₂ : cont_diff 𝕜 n f₂) :
cont_diff 𝕜 n (λ x, g (f₁ x, f₂ x)) | hg.comp $ hf₁.prod hf₂ | lemma | cont_diff.comp₂ | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.comp₃ {g : E₁ × E₂ × E₃ → G} {f₁ : F → E₁} {f₂ : F → E₂} {f₃ : F → E₃}
(hg : cont_diff 𝕜 n g) (hf₁ : cont_diff 𝕜 n f₁) (hf₂ : cont_diff 𝕜 n f₂)
(hf₃ : cont_diff 𝕜 n f₃) : cont_diff 𝕜 n (λ x, g (f₁ x, f₂ x, f₃ x)) | hg.comp₂ hf₁ $ hf₂.prod hf₃ | lemma | cont_diff.comp₃ | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.comp_cont_diff_on₂ {g : E₁ × E₂ → G} {f₁ : F → E₁} {f₂ : F → E₂} {s : set F}
(hg : cont_diff 𝕜 n g) (hf₁ : cont_diff_on 𝕜 n f₁ s) (hf₂ : cont_diff_on 𝕜 n f₂ s) :
cont_diff_on 𝕜 n (λ x, g (f₁ x, f₂ x)) s | hg.comp_cont_diff_on $ hf₁.prod hf₂ | lemma | cont_diff.comp_cont_diff_on₂ | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.comp_cont_diff_on₃ {g : E₁ × E₂ × E₃ → G} {f₁ : F → E₁} {f₂ : F → E₂} {f₃ : F → E₃}
{s : set F} (hg : cont_diff 𝕜 n g) (hf₁ : cont_diff_on 𝕜 n f₁ s) (hf₂ : cont_diff_on 𝕜 n f₂ s)
(hf₃ : cont_diff_on 𝕜 n f₃ s) : cont_diff_on 𝕜 n (λ x, g (f₁ x, f₂ x, f₃ x)) s | hg.comp_cont_diff_on₂ hf₁ $ hf₂.prod hf₃ | lemma | cont_diff.comp_cont_diff_on₃ | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.clm_comp {g : X → F →L[𝕜] G} {f : X → E →L[𝕜] F}
(hg : cont_diff 𝕜 n g) (hf : cont_diff 𝕜 n f) :
cont_diff 𝕜 n (λ x, (g x).comp (f x)) | is_bounded_bilinear_map_comp.cont_diff.comp₂ hg hf | lemma | cont_diff.clm_comp | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.clm_comp {g : X → F →L[𝕜] G} {f : X → E →L[𝕜] F}
{s : set X} (hg : cont_diff_on 𝕜 n g s) (hf : cont_diff_on 𝕜 n f s) :
cont_diff_on 𝕜 n (λ x, (g x).comp (f x)) s | is_bounded_bilinear_map_comp.cont_diff.comp_cont_diff_on₂ hg hf | lemma | cont_diff_on.clm_comp | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.clm_apply {f : E → F →L[𝕜] G} {g : E → F} {n : ℕ∞}
(hf : cont_diff 𝕜 n f) (hg : cont_diff 𝕜 n g) :
cont_diff 𝕜 n (λ x, (f x) (g x)) | is_bounded_bilinear_map_apply.cont_diff.comp₂ hf hg | lemma | cont_diff.clm_apply | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.clm_apply {f : E → F →L[𝕜] G} {g : E → F} {n : ℕ∞}
(hf : cont_diff_on 𝕜 n f s) (hg : cont_diff_on 𝕜 n g s) :
cont_diff_on 𝕜 n (λ x, (f x) (g x)) s | is_bounded_bilinear_map_apply.cont_diff.comp_cont_diff_on₂ hf hg | lemma | cont_diff_on.clm_apply | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.smul_right {f : E → F →L[𝕜] 𝕜} {g : E → G} {n : ℕ∞}
(hf : cont_diff 𝕜 n f) (hg : cont_diff 𝕜 n g) :
cont_diff 𝕜 n (λ x, (f x).smul_right (g x)) | -- giving the following implicit type arguments speeds up elaboration significantly
(@is_bounded_bilinear_map_smul_right 𝕜 _ F _ _ G _ _).cont_diff.comp₂ hf hg | lemma | cont_diff.smul_right | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff.comp₂",
"is_bounded_bilinear_map_smul_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_prod_assoc : cont_diff 𝕜 ⊤ $ equiv.prod_assoc E F G | (linear_isometry_equiv.prod_assoc 𝕜 E F G).cont_diff | lemma | cont_diff_prod_assoc | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"equiv.prod_assoc",
"linear_isometry_equiv.prod_assoc"
] | The natural equivalence `(E × F) × G ≃ E × (F × G)` is smooth.
Warning: if you think you need this lemma, it is likely that you can simplify your proof by
reformulating the lemma that you're applying next using the tips in
Note [continuity lemma statement] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_prod_assoc_symm : cont_diff 𝕜 ⊤ $ (equiv.prod_assoc E F G).symm | (linear_isometry_equiv.prod_assoc 𝕜 E F G).symm.cont_diff | lemma | cont_diff_prod_assoc_symm | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"equiv.prod_assoc",
"linear_isometry_equiv.prod_assoc"
] | The natural equivalence `E × (F × G) ≃ (E × F) × G` is smooth.
Warning: see remarks attached to `cont_diff_prod_assoc` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at.has_fderiv_within_at_nhds {f : E → F → G} {g : E → F}
{t : set F} {n : ℕ} {x₀ : E}
(hf : cont_diff_within_at 𝕜 (n+1) (uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀))
(hg : cont_diff_within_at 𝕜 n g s x₀)
(hgt : t ∈ 𝓝[g '' s] g x₀) :
∃ v ∈ 𝓝[insert x₀ s] x₀, v ⊆ insert x₀ s ∧ ∃ f' : E → F →L[... | begin
have hst : insert x₀ s ×ˢ t ∈ 𝓝[(λ x, (x, g x)) '' s] (x₀, g x₀),
{ refine nhds_within_mono _ _ (nhds_within_prod self_mem_nhds_within hgt),
simp_rw [image_subset_iff, mk_preimage_prod, preimage_id', subset_inter_iff, subset_insert,
true_and, subset_preimage_image] },
obtain ⟨v, hv, hvs, f', hvf'... | lemma | cont_diff_within_at.has_fderiv_within_at_nhds | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at",
"continuous_linear_map.compL",
"continuous_linear_map.inr",
"has_fderiv_at_prod_mk_right",
"has_fderiv_within_at",
"mem_of_mem_nhds_within",
"nhds_within_le_iff",
"nhds_within_mono",
"nhds_within_prod",
"self_mem_nhds_within"
] | One direction of `cont_diff_within_at_succ_iff_has_fderiv_within_at`, but where all derivatives
are taken within the same set. Version for partial derivatives / functions with parameters.
If `f x` is a `C^n+1` family of functions and `g x` is a `C^n` family of points, then the
derivative of `f x` at `g x` depends i... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at.fderiv_within'' {f : E → F → G} {g : E → F}
{t : set F} {n : ℕ∞}
(hf : cont_diff_within_at 𝕜 n (function.uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀))
(hg : cont_diff_within_at 𝕜 m g s x₀)
(ht : ∀ᶠ x in 𝓝[insert x₀ s] x₀, unique_diff_within_at 𝕜 t (g x))
(hmn : m + 1 ≤ n)
(hgt : t ∈ 𝓝[g... | begin
have : ∀ k : ℕ, (k : ℕ∞) ≤ m →
cont_diff_within_at 𝕜 k (λ x, fderiv_within 𝕜 (f x) t (g x)) s x₀,
{ intros k hkm,
obtain ⟨v, hv, -, f', hvf', hf'⟩ :=
(hf.of_le $ (add_le_add_right hkm 1).trans hmn).has_fderiv_within_at_nhds (hg.of_le hkm) hgt,
refine hf'.congr_of_eventually_eq_insert _,
... | lemma | cont_diff_within_at.fderiv_within'' | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at",
"cont_diff_within_at_top",
"fderiv_within",
"le_rfl",
"le_top",
"unique_diff_within_at",
"with_top.rec_top_coe"
] | The most general lemma stating that `x ↦ fderiv_within 𝕜 (f x) t (g x)` is `C^n`
at a point within a set.
To show that `x ↦ D_yf(x,y)g(x)` (taken within `t`) is `C^m` at `x₀` within `s`, we require that
* `f` is `C^n` at `(x₀, g(x₀))` within `(s ∪ {x₀}) × t` for `n ≥ m+1`.
* `g` is `C^m` at `x₀` within `s`;
* Derivati... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at.fderiv_within' {f : E → F → G} {g : E → F}
{t : set F} {n : ℕ∞}
(hf : cont_diff_within_at 𝕜 n (function.uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀))
(hg : cont_diff_within_at 𝕜 m g s x₀)
(ht : ∀ᶠ x in 𝓝[insert x₀ s] x₀, unique_diff_within_at 𝕜 t (g x))
(hmn : m + 1 ≤ n)
(hst : s ⊆ g ⁻¹'... | hf.fderiv_within'' hg ht hmn $ mem_of_superset self_mem_nhds_within $ image_subset_iff.mpr hst | lemma | cont_diff_within_at.fderiv_within' | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at",
"fderiv_within",
"self_mem_nhds_within",
"unique_diff_within_at"
] | A special case of `cont_diff_within_at.fderiv_within''` where we require that `s ⊆ g⁻¹(t)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at.fderiv_within {f : E → F → G} {g : E → F}
{t : set F} {n : ℕ∞}
(hf : cont_diff_within_at 𝕜 n (function.uncurry f) (s ×ˢ t) (x₀, g x₀))
(hg : cont_diff_within_at 𝕜 m g s x₀)
(ht : unique_diff_on 𝕜 t)
(hmn : m + 1 ≤ n) (hx₀ : x₀ ∈ s)
(hst : s ⊆ g ⁻¹' t) :
cont_diff_within_at 𝕜 m (λ x... | begin
rw [← insert_eq_self.mpr hx₀] at hf,
refine hf.fderiv_within' hg _ hmn hst,
rw [insert_eq_self.mpr hx₀],
exact eventually_of_mem self_mem_nhds_within (λ x hx, ht _ (hst hx))
end | lemma | cont_diff_within_at.fderiv_within | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at",
"fderiv_within",
"self_mem_nhds_within",
"unique_diff_on"
] | A special case of `cont_diff_within_at.fderiv_within'` where we require that `x₀ ∈ s` and there
are unique derivatives everywhere within `t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at.fderiv_within_apply {f : E → F → G} {g k : E → F}
{t : set F} {n : ℕ∞}
(hf : cont_diff_within_at 𝕜 n (function.uncurry f) (s ×ˢ t) (x₀, g x₀))
(hg : cont_diff_within_at 𝕜 m g s x₀)
(hk : cont_diff_within_at 𝕜 m k s x₀)
(ht : unique_diff_on 𝕜 t)
(hmn : m + 1 ≤ n) (hx₀ : x₀ ∈ s)
(hst... | (cont_diff_fst.clm_apply cont_diff_snd).cont_diff_at.comp_cont_diff_within_at x₀
((hf.fderiv_within hg ht hmn hx₀ hst).prod hk) | lemma | cont_diff_within_at.fderiv_within_apply | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at.comp_cont_diff_within_at",
"cont_diff_snd",
"cont_diff_within_at",
"fderiv_within",
"unique_diff_on"
] | `x ↦ fderiv_within 𝕜 (f x) t (g x) (k x)` is smooth at a point within a set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at.fderiv_within_right
(hf : cont_diff_within_at 𝕜 n f s x₀) (hs : unique_diff_on 𝕜 s)
(hmn : (m + 1 : ℕ∞) ≤ n) (hx₀s : x₀ ∈ s) :
cont_diff_within_at 𝕜 m (fderiv_within 𝕜 f s) s x₀ | cont_diff_within_at.fderiv_within
(cont_diff_within_at.comp (x₀, x₀) hf cont_diff_within_at_snd $ prod_subset_preimage_snd s s)
cont_diff_within_at_id hs hmn hx₀s (by rw [preimage_id']) | lemma | cont_diff_within_at.fderiv_within_right | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at",
"cont_diff_within_at.comp",
"cont_diff_within_at.fderiv_within",
"cont_diff_within_at_id",
"cont_diff_within_at_snd",
"fderiv_within",
"unique_diff_on"
] | `fderiv_within 𝕜 f s` is smooth at `x₀` within `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.fderiv {f : E → F → G} {g : E → F} {n : ℕ∞}
(hf : cont_diff_at 𝕜 n (function.uncurry f) (x₀, g x₀))
(hg : cont_diff_at 𝕜 m g x₀)
(hmn : m + 1 ≤ n) :
cont_diff_at 𝕜 m (λ x, fderiv 𝕜 (f x) (g x)) x₀ | begin
simp_rw [← fderiv_within_univ],
refine (cont_diff_within_at.fderiv_within hf.cont_diff_within_at hg.cont_diff_within_at
unique_diff_on_univ hmn (mem_univ x₀) _).cont_diff_at univ_mem,
rw [preimage_univ]
end | lemma | cont_diff_at.fderiv | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at",
"cont_diff_within_at.fderiv_within",
"fderiv",
"fderiv_within_univ",
"unique_diff_on_univ"
] | `x ↦ fderiv 𝕜 (f x) (g x)` is smooth at `x₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.fderiv_right (hf : cont_diff_at 𝕜 n f x₀) (hmn : (m + 1 : ℕ∞) ≤ n) :
cont_diff_at 𝕜 m (fderiv 𝕜 f) x₀ | cont_diff_at.fderiv (cont_diff_at.comp (x₀, x₀) hf cont_diff_at_snd) cont_diff_at_id hmn | lemma | cont_diff_at.fderiv_right | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at",
"cont_diff_at.comp",
"cont_diff_at.fderiv",
"cont_diff_at_id",
"cont_diff_at_snd",
"fderiv"
] | `fderiv 𝕜 f` is smooth at `x₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.fderiv {f : E → F → G} {g : E → F} {n m : ℕ∞}
(hf : cont_diff 𝕜 m $ function.uncurry f) (hg : cont_diff 𝕜 n g) (hnm : n + 1 ≤ m) :
cont_diff 𝕜 n (λ x, fderiv 𝕜 (f x) (g x)) | cont_diff_iff_cont_diff_at.mpr $ λ x, hf.cont_diff_at.fderiv hg.cont_diff_at hnm | lemma | cont_diff.fderiv | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"fderiv"
] | `x ↦ fderiv 𝕜 (f x) (g x)` is smooth. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.fderiv_right (hf : cont_diff 𝕜 n f) (hmn : (m + 1 : ℕ∞) ≤ n) :
cont_diff 𝕜 m (fderiv 𝕜 f) | cont_diff_iff_cont_diff_at.mpr $ λ x, hf.cont_diff_at.fderiv_right hmn | lemma | cont_diff.fderiv_right | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"fderiv"
] | `fderiv 𝕜 f` is smooth. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous.fderiv {f : E → F → G} {g : E → F} {n : ℕ∞}
(hf : cont_diff 𝕜 n $ function.uncurry f) (hg : continuous g) (hn : 1 ≤ n) :
continuous (λ x, fderiv 𝕜 (f x) (g x)) | (hf.fderiv (cont_diff_zero.mpr hg) hn).continuous | lemma | continuous.fderiv | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"continuous",
"fderiv"
] | `x ↦ fderiv 𝕜 (f x) (g x)` is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.fderiv_apply {f : E → F → G} {g k : E → F} {n m : ℕ∞}
(hf : cont_diff 𝕜 m $ function.uncurry f) (hg : cont_diff 𝕜 n g) (hk : cont_diff 𝕜 n k)
(hnm : n + 1 ≤ m) :
cont_diff 𝕜 n (λ x, fderiv 𝕜 (f x) (g x) (k x)) | (hf.fderiv hg hnm).clm_apply hk | lemma | cont_diff.fderiv_apply | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"fderiv"
] | `x ↦ fderiv 𝕜 (f x) (g x) (k x)` is smooth. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_fderiv_within_apply {m n : ℕ∞} {s : set E}
{f : E → F} (hf : cont_diff_on 𝕜 n f s) (hs : unique_diff_on 𝕜 s) (hmn : m + 1 ≤ n) :
cont_diff_on 𝕜 m (λp : E × E, (fderiv_within 𝕜 f s p.1 : E →L[𝕜] F) p.2) (s ×ˢ univ) | ((hf.fderiv_within hs hmn).comp cont_diff_on_fst (prod_subset_preimage_fst _ _)).clm_apply
cont_diff_on_snd | lemma | cont_diff_on_fderiv_within_apply | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on",
"cont_diff_on_fst",
"cont_diff_on_snd",
"fderiv_within",
"unique_diff_on"
] | The bundled derivative of a `C^{n+1}` function is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.continuous_on_fderiv_within_apply
(hf : cont_diff_on 𝕜 n f s) (hs : unique_diff_on 𝕜 s) (hn : 1 ≤ n) :
continuous_on (λp : E × E, (fderiv_within 𝕜 f s p.1 : E → F) p.2) (s ×ˢ univ) | (cont_diff_on_fderiv_within_apply hf hs $ by rwa [zero_add]).continuous_on | lemma | cont_diff_on.continuous_on_fderiv_within_apply | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on",
"cont_diff_on_fderiv_within_apply",
"continuous_on",
"fderiv_within",
"unique_diff_on"
] | 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 |
cont_diff.cont_diff_fderiv_apply {f : E → F}
(hf : cont_diff 𝕜 n f) (hmn : m + 1 ≤ n) :
cont_diff 𝕜 m (λp : E × E, (fderiv 𝕜 f p.1 : E →L[𝕜] F) p.2) | begin
rw ← cont_diff_on_univ at ⊢ hf,
rw [← fderiv_within_univ, ← univ_prod_univ],
exact cont_diff_on_fderiv_within_apply hf unique_diff_on_univ hmn
end | lemma | cont_diff.cont_diff_fderiv_apply | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_on_fderiv_within_apply",
"cont_diff_on_univ",
"fderiv",
"fderiv_within_univ",
"unique_diff_on_univ"
] | The bundled derivative of a `C^{n+1}` function is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to_on_pi :
has_ftaylor_series_up_to_on n (λ x i, φ i x)
(λ x m, continuous_multilinear_map.pi (λ i, p' i x m)) s ↔
∀ i, has_ftaylor_series_up_to_on n (φ i) (p' i) s | begin
set pr := @continuous_linear_map.proj 𝕜 _ ι F' _ _ _,
letI : Π (m : ℕ) (i : ι), normed_space 𝕜 (E [×m]→L[𝕜] (F' i)) := λ m i, infer_instance,
set L : Π m : ℕ, (Π i, E [×m]→L[𝕜] (F' i)) ≃ₗᵢ[𝕜] (E [×m]→L[𝕜] (Π i, F' i)) :=
λ m, continuous_multilinear_map.piₗᵢ _ _,
refine ⟨λ h i, _, λ h, ⟨λ x hx, _... | lemma | has_ftaylor_series_up_to_on_pi | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont",
"continuous.comp_continuous_on",
"continuous_linear_map.proj",
"continuous_multilinear_map.pi",
"continuous_multilinear_map.piₗᵢ",
"fderiv_within",
"has_fderiv_at.comp_has_fderiv_within_at",
"has_ftaylor_series_up_to_on",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to_on_pi' :
has_ftaylor_series_up_to_on n Φ P' s ↔
∀ i, has_ftaylor_series_up_to_on n (λ x, Φ x i)
(λ x m, (@continuous_linear_map.proj 𝕜 _ ι F' _ _ _ i).comp_continuous_multilinear_map
(P' x m)) s | by { convert has_ftaylor_series_up_to_on_pi, ext, refl } | lemma | has_ftaylor_series_up_to_on_pi' | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"continuous_linear_map.proj",
"has_ftaylor_series_up_to_on",
"has_ftaylor_series_up_to_on_pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at_pi :
cont_diff_within_at 𝕜 n Φ s x ↔
∀ i, cont_diff_within_at 𝕜 n (λ x, Φ x i) s x | begin
set pr := @continuous_linear_map.proj 𝕜 _ ι F' _ _ _,
refine ⟨λ h i, h.continuous_linear_map_comp (pr i), λ h m hm, _⟩,
choose u hux p hp using λ i, h i m hm,
exact ⟨⋂ i, u i, filter.Inter_mem.2 hux, _,
has_ftaylor_series_up_to_on_pi.2 (λ i, (hp i).mono $ Inter_subset _ _)⟩,
end | lemma | cont_diff_within_at_pi | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at",
"continuous_linear_map.proj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_pi :
cont_diff_on 𝕜 n Φ s ↔ ∀ i, cont_diff_on 𝕜 n (λ x, Φ x i) s | ⟨λ h i x hx, cont_diff_within_at_pi.1 (h x hx) _,
λ h x hx, cont_diff_within_at_pi.2 (λ i, h i x hx)⟩ | lemma | cont_diff_on_pi | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_pi :
cont_diff_at 𝕜 n Φ x ↔ ∀ i, cont_diff_at 𝕜 n (λ x, Φ x i) x | cont_diff_within_at_pi | lemma | cont_diff_at_pi | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at",
"cont_diff_within_at_pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_pi :
cont_diff 𝕜 n Φ ↔ ∀ i, cont_diff 𝕜 n (λ x, Φ x i) | by simp only [← cont_diff_on_univ, cont_diff_on_pi] | lemma | cont_diff_pi | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_on_pi",
"cont_diff_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_apply (i : ι) : cont_diff 𝕜 n (λ (f : ι → E), f i) | cont_diff_pi.mp cont_diff_id i | lemma | cont_diff_apply | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_apply_apply (i : ι) (j : ι') : cont_diff 𝕜 n (λ (f : ι → ι' → E), f i j) | cont_diff_pi.mp (cont_diff_apply 𝕜 (ι' → E) i) j | lemma | cont_diff_apply_apply | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"cont_diff_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_add : cont_diff 𝕜 n (λp : F × F, p.1 + p.2) | (is_bounded_linear_map.fst.add is_bounded_linear_map.snd).cont_diff | lemma | cont_diff_add | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"is_bounded_linear_map.snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.add {s : set E} {f g : E → F}
(hf : cont_diff_within_at 𝕜 n f s x) (hg : cont_diff_within_at 𝕜 n g s x) :
cont_diff_within_at 𝕜 n (λx, f x + g x) s x | cont_diff_add.cont_diff_within_at.comp x (hf.prod hg) subset_preimage_univ | lemma | cont_diff_within_at.add | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at"
] | The sum of two `C^n` functions within a set at a point is `C^n` within this set
at this point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.add {f g : E → F} (hf : cont_diff_at 𝕜 n f x) (hg : cont_diff_at 𝕜 n g x) :
cont_diff_at 𝕜 n (λx, f x + g x) x | by rw [← cont_diff_within_at_univ] at *; exact hf.add hg | lemma | cont_diff_at.add | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at",
"cont_diff_within_at_univ"
] | The sum of two `C^n` functions at a point is `C^n` at this point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.add {f g : E → F} (hf : cont_diff 𝕜 n f) (hg : cont_diff 𝕜 n g) :
cont_diff 𝕜 n (λx, f x + g x) | cont_diff_add.comp (hf.prod hg) | lemma | cont_diff.add | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff"
] | The sum of two `C^n`functions is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.add {s : set E} {f g : E → F}
(hf : cont_diff_on 𝕜 n f s) (hg : cont_diff_on 𝕜 n g s) :
cont_diff_on 𝕜 n (λx, f x + g x) s | λ x hx, (hf x hx).add (hg x hx) | lemma | cont_diff_on.add | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on"
] | The sum of two `C^n` functions on a domain is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_fderiv_within_add_apply {f g : E → F}
(hf : cont_diff_on 𝕜 i f s) (hg : cont_diff_on 𝕜 i g s) (hu : unique_diff_on 𝕜 s)
(hx : x ∈ s) :
iterated_fderiv_within 𝕜 i (f + g) s x =
iterated_fderiv_within 𝕜 i f s x + iterated_fderiv_within 𝕜 i g s x | begin
induction i with i hi generalizing x,
{ ext h, simp },
{ ext h,
have hi' : (i : ℕ∞) < i+1 :=
with_top.coe_lt_coe.mpr (nat.lt_succ_self _),
have hdf : differentiable_on 𝕜 (iterated_fderiv_within 𝕜 i f s) s :=
hf.differentiable_on_iterated_fderiv_within hi' hu,
have hdg : differentia... | lemma | iterated_fderiv_within_add_apply | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on",
"differentiable_on",
"fderiv_within",
"fderiv_within_add",
"fderiv_within_congr'",
"fin.tail",
"iterated_fderiv_within",
"unique_diff_on"
] | The iterated derivative of the sum of two functions is the sum of the iterated derivatives.
See also `iterated_fderiv_within_add_apply'`, which uses the spelling `(λ x, f x + g x)`
instead of `f + g`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_fderiv_within_add_apply' {f g : E → F}
(hf : cont_diff_on 𝕜 i f s) (hg : cont_diff_on 𝕜 i g s) (hu : unique_diff_on 𝕜 s)
(hx : x ∈ s) :
iterated_fderiv_within 𝕜 i (λ x, f x + g x) s x =
iterated_fderiv_within 𝕜 i f s x + iterated_fderiv_within 𝕜 i g s x | iterated_fderiv_within_add_apply hf hg hu hx | lemma | iterated_fderiv_within_add_apply' | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on",
"iterated_fderiv_within",
"iterated_fderiv_within_add_apply",
"unique_diff_on"
] | The iterated derivative of the sum of two functions is the sum of the iterated derivatives.
This is the same as `iterated_fderiv_within_add_apply`, but using the spelling `(λ x, f x + g x)`
instead of `f + g`, which can be handy for some rewrites.
TODO: use one form consistently. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_fderiv_add_apply {i : ℕ} {f g : E → F} (hf : cont_diff 𝕜 i f)
(hg : cont_diff 𝕜 i g) :
iterated_fderiv 𝕜 i (f + g) x = iterated_fderiv 𝕜 i f x + iterated_fderiv 𝕜 i g x | begin
simp_rw [←cont_diff_on_univ, ←iterated_fderiv_within_univ] at hf hg ⊢,
exact iterated_fderiv_within_add_apply hf hg unique_diff_on_univ (set.mem_univ _),
end | lemma | iterated_fderiv_add_apply | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"iterated_fderiv",
"iterated_fderiv_within_add_apply",
"set.mem_univ",
"unique_diff_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_add_apply' {i : ℕ} {f g : E → F} (hf : cont_diff 𝕜 i f)
(hg : cont_diff 𝕜 i g) :
iterated_fderiv 𝕜 i (λ x, f x + g x) x = iterated_fderiv 𝕜 i f x + iterated_fderiv 𝕜 i g x | iterated_fderiv_add_apply hf hg | lemma | iterated_fderiv_add_apply' | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"iterated_fderiv",
"iterated_fderiv_add_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_neg : cont_diff 𝕜 n (λp : F, -p) | is_bounded_linear_map.id.neg.cont_diff | lemma | cont_diff_neg | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.neg {s : set E} {f : E → F}
(hf : cont_diff_within_at 𝕜 n f s x) : cont_diff_within_at 𝕜 n (λx, -f x) s x | cont_diff_neg.cont_diff_within_at.comp x hf subset_preimage_univ | lemma | cont_diff_within_at.neg | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_within_at"
] | The negative of a `C^n` function within a domain at a point is `C^n` within this domain at
this point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.neg {f : E → F}
(hf : cont_diff_at 𝕜 n f x) : cont_diff_at 𝕜 n (λx, -f x) x | by rw ← cont_diff_within_at_univ at *; exact hf.neg | lemma | cont_diff_at.neg | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at",
"cont_diff_within_at_univ"
] | The negative of a `C^n` function at a point is `C^n` at this point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.neg {f : E → F} (hf : cont_diff 𝕜 n f) : cont_diff 𝕜 n (λx, -f x) | cont_diff_neg.comp hf | lemma | cont_diff.neg | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff"
] | The negative of a `C^n`function is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.neg {s : set E} {f : E → F}
(hf : cont_diff_on 𝕜 n f s) : cont_diff_on 𝕜 n (λx, -f x) s | λ x hx, (hf x hx).neg | lemma | cont_diff_on.neg | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on"
] | The negative of a `C^n` function on a domain is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_fderiv_within_neg_apply {f : E → F} (hu : unique_diff_on 𝕜 s) (hx : x ∈ s) :
iterated_fderiv_within 𝕜 i (-f) s x = -iterated_fderiv_within 𝕜 i f s x | begin
induction i with i hi generalizing x,
{ ext h, simp },
{ ext h,
have hi' : (i : ℕ∞) < i+1 :=
with_top.coe_lt_coe.mpr (nat.lt_succ_self _),
calc iterated_fderiv_within 𝕜 (i+1) (-f) s x h
= fderiv_within 𝕜 (iterated_fderiv_within 𝕜 i (-f) s) s x (h 0) (fin.tail h) : rfl
... = fder... | lemma | iterated_fderiv_within_neg_apply | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"fderiv_within",
"fderiv_within_congr'",
"fderiv_within_neg",
"fin.tail",
"iterated_fderiv_within",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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