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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unique_diff_within_at.mono_nhds (h : unique_diff_within_at 𝕜 s x)
(st : 𝓝[s] x ≤ 𝓝[t] x) :
unique_diff_within_at 𝕜 t x | begin
simp only [unique_diff_within_at_iff] at *,
rw [mem_closure_iff_nhds_within_ne_bot] at h ⊢,
exact ⟨h.1.mono $ submodule.span_mono $ tangent_cone_mono_nhds st,
h.2.mono st⟩
end | lemma | unique_diff_within_at.mono_nhds | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"mem_closure_iff_nhds_within_ne_bot",
"submodule.span_mono",
"tangent_cone_mono_nhds",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_within_at.mono (h : unique_diff_within_at 𝕜 s x) (st : s ⊆ t) :
unique_diff_within_at 𝕜 t x | h.mono_nhds $ nhds_within_mono _ st | lemma | unique_diff_within_at.mono | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"nhds_within_mono",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_within_at_congr (st : 𝓝[s] x = 𝓝[t] x) :
unique_diff_within_at 𝕜 s x ↔ unique_diff_within_at 𝕜 t x | ⟨λ h, h.mono_nhds $ le_of_eq st, λ h, h.mono_nhds $ le_of_eq st.symm⟩ | lemma | unique_diff_within_at_congr | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_within_at_inter (ht : t ∈ 𝓝 x) :
unique_diff_within_at 𝕜 (s ∩ t) x ↔ unique_diff_within_at 𝕜 s x | unique_diff_within_at_congr $ (nhds_within_restrict' _ ht).symm | lemma | unique_diff_within_at_inter | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"nhds_within_restrict'",
"unique_diff_within_at",
"unique_diff_within_at_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_within_at.inter (hs : unique_diff_within_at 𝕜 s x) (ht : t ∈ 𝓝 x) :
unique_diff_within_at 𝕜 (s ∩ t) x | (unique_diff_within_at_inter ht).2 hs | lemma | unique_diff_within_at.inter | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"unique_diff_within_at",
"unique_diff_within_at_inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_within_at_inter' (ht : t ∈ 𝓝[s] x) :
unique_diff_within_at 𝕜 (s ∩ t) x ↔ unique_diff_within_at 𝕜 s x | unique_diff_within_at_congr $ (nhds_within_restrict'' _ ht).symm | lemma | unique_diff_within_at_inter' | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"nhds_within_restrict''",
"unique_diff_within_at",
"unique_diff_within_at_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_within_at.inter' (hs : unique_diff_within_at 𝕜 s x) (ht : t ∈ 𝓝[s] x) :
unique_diff_within_at 𝕜 (s ∩ t) x | (unique_diff_within_at_inter' ht).2 hs | lemma | unique_diff_within_at.inter' | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"unique_diff_within_at",
"unique_diff_within_at_inter'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_within_at_of_mem_nhds (h : s ∈ 𝓝 x) : unique_diff_within_at 𝕜 s x | by simpa only [univ_inter] using unique_diff_within_at_univ.inter h | lemma | unique_diff_within_at_of_mem_nhds | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.unique_diff_within_at (hs : is_open s) (xs : x ∈ s) : unique_diff_within_at 𝕜 s x | unique_diff_within_at_of_mem_nhds (is_open.mem_nhds hs xs) | lemma | is_open.unique_diff_within_at | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"is_open",
"is_open.mem_nhds",
"unique_diff_within_at",
"unique_diff_within_at_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_on.inter (hs : unique_diff_on 𝕜 s) (ht : is_open t) : unique_diff_on 𝕜 (s ∩ t) | λx hx, (hs x hx.1).inter (is_open.mem_nhds ht hx.2) | lemma | unique_diff_on.inter | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"is_open",
"is_open.mem_nhds",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.unique_diff_on (hs : is_open s) : unique_diff_on 𝕜 s | λx hx, is_open.unique_diff_within_at hs hx | lemma | is_open.unique_diff_on | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"is_open",
"is_open.unique_diff_within_at",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_within_at.prod {t : set F} {y : F}
(hs : unique_diff_within_at 𝕜 s x) (ht : unique_diff_within_at 𝕜 t y) :
unique_diff_within_at 𝕜 (s ×ˢ t) (x, y) | begin
rw [unique_diff_within_at_iff] at ⊢ hs ht,
rw [closure_prod_eq],
refine ⟨_, hs.2, ht.2⟩,
have : _ ≤ submodule.span 𝕜 (tangent_cone_at 𝕜 (s ×ˢ t) (x, y)) :=
submodule.span_mono (union_subset (subset_tangent_cone_prod_left ht.2)
(subset_tangent_cone_prod_right hs.2)),
rw [linear_map.span_inl_u... | lemma | unique_diff_within_at.prod | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"closure_prod_eq",
"linear_map.span_inl_union_inr",
"set_like.le_def",
"submodule.span",
"submodule.span_mono",
"subset_tangent_cone_prod_left",
"subset_tangent_cone_prod_right",
"tangent_cone_at",
"unique_diff_within_at"
] | The product of two sets of unique differentiability at points `x` and `y` has unique
differentiability at `(x, y)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_diff_within_at.univ_pi (ι : Type*) [finite ι] (E : ι → Type*)
[Π i, normed_add_comm_group (E i)] [Π i, normed_space 𝕜 (E i)]
(s : Π i, set (E i)) (x : Π i, E i) (h : ∀ i, unique_diff_within_at 𝕜 (s i) (x i)) :
unique_diff_within_at 𝕜 (set.pi univ s) x | begin
classical,
simp only [unique_diff_within_at_iff, closure_pi_set] at h ⊢,
refine ⟨(dense_pi univ (λ i _, (h i).1)).mono _, λ i _, (h i).2⟩,
norm_cast,
simp only [← submodule.supr_map_single, supr_le_iff, linear_map.map_span, submodule.span_le,
← maps_to'],
exact λ i, (maps_to_tangent_cone_pi $ λ j ... | lemma | unique_diff_within_at.univ_pi | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"closure_pi_set",
"dense_pi",
"finite",
"maps_to_tangent_cone_pi",
"normed_add_comm_group",
"normed_space",
"set.pi",
"submodule.span_le",
"submodule.subset_span",
"submodule.supr_map_single",
"supr_le_iff",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_within_at.pi (ι : Type*) [finite ι] (E : ι → Type*)
[Π i, normed_add_comm_group (E i)] [Π i, normed_space 𝕜 (E i)]
(s : Π i, set (E i)) (x : Π i, E i) (I : set ι)
(h : ∀ i ∈ I, unique_diff_within_at 𝕜 (s i) (x i)) :
unique_diff_within_at 𝕜 (set.pi I s) x | begin
classical,
rw [← set.univ_pi_piecewise],
refine unique_diff_within_at.univ_pi _ _ _ _ (λ i, _),
by_cases hi : i ∈ I; simp [*, unique_diff_within_at_univ],
end | lemma | unique_diff_within_at.pi | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"finite",
"normed_add_comm_group",
"normed_space",
"set.pi",
"set.univ_pi_piecewise",
"unique_diff_within_at",
"unique_diff_within_at.univ_pi",
"unique_diff_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_on.prod {t : set F} (hs : unique_diff_on 𝕜 s) (ht : unique_diff_on 𝕜 t) :
unique_diff_on 𝕜 (s ×ˢ t) | λ ⟨x, y⟩ h, unique_diff_within_at.prod (hs x h.1) (ht y h.2) | lemma | unique_diff_on.prod | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"unique_diff_on",
"unique_diff_within_at.prod"
] | The product of two sets of unique differentiability is a set of unique differentiability. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_diff_on.pi (ι : Type*) [finite ι] (E : ι → Type*)
[Π i, normed_add_comm_group (E i)] [Π i, normed_space 𝕜 (E i)]
(s : Π i, set (E i)) (I : set ι) (h : ∀ i ∈ I, unique_diff_on 𝕜 (s i)) :
unique_diff_on 𝕜 (set.pi I s) | λ x hx, unique_diff_within_at.pi _ _ _ _ _ $ λ i hi, h i hi (x i) (hx i hi) | lemma | unique_diff_on.pi | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"finite",
"normed_add_comm_group",
"normed_space",
"set.pi",
"unique_diff_on",
"unique_diff_within_at.pi"
] | The finite product of a family of sets of unique differentiability is a set of unique
differentiability. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_diff_on.univ_pi (ι : Type*) [finite ι] (E : ι → Type*)
[Π i, normed_add_comm_group (E i)] [Π i, normed_space 𝕜 (E i)]
(s : Π i, set (E i)) (h : ∀ i, unique_diff_on 𝕜 (s i)) :
unique_diff_on 𝕜 (set.pi univ s) | unique_diff_on.pi _ _ _ _ $ λ i _, h i | lemma | unique_diff_on.univ_pi | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"finite",
"normed_add_comm_group",
"normed_space",
"set.pi",
"unique_diff_on",
"unique_diff_on.pi"
] | The finite product of a family of sets of unique differentiability is a set of unique
differentiability. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_diff_within_at_convex {s : set G} (conv : convex ℝ s) (hs : (interior s).nonempty)
{x : G} (hx : x ∈ closure s) : unique_diff_within_at ℝ s x | begin
rcases hs with ⟨y, hy⟩,
suffices : y - x ∈ interior (tangent_cone_at ℝ s x),
{ refine ⟨dense.of_closure _, hx⟩,
simp [(submodule.span ℝ (tangent_cone_at ℝ s x)).eq_top_of_nonempty_interior'
⟨y - x, interior_mono submodule.subset_span this⟩] },
rw [mem_interior_iff_mem_nhds],
replace hy : inter... | theorem | unique_diff_within_at_convex | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"closure",
"convex",
"interior",
"interior_mono",
"interior_subset",
"is_open.mem_nhds",
"is_open_interior",
"mem_interior_iff_mem_nhds",
"mem_tangent_cone_of_open_segment_subset",
"submodule.span",
"submodule.subset_span",
"tangent_cone_at",
"unique_diff_within_at"
] | In a real vector space, a convex set with nonempty interior is a set of unique
differentiability at every point of its closure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_diff_on_convex {s : set G} (conv : convex ℝ s) (hs : (interior s).nonempty) :
unique_diff_on ℝ s | λ x xs, unique_diff_within_at_convex conv hs (subset_closure xs) | theorem | unique_diff_on_convex | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"convex",
"interior",
"subset_closure",
"unique_diff_on",
"unique_diff_within_at_convex"
] | In a real vector space, a convex set with nonempty interior is a set of unique
differentiability. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_diff_on_Ici (a : ℝ) : unique_diff_on ℝ (Ici a) | unique_diff_on_convex (convex_Ici a) $ by simp only [interior_Ici, nonempty_Ioi] | lemma | unique_diff_on_Ici | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"convex_Ici",
"interior_Ici",
"unique_diff_on",
"unique_diff_on_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_on_Iic (a : ℝ) : unique_diff_on ℝ (Iic a) | unique_diff_on_convex (convex_Iic a) $ by simp only [interior_Iic, nonempty_Iio] | lemma | unique_diff_on_Iic | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"convex_Iic",
"interior_Iic",
"unique_diff_on",
"unique_diff_on_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_on_Ioi (a : ℝ) : unique_diff_on ℝ (Ioi a) | is_open_Ioi.unique_diff_on | lemma | unique_diff_on_Ioi | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_on_Iio (a : ℝ) : unique_diff_on ℝ (Iio a) | is_open_Iio.unique_diff_on | lemma | unique_diff_on_Iio | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_on_Icc {a b : ℝ} (hab : a < b) : unique_diff_on ℝ (Icc a b) | unique_diff_on_convex (convex_Icc a b) $ by simp only [interior_Icc, nonempty_Ioo, hab] | lemma | unique_diff_on_Icc | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"convex_Icc",
"interior_Icc",
"unique_diff_on",
"unique_diff_on_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_on_Ico (a b : ℝ) : unique_diff_on ℝ (Ico a b) | if hab : a < b
then unique_diff_on_convex (convex_Ico a b) $ by simp only [interior_Ico, nonempty_Ioo, hab]
else by simp only [Ico_eq_empty hab, unique_diff_on_empty] | lemma | unique_diff_on_Ico | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"convex_Ico",
"interior_Ico",
"unique_diff_on",
"unique_diff_on_convex",
"unique_diff_on_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_on_Ioc (a b : ℝ) : unique_diff_on ℝ (Ioc a b) | if hab : a < b
then unique_diff_on_convex (convex_Ioc a b) $ by simp only [interior_Ioc, nonempty_Ioo, hab]
else by simp only [Ioc_eq_empty hab, unique_diff_on_empty] | lemma | unique_diff_on_Ioc | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"convex_Ioc",
"interior_Ioc",
"unique_diff_on",
"unique_diff_on_convex",
"unique_diff_on_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_on_Ioo (a b : ℝ) : unique_diff_on ℝ (Ioo a b) | is_open_Ioo.unique_diff_on | lemma | unique_diff_on_Ioo | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_on_Icc_zero_one : unique_diff_on ℝ (Icc (0:ℝ) 1) | unique_diff_on_Icc zero_lt_one | lemma | unique_diff_on_Icc_zero_one | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"unique_diff_on",
"unique_diff_on_Icc",
"zero_lt_one"
] | The real interval `[0, 1]` is a set of unique differentiability. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
unique_diff_within_at_Ioo {a b t : ℝ} (ht : t ∈ set.Ioo a b) :
unique_diff_within_at ℝ (set.Ioo a b) t | is_open.unique_diff_within_at is_open_Ioo ht | lemma | unique_diff_within_at_Ioo | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"is_open.unique_diff_within_at",
"is_open_Ioo",
"set.Ioo",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_within_at_Ioi (a : ℝ) : unique_diff_within_at ℝ (Ioi a) a | unique_diff_within_at_convex (convex_Ioi a) (by simp) (by simp) | lemma | unique_diff_within_at_Ioi | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"convex_Ioi",
"unique_diff_within_at",
"unique_diff_within_at_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_within_at_Iio (a : ℝ) : unique_diff_within_at ℝ (Iio a) a | unique_diff_within_at_convex (convex_Iio a) (by simp) (by simp) | lemma | unique_diff_within_at_Iio | analysis.calculus | src/analysis/calculus/tangent_cone.lean | [
"analysis.convex.topology",
"analysis.normed_space.basic",
"analysis.specific_limits.basic"
] | [
"convex_Iio",
"unique_diff_within_at",
"unique_diff_within_at_convex"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
taylor_coeff_within (f : ℝ → E) (k : ℕ) (s : set ℝ) (x₀ : ℝ) : E | (k! : ℝ)⁻¹ • (iterated_deriv_within k f s x₀) | def | taylor_coeff_within | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"iterated_deriv_within"
] | The `k`th coefficient of the Taylor polynomial. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
taylor_within (f : ℝ → E) (n : ℕ) (s : set ℝ) (x₀ : ℝ) : polynomial_module ℝ E | (finset.range (n+1)).sum (λ k,
polynomial_module.comp (polynomial.X - polynomial.C x₀)
(polynomial_module.single ℝ k (taylor_coeff_within f k s x₀))) | def | taylor_within | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"finset.range",
"polynomial.C",
"polynomial.X",
"polynomial_module",
"polynomial_module.comp",
"polynomial_module.single",
"taylor_coeff_within"
] | The Taylor polynomial with derivatives inside of a set `s`.
The Taylor polynomial is given by
$$∑_{k=0}^n \frac{(x - x₀)^k}{k!} f^{(k)}(x₀),$$
where $f^{(k)}(x₀)$ denotes the iterated derivative in the set `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
taylor_within_eval (f : ℝ → E) (n : ℕ) (s : set ℝ) (x₀ x : ℝ) : E | polynomial_module.eval x (taylor_within f n s x₀) | def | taylor_within_eval | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"polynomial_module.eval",
"taylor_within"
] | The Taylor polynomial with derivatives inside of a set `s` considered as a function `ℝ → E` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
taylor_within_succ (f : ℝ → E) (n : ℕ) (s : set ℝ) (x₀ : ℝ) :
taylor_within f (n+1) s x₀ = taylor_within f n s x₀
+ polynomial_module.comp (polynomial.X - polynomial.C x₀)
(polynomial_module.single ℝ (n+1) (taylor_coeff_within f (n+1) s x₀)) | begin
dunfold taylor_within,
rw finset.sum_range_succ,
end | lemma | taylor_within_succ | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"polynomial.C",
"polynomial.X",
"polynomial_module.comp",
"polynomial_module.single",
"taylor_coeff_within",
"taylor_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
taylor_within_eval_succ (f : ℝ → E) (n : ℕ) (s : set ℝ) (x₀ x : ℝ) :
taylor_within_eval f (n+1) s x₀ x = taylor_within_eval f n s x₀ x
+ (((n + 1 : ℝ) * n!)⁻¹ * (x - x₀)^(n+1)) • iterated_deriv_within (n + 1) f s x₀ | begin
simp_rw [taylor_within_eval, taylor_within_succ, linear_map.map_add, polynomial_module.comp_eval],
congr,
simp only [polynomial.eval_sub, polynomial.eval_X, polynomial.eval_C,
polynomial_module.eval_single, mul_inv_rev],
dunfold taylor_coeff_within,
rw [←mul_smul, mul_comm, nat.factorial_succ, nat.c... | lemma | taylor_within_eval_succ | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"iterated_deriv_within",
"linear_map.map_add",
"mul_comm",
"mul_inv_rev",
"nat.cast_add",
"nat.cast_mul",
"nat.cast_one",
"nat.factorial_succ",
"polynomial.eval_C",
"polynomial.eval_X",
"polynomial.eval_sub",
"polynomial_module.comp_eval",
"polynomial_module.eval_single",
"taylor_coeff_wit... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
taylor_within_zero_eval (f : ℝ → E) (s : set ℝ) (x₀ x : ℝ) :
taylor_within_eval f 0 s x₀ x = f x₀ | begin
dunfold taylor_within_eval,
dunfold taylor_within,
dunfold taylor_coeff_within,
simp,
end | lemma | taylor_within_zero_eval | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"taylor_coeff_within",
"taylor_within",
"taylor_within_eval"
] | The Taylor polynomial of order zero evaluates to `f x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
taylor_within_eval_self (f : ℝ → E) (n : ℕ) (s : set ℝ) (x₀ : ℝ) :
taylor_within_eval f n s x₀ x₀ = f x₀ | begin
induction n with k hk,
{ exact taylor_within_zero_eval _ _ _ _},
simp [hk]
end | lemma | taylor_within_eval_self | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"taylor_within_eval",
"taylor_within_zero_eval"
] | Evaluating the Taylor polynomial at `x = x₀` yields `f x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
taylor_within_apply (f : ℝ → E) (n : ℕ) (s : set ℝ) (x₀ x : ℝ) :
taylor_within_eval f n s x₀ x = ∑ k in finset.range (n+1),
((k! : ℝ)⁻¹ * (x - x₀)^k) • iterated_deriv_within k f s x₀ | begin
induction n with k hk,
{ simp },
rw [taylor_within_eval_succ, finset.sum_range_succ, hk],
simp,
end | lemma | taylor_within_apply | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"finset.range",
"iterated_deriv_within",
"taylor_within_eval",
"taylor_within_eval_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_taylor_within_eval {f : ℝ → E} {x : ℝ} {n : ℕ} {s : set ℝ}
(hs : unique_diff_on ℝ s) (hf : cont_diff_on ℝ n f s) :
continuous_on (λ t, taylor_within_eval f n s t x) s | begin
simp_rw taylor_within_apply,
refine continuous_on_finset_sum (finset.range (n+1)) (λ i hi, _),
refine (continuous_on_const.mul ((continuous_on_const.sub continuous_on_id).pow _)).smul _,
rw cont_diff_on_iff_continuous_on_differentiable_on_deriv hs at hf,
cases hf,
specialize hf_left i,
simp only [fi... | lemma | continuous_on_taylor_within_eval | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"cont_diff_on",
"cont_diff_on_iff_continuous_on_differentiable_on_deriv",
"continuous_on",
"continuous_on_id",
"finset.mem_range",
"finset.range",
"taylor_within_apply",
"taylor_within_eval",
"unique_diff_on",
"with_top.coe_le_coe"
] | If `f` is `n` times continuous differentiable on a set `s`, then the Taylor polynomial
`taylor_within_eval f n s x₀ x` is continuous in `x₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
monomial_has_deriv_aux (t x : ℝ) (n : ℕ) :
has_deriv_at (λ y, (x - y)^(n+1)) (-(n+1) * (x - t)^n) t | begin
simp_rw sub_eq_neg_add,
rw [←neg_one_mul, mul_comm (-1 : ℝ), mul_assoc, mul_comm (-1 : ℝ), ←mul_assoc],
convert @has_deriv_at.pow _ _ _ _ _ (n+1) ((has_deriv_at_id t).neg.add_const x),
simp only [nat.cast_add, nat.cast_one],
end | lemma | monomial_has_deriv_aux | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"has_deriv_at",
"has_deriv_at.pow",
"has_deriv_at_id",
"mul_assoc",
"mul_comm",
"nat.cast_add",
"nat.cast_one"
] | Helper lemma for calculating the derivative of the monomial that appears in Taylor expansions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_within_at_taylor_coeff_within {f : ℝ → E} {x y : ℝ} {k : ℕ} {s t : set ℝ}
(ht : unique_diff_within_at ℝ t y) (hs : s ∈ 𝓝[t] y)
(hf : differentiable_within_at ℝ (iterated_deriv_within (k+1) f s) s y) :
has_deriv_within_at (λ z,
(((k+1 : ℝ) * k!)⁻¹ * (x - z)^(k+1)) • iterated_deriv_within (k+1) f s z... | begin
replace hf : has_deriv_within_at (iterated_deriv_within (k+1) f s)
(iterated_deriv_within (k+2) f s y) t y :=
begin
convert (hf.mono_of_mem hs).has_deriv_within_at,
rw iterated_deriv_within_succ (ht.mono_nhds (nhds_within_le_iff.mpr hs)),
exact (deriv_within_of_mem hs ht hf).symm
end,
have... | lemma | has_deriv_within_at_taylor_coeff_within | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"deriv_within_of_mem",
"differentiable_within_at",
"has_deriv_within_at",
"has_deriv_within_at.const_mul",
"iterated_deriv_within",
"iterated_deriv_within_succ",
"monomial_has_deriv_aux",
"nat.cast_add_one_ne_zero",
"nat.factorial_ne_zero",
"neg_div",
"neg_smul",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_taylor_within_eval {f : ℝ → E} {x y : ℝ} {n : ℕ} {s s' : set ℝ}
(hs'_unique : unique_diff_within_at ℝ s' y) (hs_unique : unique_diff_on ℝ s)
(hs' : s' ∈ 𝓝[s] y) (hy : y ∈ s') (h : s' ⊆ s)
(hf : cont_diff_on ℝ n f s)
(hf' : differentiable_within_at ℝ (iterated_deriv_within n f s) s y) :
ha... | begin
induction n with k hk,
{ simp only [taylor_within_zero_eval, nat.factorial_zero, nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul],
simp only [iterated_deriv_within_zero] at hf',
rw iterated_deriv_within_one (hs_unique _ (h hy)),
exact hf'.has_deriv_within_at.mono h },
simp_rw ... | lemma | has_deriv_within_at_taylor_within_eval | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"cont_diff_on",
"differentiable_on",
"differentiable_on.mono",
"differentiable_within_at",
"has_deriv_within_at",
"has_deriv_within_at_taylor_coeff_within",
"inv_one",
"iterated_deriv_within",
"iterated_deriv_within_one",
"iterated_deriv_within_zero",
"mul_one",
"nat.cast_add",
"nat.cast_mul... | Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for arbitrary sets | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
taylor_within_eval_has_deriv_at_Ioo {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ}
(hx : a < b) (ht : t ∈ Ioo a b)
(hf : cont_diff_on ℝ n f (Icc a b))
(hf' : differentiable_on ℝ (iterated_deriv_within n f (Icc a b)) (Ioo a b)) :
has_deriv_at (λ y, taylor_within_eval f n (Icc a b) y x)
(((n! : ℝ)⁻¹ * (x - t)^n) • (... | have h_nhds : Ioo a b ∈ 𝓝 t := is_open_Ioo.mem_nhds ht,
have h_nhds' : Ioo a b ∈ 𝓝[Icc a b] t := nhds_within_le_nhds h_nhds,
(has_deriv_within_at_taylor_within_eval (unique_diff_within_at_Ioo ht) (unique_diff_on_Icc hx)
h_nhds' ht Ioo_subset_Icc_self hf $ (hf' t ht).mono_of_mem h_nhds').has_deriv_at h_nhds | lemma | taylor_within_eval_has_deriv_at_Ioo | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"cont_diff_on",
"differentiable_on",
"has_deriv_at",
"has_deriv_within_at_taylor_within_eval",
"iterated_deriv_within",
"nhds_within_le_nhds",
"taylor_within_eval",
"unique_diff_on_Icc",
"unique_diff_within_at_Ioo"
] | Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for open intervals | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_within_taylor_within_eval_at_Icc {f : ℝ → E} {a b t : ℝ} (x : ℝ) {n : ℕ}
(hx : a < b) (ht : t ∈ Icc a b) (hf : cont_diff_on ℝ n f (Icc a b))
(hf' : differentiable_on ℝ (iterated_deriv_within n f (Icc a b)) (Icc a b)) :
has_deriv_within_at (λ y, taylor_within_eval f n (Icc a b) y x)
(((n! : ℝ)⁻¹ * (x... | has_deriv_within_at_taylor_within_eval (unique_diff_on_Icc hx t ht) (unique_diff_on_Icc hx)
self_mem_nhds_within ht rfl.subset hf (hf' t ht) | lemma | has_deriv_within_taylor_within_eval_at_Icc | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"cont_diff_on",
"differentiable_on",
"has_deriv_within_at",
"has_deriv_within_at_taylor_within_eval",
"iterated_deriv_within",
"self_mem_nhds_within",
"taylor_within_eval",
"unique_diff_on_Icc"
] | Calculate the derivative of the Taylor polynomial with respect to `x₀`.
Version for closed intervals | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
taylor_mean_remainder {f : ℝ → ℝ} {g g' : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x)
(hf : cont_diff_on ℝ n f (Icc x₀ x))
(hf' : differentiable_on ℝ (iterated_deriv_within n f (Icc x₀ x)) (Ioo x₀ x))
(gcont : continuous_on g (Icc x₀ x))
(gdiff : ∀ (x_1 : ℝ), x_1 ∈ Ioo x₀ x → has_deriv_at g (g' x_1) x_1)
(g'_ne :... | begin
-- We apply the mean value theorem
rcases exists_ratio_has_deriv_at_eq_ratio_slope (λ t, taylor_within_eval f n (Icc x₀ x) t x)
(λ t, ((n! : ℝ)⁻¹ * (x - t)^n) • (iterated_deriv_within (n+1) f (Icc x₀ x) t)) hx
(continuous_on_taylor_within_eval (unique_diff_on_Icc hx) hf)
(λ _ hy, taylor_within_eva... | lemma | taylor_mean_remainder | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"cont_diff_on",
"continuous_on",
"continuous_on_taylor_within_eval",
"differentiable_on",
"exists_ratio_has_deriv_at_eq_ratio_slope",
"has_deriv_at",
"iterated_deriv_within",
"mul_comm",
"mul_div_cancel",
"ring",
"taylor_within_eval",
"taylor_within_eval_has_deriv_at_Ioo",
"taylor_within_eva... | **Taylor's theorem** with the general mean value form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`, and `g` is a differentiable function on
`Ioo x₀ x` and continuous on `Icc x₀ x`. Then there exist... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x)
(hf : cont_diff_on ℝ n f (Icc x₀ x))
(hf' : differentiable_on ℝ (iterated_deriv_within n f (Icc x₀ x)) (Ioo x₀ x)) :
∃ (x' : ℝ) (hx' : x' ∈ Ioo x₀ x), f x - taylor_within_eval f n (Icc x₀ x) x₀ x =
(iterated_deriv_within (n+1) f (Icc x₀... | begin
have gcont : continuous_on (λ (t : ℝ), (x - t) ^ (n + 1)) (Icc x₀ x) :=
by { refine continuous.continuous_on _, continuity },
have xy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y)^n ≠ 0 :=
begin
intros y hy,
refine pow_ne_zero _ _,
rw [mem_Ioo] at hy,
rw sub_ne_zero,
exact hy.2.ne.symm,
end... | lemma | taylor_mean_remainder_lagrange | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"cont_diff_on",
"continuity",
"continuous.continuous_on",
"continuous_on",
"differentiable_on",
"iterated_deriv_within",
"monomial_has_deriv_aux",
"mul_ne_zero",
"mul_neg",
"nat.cast_add_one_ne_zero",
"neg_div",
"neg_mul",
"pow_ne_zero",
"ring",
"taylor_mean_remainder",
"taylor_within_... | **Taylor's theorem** with the Lagrange form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable in the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists a `x' ∈ Ioo x₀ x` such that
$$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x₀)^{n+1}}... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
taylor_mean_remainder_cauchy {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x)
(hf : cont_diff_on ℝ n f (Icc x₀ x))
(hf' : differentiable_on ℝ (iterated_deriv_within n f (Icc x₀ x)) (Ioo x₀ x)) :
∃ (x' : ℝ) (hx' : x' ∈ Ioo x₀ x), f x - taylor_within_eval f n (Icc x₀ x) x₀ x =
(iterated_deriv_within (n+1) f (Icc x₀ x... | begin
have gcont : continuous_on id (Icc x₀ x) := continuous.continuous_on (by continuity),
have gdiff : (∀ (x_1 : ℝ), x_1 ∈ Ioo x₀ x → has_deriv_at id
((λ (t : ℝ), (1 : ℝ)) x_1) x_1) := λ _ _, has_deriv_at_id _,
-- We apply the general theorem with g = id
rcases taylor_mean_remainder hx hf hf' gcont gdiff ... | lemma | taylor_mean_remainder_cauchy | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"cont_diff_on",
"continuity",
"continuous.continuous_on",
"continuous_on",
"differentiable_on",
"has_deriv_at",
"has_deriv_at_id",
"iterated_deriv_within",
"ring",
"taylor_mean_remainder",
"taylor_within_eval"
] | **Taylor's theorem** with the Cauchy form of the remainder.
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc x₀ x` and
`n+1`-times differentiable on the open set `Ioo x₀ x`. Then there exists a `x' ∈ Ioo x₀ x` such that
$$f(x) - (P_n f)(x₀, x) = \frac{f^{(n+1)}(x') (x - x')^n (x-x₀)... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
taylor_mean_remainder_bound {f : ℝ → E} {a b C x : ℝ} {n : ℕ}
(hab : a ≤ b) (hf : cont_diff_on ℝ (n+1) f (Icc a b)) (hx : x ∈ Icc a b)
(hC : ∀ y ∈ Icc a b, ‖iterated_deriv_within (n + 1) f (Icc a b) y‖ ≤ C) :
‖f x - taylor_within_eval f n (Icc a b) a x‖ ≤ C * (x - a)^(n+1) / n! | begin
rcases eq_or_lt_of_le hab with rfl|h,
{ rw [Icc_self, mem_singleton_iff] at hx,
simp [hx] },
-- The nth iterated derivative is differentiable
have hf' : differentiable_on ℝ (iterated_deriv_within n f (Icc a b)) (Icc a b) :=
hf.differentiable_on_iterated_deriv_within (with_top.coe_lt_coe.mpr n.lt_suc... | lemma | taylor_mean_remainder_bound | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"abs_inv",
"abs_mul",
"abs_of_nonneg",
"abs_pow",
"cont_diff_on",
"differentiable_on",
"eq_or_lt_of_le",
"has_deriv_within_at",
"has_deriv_within_taylor_within_eval_at_Icc",
"iterated_deriv_within",
"mul_le_mul",
"nat.abs_cast",
"norm_image_sub_le_of_norm_deriv_le_segment'",
"norm_smul",
... | **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
The difference of `f` and its `n`-th Taylor polynomial can be estimated by
`C * (x - a)^(n+1) / n!` where `C` is a bound for the `n+1`-th iterated derivative of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
exists_taylor_mean_remainder_bound {f : ℝ → E} {a b : ℝ} {n : ℕ}
(hab : a ≤ b) (hf : cont_diff_on ℝ (n+1) f (Icc a b)) :
∃ C, ∀ x ∈ Icc a b, ‖f x - taylor_within_eval f n (Icc a b) a x‖ ≤ C * (x - a)^(n+1) | begin
rcases eq_or_lt_of_le hab with rfl|h,
{ refine ⟨0, λ x hx, _⟩,
have : a = x, by simpa [← le_antisymm_iff] using hx,
simp [← this] },
-- We estimate by the supremum of the norm of the iterated derivative
let g : ℝ → ℝ := λ y, ‖iterated_deriv_within (n + 1) f (Icc a b) y‖,
use [has_Sup.Sup (g '' I... | lemma | exists_taylor_mean_remainder_bound | analysis.calculus | src/analysis/calculus/taylor.lean | [
"analysis.calculus.iterated_deriv",
"analysis.calculus.mean_value",
"data.polynomial.module"
] | [
"cont_diff_on",
"div_mul_eq_mul_div₀",
"eq_or_lt_of_le",
"taylor_mean_remainder_bound",
"taylor_within_eval",
"unique_diff_on_Icc"
] | **Taylor's theorem** with a polynomial bound on the remainder
We assume that `f` is `n+1`-times continuously differentiable on the closed set `Icc a b`.
There exists a constant `C` such that for all `x ∈ Icc a b` the difference of `f` and its `n`-th
Taylor polynomial can be estimated by `C * (x - a)^(n+1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_cauchy_seq_on_filter_of_fderiv
(hf' : uniform_cauchy_seq_on_filter f' l (𝓝 x))
(hf : ∀ᶠ (n : ι × E) in (l ×ᶠ 𝓝 x), has_fderiv_at (f n.1) (f' n.1 n.2) n.2)
(hfg : cauchy (map (λ n, f n x) l)) :
uniform_cauchy_seq_on_filter f l (𝓝 x) | begin
letI : normed_space ℝ E, from normed_space.restrict_scalars ℝ 𝕜 _,
rw seminormed_add_group.uniform_cauchy_seq_on_filter_iff_tendsto_uniformly_on_filter_zero at
hf' ⊢,
suffices : tendsto_uniformly_on_filter
(λ (n : ι × ι) (z : E), f n.1 z - f n.2 z - (f n.1 x - f n.2 x)) 0 (l ×ᶠ l) (𝓝 x) ∧
ten... | lemma | uniform_cauchy_seq_on_filter_of_fderiv | analysis.calculus | src/analysis/calculus/uniform_limits_deriv.lean | [
"analysis.calculus.mean_value",
"analysis.normed_space.is_R_or_C",
"order.filter.curry"
] | [
"cauchy",
"convex.norm_image_sub_le_of_norm_has_fderiv_within_le",
"convex_ball",
"has_fderiv_at",
"has_fderiv_within_at",
"metric.ball",
"metric.mem_ball",
"metric.mem_ball_self",
"metric.tendsto_uniformly_on_filter_iff",
"mul_lt_iff_lt_one_right",
"normed_space",
"normed_space.restrict_scala... | If a sequence of functions real or complex functions are eventually differentiable on a
neighborhood of `x`, they are Cauchy _at_ `x`, and their derivatives
are a uniform Cauchy sequence in a neighborhood of `x`, then the functions form a uniform Cauchy
sequence in a neighborhood of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_cauchy_seq_on_ball_of_fderiv
{r : ℝ} (hf' : uniform_cauchy_seq_on f' l (metric.ball x r))
(hf : ∀ n : ι, ∀ y : E, y ∈ metric.ball x r → has_fderiv_at (f n) (f' n y) y)
(hfg : cauchy (map (λ n, f n x) l)) :
uniform_cauchy_seq_on f l (metric.ball x r) | begin
letI : normed_space ℝ E, from normed_space.restrict_scalars ℝ 𝕜 _,
haveI : ne_bot l, from (cauchy_map_iff.1 hfg).1,
rcases le_or_lt r 0 with hr|hr,
{ simp only [metric.ball_eq_empty.2 hr, uniform_cauchy_seq_on, set.mem_empty_iff_false,
is_empty.forall_iff, eventually_const, implies_true_iff] },
r... | lemma | uniform_cauchy_seq_on_ball_of_fderiv | analysis.calculus | src/analysis/calculus/uniform_limits_deriv.lean | [
"analysis.calculus.mean_value",
"analysis.normed_space.is_R_or_C",
"order.filter.curry"
] | [
"cauchy",
"convex.norm_image_sub_le_of_norm_has_fderiv_within_le",
"convex_ball",
"exists_pos_mul_lt",
"has_fderiv_at",
"has_fderiv_within_at",
"is_empty.forall_iff",
"metric.ball",
"metric.mem_ball_self",
"metric.tendsto_uniformly_on_iff",
"mul_comm",
"mul_lt_mul'",
"normed_space",
"norme... | A variant of the second fundamental theorem of calculus (FTC-2): If a sequence of functions
between real or complex normed spaces are differentiable on a ball centered at `x`, they
form a Cauchy sequence _at_ `x`, and their derivatives are Cauchy uniformly on the ball, then the
functions form a uniform Cauchy sequence ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cauchy_map_of_uniform_cauchy_seq_on_fderiv
{s : set E} (hs : is_open s) (h's : is_preconnected s)
(hf' : uniform_cauchy_seq_on f' l s)
(hf : ∀ n : ι, ∀ y : E, y ∈ s → has_fderiv_at (f n) (f' n y) y)
{x₀ x : E} (hx₀ : x₀ ∈ s) (hx : x ∈ s)
(hfg : cauchy (map (λ n, f n x₀) l)) :
cauchy (map (λ n, f n x) l) | begin
haveI : ne_bot l, from (cauchy_map_iff.1 hfg).1,
let t := {y | y ∈ s ∧ cauchy (map (λ n, f n y) l)},
suffices H : s ⊆ t, from (H hx).2,
have A : ∀ x ε, x ∈ t → metric.ball x ε ⊆ s → metric.ball x ε ⊆ t,
from λ x ε xt hx y hy, ⟨hx hy, (uniform_cauchy_seq_on_ball_of_fderiv (hf'.mono hx)
(λ n y hy, hf ... | lemma | cauchy_map_of_uniform_cauchy_seq_on_fderiv | analysis.calculus | src/analysis/calculus/uniform_limits_deriv.lean | [
"analysis.calculus.mean_value",
"analysis.normed_space.is_R_or_C",
"order.filter.curry"
] | [
"cauchy",
"closure",
"dist_comm",
"half_pos",
"has_fderiv_at",
"is_open",
"is_preconnected",
"metric.ball",
"metric.ball_subset_ball'",
"metric.is_open_iff",
"uniform_cauchy_seq_on",
"uniform_cauchy_seq_on_ball_of_fderiv"
] | If a sequence of functions between real or complex normed spaces are differentiable on a
preconnected open set, they form a Cauchy sequence _at_ `x`, and their derivatives are Cauchy
uniformly on the set, then the functions form a Cauchy sequence at any point in the set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
difference_quotients_converge_uniformly
(hf' : tendsto_uniformly_on_filter f' g' l (𝓝 x))
(hf : ∀ᶠ (n : ι × E) in (l ×ᶠ 𝓝 x), has_fderiv_at (f n.1) (f' n.1 n.2) n.2)
(hfg : ∀ᶠ (y : E) in 𝓝 x, tendsto (λ n, f n y) l (𝓝 (g y))) :
tendsto_uniformly_on_filter
(λ n : ι, λ y : E, (‖y - x‖⁻¹ : 𝕜) • (f n y - f... | begin
letI : normed_space ℝ E, from normed_space.restrict_scalars ℝ 𝕜 _,
rcases eq_or_ne l ⊥ with hl|hl,
{ simp only [hl, tendsto_uniformly_on_filter, bot_prod, eventually_bot, implies_true_iff] },
haveI : ne_bot l := ⟨hl⟩,
refine uniform_cauchy_seq_on_filter.tendsto_uniformly_on_filter_of_tendsto _
((hf... | lemma | difference_quotients_converge_uniformly | analysis.calculus | src/analysis/calculus/uniform_limits_deriv.lean | [
"analysis.calculus.mean_value",
"analysis.normed_space.is_R_or_C",
"order.filter.curry"
] | [
"convex.norm_image_sub_le_of_norm_has_fderiv_within_le",
"convex_ball",
"eq_or_ne",
"exists_pos_rat_lt",
"has_fderiv_at",
"has_fderiv_within_at",
"inv_mul_le_iff",
"is_R_or_C.norm_coe_norm",
"metric.ball",
"metric.mem_ball_self",
"metric.tendsto_uniformly_on_filter_iff",
"mul_comm",
"norm_in... | If `f_n → g` pointwise and the derivatives `(f_n)' → h` _uniformly_ converge, then
in fact for a fixed `y`, the difference quotients `‖z - y‖⁻¹ • (f_n z - f_n y)` converge
_uniformly_ to `‖z - y‖⁻¹ • (g z - g y)` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_of_tendsto_uniformly_on_filter [ne_bot l]
(hf' : tendsto_uniformly_on_filter f' g' l (𝓝 x))
(hf : ∀ᶠ (n : ι × E) in (l ×ᶠ 𝓝 x), has_fderiv_at (f n.1) (f' n.1 n.2) n.2)
(hfg : ∀ᶠ y in 𝓝 x, tendsto (λ n, f n y) l (𝓝 (g y))) :
has_fderiv_at g (g' x) x | begin
-- The proof strategy follows several steps:
-- 1. The quantifiers in the definition of the derivative are
-- `∀ ε > 0, ∃δ > 0, ∀y ∈ B_δ(x)`. We will introduce a quantifier in the middle:
-- `∀ ε > 0, ∃N, ∀n ≥ N, ∃δ > 0, ∀y ∈ B_δ(x)` which will allow us to introduce the `f(') n`
-- 2. The ... | lemma | has_fderiv_at_of_tendsto_uniformly_on_filter | analysis.calculus | src/analysis/calculus/uniform_limits_deriv.lean | [
"analysis.calculus.mean_value",
"analysis.normed_space.is_R_or_C",
"order.filter.curry"
] | [
"abs_inv",
"abs_norm",
"continuous_linear_map.coe_sub'",
"difference_quotients_converge_uniformly",
"forall_const",
"has_fderiv_at",
"has_fderiv_at_iff_tendsto",
"imp_self",
"inv_mul_le_iff",
"is_R_or_C.norm_coe_norm",
"is_R_or_C.norm_of_real",
"is_R_or_C.of_real_inv",
"metric.tendsto_nhds",... | `(d/dx) lim_{n → ∞} f n x = lim_{n → ∞} f' n x` when the `f' n` converge
_uniformly_ to their limit at `x`.
In words the assumptions mean the following:
* `hf'`: The `f'` converge "uniformly at" `x` to `g'`. This does not mean that the `f' n` even
converge away from `x`!
* `hf`: For all `(y, n)` with `y` suffi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_of_tendsto_locally_uniformly_on [ne_bot l] {s : set E} (hs : is_open s)
(hf' : tendsto_locally_uniformly_on f' g' l s)
(hf : ∀ n, ∀ x ∈ s, has_fderiv_at (f n) (f' n x) x)
(hfg : ∀ x ∈ s, tendsto (λ n, f n x) l (𝓝 (g x)))
(hx : x ∈ s) :
has_fderiv_at g (g' x) x | begin
have h1 : s ∈ 𝓝 x := hs.mem_nhds hx,
have h3 : set.univ ×ˢ s ∈ l ×ᶠ 𝓝 x := by simp only [h1, prod_mem_prod_iff, univ_mem, and_self],
have h4 : ∀ᶠ (n : ι × E) in l ×ᶠ 𝓝 x, has_fderiv_at (f n.1) (f' n.1 n.2) n.2,
from eventually_of_mem h3 (λ ⟨n, z⟩ ⟨hn, hz⟩, hf n z hz),
refine has_fderiv_at_of_tendst... | lemma | has_fderiv_at_of_tendsto_locally_uniformly_on | analysis.calculus | src/analysis/calculus/uniform_limits_deriv.lean | [
"analysis.calculus.mean_value",
"analysis.normed_space.is_R_or_C",
"order.filter.curry"
] | [
"has_fderiv_at",
"has_fderiv_at_of_tendsto_uniformly_on_filter",
"is_open",
"is_open.nhds_within_eq",
"tendsto_locally_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_of_tendsto_locally_uniformly_on' [ne_bot l] {s : set E} (hs : is_open s)
(hf' : tendsto_locally_uniformly_on (fderiv 𝕜 ∘ f) g' l s)
(hf : ∀ n, differentiable_on 𝕜 (f n) s)
(hfg : ∀ x ∈ s, tendsto (λ n, f n x) l (𝓝 (g x)))
(hx : x ∈ s) :
has_fderiv_at g (g' x) x | begin
refine has_fderiv_at_of_tendsto_locally_uniformly_on hs hf' (λ n z hz, _) hfg hx,
exact ((hf n z hz).differentiable_at (hs.mem_nhds hz)).has_fderiv_at
end | lemma | has_fderiv_at_of_tendsto_locally_uniformly_on' | analysis.calculus | src/analysis/calculus/uniform_limits_deriv.lean | [
"analysis.calculus.mean_value",
"analysis.normed_space.is_R_or_C",
"order.filter.curry"
] | [
"differentiable_at",
"differentiable_on",
"fderiv",
"has_fderiv_at",
"has_fderiv_at_of_tendsto_locally_uniformly_on",
"is_open",
"tendsto_locally_uniformly_on"
] | A slight variant of `has_fderiv_at_of_tendsto_locally_uniformly_on` with the assumption stated
in terms of `differentiable_on` rather than `has_fderiv_at`. This makes a few proofs nicer in
complex analysis where holomorphicity is assumed but the derivative is not known a priori. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_of_tendsto_uniformly_on [ne_bot l]
{s : set E} (hs : is_open s)
(hf' : tendsto_uniformly_on f' g' l s)
(hf : ∀ (n : ι), ∀ (x : E), x ∈ s → has_fderiv_at (f n) (f' n x) x)
(hfg : ∀ (x : E), x ∈ s → tendsto (λ n, f n x) l (𝓝 (g x))) :
∀ (x : E), x ∈ s → has_fderiv_at g (g' x) x | λ x, has_fderiv_at_of_tendsto_locally_uniformly_on hs hf'.tendsto_locally_uniformly_on hf hfg | lemma | has_fderiv_at_of_tendsto_uniformly_on | analysis.calculus | src/analysis/calculus/uniform_limits_deriv.lean | [
"analysis.calculus.mean_value",
"analysis.normed_space.is_R_or_C",
"order.filter.curry"
] | [
"has_fderiv_at",
"has_fderiv_at_of_tendsto_locally_uniformly_on",
"is_open",
"tendsto_uniformly_on"
] | `(d/dx) lim_{n → ∞} f n x = lim_{n → ∞} f' n x` when the `f' n` converge
_uniformly_ to their limit on an open set containing `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_of_tendsto_uniformly [ne_bot l]
(hf' : tendsto_uniformly f' g' l)
(hf : ∀ (n : ι), ∀ (x : E), has_fderiv_at (f n) (f' n x) x)
(hfg : ∀ (x : E), tendsto (λ n, f n x) l (𝓝 (g x))) :
∀ (x : E), has_fderiv_at g (g' x) x | begin
intros x,
have hf : ∀ (n : ι), ∀ (x : E), x ∈ set.univ → has_fderiv_at (f n) (f' n x) x, { simp [hf], },
have hfg : ∀ (x : E), x ∈ set.univ → tendsto (λ n, f n x) l (𝓝 (g x)), { simp [hfg], },
have hf' : tendsto_uniformly_on f' g' l set.univ, { rwa tendsto_uniformly_on_univ, },
refine has_fderiv_at_of_... | lemma | has_fderiv_at_of_tendsto_uniformly | analysis.calculus | src/analysis/calculus/uniform_limits_deriv.lean | [
"analysis.calculus.mean_value",
"analysis.normed_space.is_R_or_C",
"order.filter.curry"
] | [
"has_fderiv_at",
"has_fderiv_at_of_tendsto_uniformly_on",
"is_open_univ",
"set.mem_univ",
"tendsto_uniformly",
"tendsto_uniformly_on",
"tendsto_uniformly_on_univ"
] | `(d/dx) lim_{n → ∞} f n x = lim_{n → ∞} f' n x` when the `f' n` converge
_uniformly_ to their limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_cauchy_seq_on_filter.one_smul_right {l' : filter 𝕜}
(hf' : uniform_cauchy_seq_on_filter f' l l') :
uniform_cauchy_seq_on_filter (λ n, λ z, (1 : 𝕜 →L[𝕜] 𝕜).smul_right (f' n z)) l l' | begin
-- The tricky part of this proof is that operator norms are written in terms of `≤` whereas
-- metrics are written in terms of `<`. So we need to shrink `ε` utilizing the archimedean
-- property of `ℝ`
rw [seminormed_add_group.uniform_cauchy_seq_on_filter_iff_tendsto_uniformly_on_filter_zero,
metri... | lemma | uniform_cauchy_seq_on_filter.one_smul_right | analysis.calculus | src/analysis/calculus/uniform_limits_deriv.lean | [
"analysis.calculus.mean_value",
"analysis.normed_space.is_R_or_C",
"order.filter.curry"
] | [
"continuous_linear_map.coe_sub'",
"continuous_linear_map.one_apply",
"continuous_linear_map.op_norm_le_bound",
"continuous_linear_map.smul_right_apply",
"exists_between",
"filter",
"metric.tendsto_uniformly_on_filter_iff",
"mul_comm",
"mul_le_mul",
"norm_smul",
"uniform_cauchy_seq_on_filter"
] | If our derivatives converge uniformly, then the Fréchet derivatives converge uniformly | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_cauchy_seq_on_filter_of_deriv
(hf' : uniform_cauchy_seq_on_filter f' l (𝓝 x))
(hf : ∀ᶠ (n : ι × 𝕜) in (l ×ᶠ 𝓝 x), has_deriv_at (f n.1) (f' n.1 n.2) n.2)
(hfg : cauchy (map (λ n, f n x) l)) :
uniform_cauchy_seq_on_filter f l (𝓝 x) | begin
simp_rw has_deriv_at_iff_has_fderiv_at at hf,
exact uniform_cauchy_seq_on_filter_of_fderiv
hf'.one_smul_right hf hfg,
end | lemma | uniform_cauchy_seq_on_filter_of_deriv | analysis.calculus | src/analysis/calculus/uniform_limits_deriv.lean | [
"analysis.calculus.mean_value",
"analysis.normed_space.is_R_or_C",
"order.filter.curry"
] | [
"cauchy",
"has_deriv_at",
"has_deriv_at_iff_has_fderiv_at",
"uniform_cauchy_seq_on_filter",
"uniform_cauchy_seq_on_filter_of_fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_cauchy_seq_on_ball_of_deriv
{r : ℝ} (hf' : uniform_cauchy_seq_on f' l (metric.ball x r))
(hf : ∀ n : ι, ∀ y : 𝕜, y ∈ metric.ball x r → has_deriv_at (f n) (f' n y) y)
(hfg : cauchy (map (λ n, f n x) l)) :
uniform_cauchy_seq_on f l (metric.ball x r) | begin
simp_rw has_deriv_at_iff_has_fderiv_at at hf,
rw uniform_cauchy_seq_on_iff_uniform_cauchy_seq_on_filter at hf',
have hf' : uniform_cauchy_seq_on (λ n, λ z, (1 : 𝕜 →L[𝕜] 𝕜).smul_right (f' n z)) l
(metric.ball x r),
{ rw uniform_cauchy_seq_on_iff_uniform_cauchy_seq_on_filter,
exact hf'.one_smul_r... | lemma | uniform_cauchy_seq_on_ball_of_deriv | analysis.calculus | src/analysis/calculus/uniform_limits_deriv.lean | [
"analysis.calculus.mean_value",
"analysis.normed_space.is_R_or_C",
"order.filter.curry"
] | [
"cauchy",
"has_deriv_at",
"has_deriv_at_iff_has_fderiv_at",
"metric.ball",
"uniform_cauchy_seq_on",
"uniform_cauchy_seq_on_ball_of_fderiv",
"uniform_cauchy_seq_on_iff_uniform_cauchy_seq_on_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_of_tendsto_uniformly_on_filter [ne_bot l]
(hf' : tendsto_uniformly_on_filter f' g' l (𝓝 x))
(hf : ∀ᶠ (n : ι × 𝕜) in (l ×ᶠ 𝓝 x), has_deriv_at (f n.1) (f' n.1 n.2) n.2)
(hfg : ∀ᶠ y in 𝓝 x, tendsto (λ n, f n y) l (𝓝 (g y))) :
has_deriv_at g (g' x) x | begin
-- The first part of the proof rewrites `hf` and the goal to be functions so that Lean
-- can recognize them when we apply `has_fderiv_at_of_tendsto_uniformly_on_filter`
let F' := (λ n, λ z, (1 : 𝕜 →L[𝕜] 𝕜).smul_right (f' n z)),
let G' := λ z, (1 : 𝕜 →L[𝕜] 𝕜).smul_right (g' z),
simp_rw has_deriv_a... | lemma | has_deriv_at_of_tendsto_uniformly_on_filter | analysis.calculus | src/analysis/calculus/uniform_limits_deriv.lean | [
"analysis.calculus.mean_value",
"analysis.normed_space.is_R_or_C",
"order.filter.curry"
] | [
"continuous_linear_map.coe_sub'",
"continuous_linear_map.one_apply",
"continuous_linear_map.op_norm_le_bound",
"continuous_linear_map.smul_right_apply",
"exists_between",
"has_deriv_at",
"has_deriv_at_iff_has_fderiv_at",
"has_fderiv_at_of_tendsto_uniformly_on_filter",
"metric.tendsto_uniformly_on_fi... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_of_tendsto_locally_uniformly_on [ne_bot l] {s : set 𝕜} (hs : is_open s)
(hf' : tendsto_locally_uniformly_on f' g' l s)
(hf : ∀ᶠ n in l, ∀ x ∈ s, has_deriv_at (f n) (f' n x) x)
(hfg : ∀ x ∈ s, tendsto (λ n, f n x) l (𝓝 (g x)))
(hx : x ∈ s) :
has_deriv_at g (g' x) x | begin
have h1 : s ∈ 𝓝 x := hs.mem_nhds hx,
have h2 : ∀ᶠ (n : ι × 𝕜) in l ×ᶠ 𝓝 x, has_deriv_at (f n.1) (f' n.1 n.2) n.2,
from eventually_prod_iff.2 ⟨_, hf, λ x, x ∈ s, h1, λ n, id⟩,
refine has_deriv_at_of_tendsto_uniformly_on_filter _ h2 (eventually_of_mem h1 hfg),
simpa [is_open.nhds_within_eq hs hx] usi... | lemma | has_deriv_at_of_tendsto_locally_uniformly_on | analysis.calculus | src/analysis/calculus/uniform_limits_deriv.lean | [
"analysis.calculus.mean_value",
"analysis.normed_space.is_R_or_C",
"order.filter.curry"
] | [
"has_deriv_at",
"has_deriv_at_of_tendsto_uniformly_on_filter",
"is_open",
"is_open.nhds_within_eq",
"tendsto_locally_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_of_tendsto_locally_uniformly_on' [ne_bot l] {s : set 𝕜} (hs : is_open s)
(hf' : tendsto_locally_uniformly_on (deriv ∘ f) g' l s)
(hf : ∀ᶠ n in l, differentiable_on 𝕜 (f n) s)
(hfg : ∀ x ∈ s, tendsto (λ n, f n x) l (𝓝 (g x)))
(hx : x ∈ s) :
has_deriv_at g (g' x) x | begin
refine has_deriv_at_of_tendsto_locally_uniformly_on hs hf' _ hfg hx,
filter_upwards [hf] with n h z hz using ((h z hz).differentiable_at (hs.mem_nhds hz)).has_deriv_at
end | lemma | has_deriv_at_of_tendsto_locally_uniformly_on' | analysis.calculus | src/analysis/calculus/uniform_limits_deriv.lean | [
"analysis.calculus.mean_value",
"analysis.normed_space.is_R_or_C",
"order.filter.curry"
] | [
"deriv",
"differentiable_at",
"differentiable_on",
"has_deriv_at",
"has_deriv_at_of_tendsto_locally_uniformly_on",
"is_open",
"tendsto_locally_uniformly_on"
] | A slight variant of `has_deriv_at_of_tendsto_locally_uniformly_on` with the assumption stated in
terms of `differentiable_on` rather than `has_deriv_at`. This makes a few proofs nicer in complex
analysis where holomorphicity is assumed but the derivative is not known a priori. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_of_tendsto_uniformly_on [ne_bot l]
{s : set 𝕜} (hs : is_open s)
(hf' : tendsto_uniformly_on f' g' l s)
(hf : ∀ᶠ n in l, ∀ (x : 𝕜), x ∈ s → has_deriv_at (f n) (f' n x) x)
(hfg : ∀ (x : 𝕜), x ∈ s → tendsto (λ n, f n x) l (𝓝 (g x))) :
∀ (x : 𝕜), x ∈ s → has_deriv_at g (g' x) x | λ x, has_deriv_at_of_tendsto_locally_uniformly_on hs hf'.tendsto_locally_uniformly_on hf hfg | lemma | has_deriv_at_of_tendsto_uniformly_on | analysis.calculus | src/analysis/calculus/uniform_limits_deriv.lean | [
"analysis.calculus.mean_value",
"analysis.normed_space.is_R_or_C",
"order.filter.curry"
] | [
"has_deriv_at",
"has_deriv_at_of_tendsto_locally_uniformly_on",
"is_open",
"tendsto_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_of_tendsto_uniformly [ne_bot l]
(hf' : tendsto_uniformly f' g' l)
(hf : ∀ᶠ n in l, ∀ (x : 𝕜), has_deriv_at (f n) (f' n x) x)
(hfg : ∀ (x : 𝕜), tendsto (λ n, f n x) l (𝓝 (g x))) :
∀ (x : 𝕜), has_deriv_at g (g' x) x | begin
intros x,
have hf : ∀ᶠ n in l, ∀ (x : 𝕜), x ∈ set.univ → has_deriv_at (f n) (f' n x) x,
by filter_upwards [hf] with n h x hx using h x,
have hfg : ∀ (x : 𝕜), x ∈ set.univ → tendsto (λ n, f n x) l (𝓝 (g x)), { simp [hfg], },
have hf' : tendsto_uniformly_on f' g' l set.univ, { rwa tendsto_uniformly_o... | lemma | has_deriv_at_of_tendsto_uniformly | analysis.calculus | src/analysis/calculus/uniform_limits_deriv.lean | [
"analysis.calculus.mean_value",
"analysis.normed_space.is_R_or_C",
"order.filter.curry"
] | [
"has_deriv_at",
"has_deriv_at_of_tendsto_uniformly_on",
"is_open_univ",
"set.mem_univ",
"tendsto_uniformly",
"tendsto_uniformly_on",
"tendsto_uniformly_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conformal_at_iff' {f : E → F} {x : E} :
conformal_at f x ↔
∃ (c : ℝ), 0 < c ∧ ∀ (u v : E), ⟪fderiv ℝ f x u, fderiv ℝ f x v⟫ = c * ⟪u, v⟫ | by rw [conformal_at_iff_is_conformal_map_fderiv, is_conformal_map_iff] | lemma | conformal_at_iff' | analysis.calculus.conformal | src/analysis/calculus/conformal/inner_product.lean | [
"analysis.calculus.conformal.normed_space",
"analysis.inner_product_space.conformal_linear_map"
] | [
"conformal_at",
"conformal_at_iff_is_conformal_map_fderiv",
"fderiv",
"is_conformal_map_iff"
] | A real differentiable map `f` is conformal at point `x` if and only if its
differential `fderiv ℝ f x` at that point scales every inner product by a positive scalar. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conformal_at_iff {f : E → F} {x : E} {f' : E →L[ℝ] F} (h : has_fderiv_at f f' x) :
conformal_at f x ↔ ∃ (c : ℝ), 0 < c ∧ ∀ (u v : E), ⟪f' u, f' v⟫ = c * ⟪u, v⟫ | by simp only [conformal_at_iff', h.fderiv] | lemma | conformal_at_iff | analysis.calculus.conformal | src/analysis/calculus/conformal/inner_product.lean | [
"analysis.calculus.conformal.normed_space",
"analysis.inner_product_space.conformal_linear_map"
] | [
"conformal_at",
"conformal_at_iff'",
"has_fderiv_at"
] | A real differentiable map `f` is conformal at point `x` if and only if its
differential `f'` at that point scales every inner product by a positive scalar. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conformal_factor_at {f : E → F} {x : E} (h : conformal_at f x) : ℝ | classical.some (conformal_at_iff'.mp h) | def | conformal_factor_at | analysis.calculus.conformal | src/analysis/calculus/conformal/inner_product.lean | [
"analysis.calculus.conformal.normed_space",
"analysis.inner_product_space.conformal_linear_map"
] | [
"conformal_at"
] | The conformal factor of a conformal map at some point `x`. Some authors refer to this function
as the characteristic function of the conformal map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conformal_factor_at_pos {f : E → F} {x : E} (h : conformal_at f x) :
0 < conformal_factor_at h | (classical.some_spec $ conformal_at_iff'.mp h).1 | lemma | conformal_factor_at_pos | analysis.calculus.conformal | src/analysis/calculus/conformal/inner_product.lean | [
"analysis.calculus.conformal.normed_space",
"analysis.inner_product_space.conformal_linear_map"
] | [
"conformal_at",
"conformal_factor_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conformal_factor_at_inner_eq_mul_inner' {f : E → F} {x : E}
(h : conformal_at f x) (u v : E) :
⟪(fderiv ℝ f x) u, (fderiv ℝ f x) v⟫ = (conformal_factor_at h : ℝ) * ⟪u, v⟫ | (classical.some_spec $ conformal_at_iff'.mp h).2 u v | lemma | conformal_factor_at_inner_eq_mul_inner' | analysis.calculus.conformal | src/analysis/calculus/conformal/inner_product.lean | [
"analysis.calculus.conformal.normed_space",
"analysis.inner_product_space.conformal_linear_map"
] | [
"conformal_at",
"conformal_factor_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conformal_factor_at_inner_eq_mul_inner {f : E → F} {x : E} {f' : E →L[ℝ] F}
(h : has_fderiv_at f f' x) (H : conformal_at f x) (u v : E) :
⟪f' u, f' v⟫ = (conformal_factor_at H : ℝ) * ⟪u, v⟫ | (H.differentiable_at.has_fderiv_at.unique h) ▸ conformal_factor_at_inner_eq_mul_inner' H u v | lemma | conformal_factor_at_inner_eq_mul_inner | analysis.calculus.conformal | src/analysis/calculus/conformal/inner_product.lean | [
"analysis.calculus.conformal.normed_space",
"analysis.inner_product_space.conformal_linear_map"
] | [
"conformal_at",
"conformal_factor_at",
"conformal_factor_at_inner_eq_mul_inner'",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conformal_at (f : X → Y) (x : X) | ∃ (f' : X →L[ℝ] Y), has_fderiv_at f f' x ∧ is_conformal_map f' | def | conformal_at | analysis.calculus.conformal | src/analysis/calculus/conformal/normed_space.lean | [
"analysis.normed_space.conformal_linear_map",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_fderiv_at",
"is_conformal_map"
] | A map `f` is said to be conformal if it has a conformal differential `f'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conformal_at_id (x : X) : conformal_at id x | ⟨id ℝ X, has_fderiv_at_id _, is_conformal_map_id⟩ | lemma | conformal_at_id | analysis.calculus.conformal | src/analysis/calculus/conformal/normed_space.lean | [
"analysis.normed_space.conformal_linear_map",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"conformal_at",
"has_fderiv_at_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conformal_at_const_smul {c : ℝ} (h : c ≠ 0) (x : X) :
conformal_at (λ (x': X), c • x') x | ⟨c • continuous_linear_map.id ℝ X,
(has_fderiv_at_id x).const_smul c, is_conformal_map_const_smul h⟩ | lemma | conformal_at_const_smul | analysis.calculus.conformal | src/analysis/calculus/conformal/normed_space.lean | [
"analysis.normed_space.conformal_linear_map",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"conformal_at",
"continuous_linear_map.id",
"has_fderiv_at_id",
"is_conformal_map_const_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subsingleton.conformal_at [subsingleton X] (f : X → Y) (x : X) :
conformal_at f x | ⟨0, has_fderiv_at_of_subsingleton _ _, is_conformal_map_of_subsingleton _⟩ | lemma | subsingleton.conformal_at | analysis.calculus.conformal | src/analysis/calculus/conformal/normed_space.lean | [
"analysis.normed_space.conformal_linear_map",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"conformal_at",
"has_fderiv_at_of_subsingleton",
"is_conformal_map_of_subsingleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conformal_at_iff_is_conformal_map_fderiv {f : X → Y} {x : X} :
conformal_at f x ↔ is_conformal_map (fderiv ℝ f x) | begin
split,
{ rintros ⟨f', hf, hf'⟩,
rwa hf.fderiv },
{ intros H,
by_cases h : differentiable_at ℝ f x,
{ exact ⟨fderiv ℝ f x, h.has_fderiv_at, H⟩, },
{ nontriviality X,
exact absurd (fderiv_zero_of_not_differentiable_at h) H.ne_zero } },
end | lemma | conformal_at_iff_is_conformal_map_fderiv | analysis.calculus.conformal | src/analysis/calculus/conformal/normed_space.lean | [
"analysis.normed_space.conformal_linear_map",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"conformal_at",
"differentiable_at",
"fderiv",
"fderiv_zero_of_not_differentiable_at",
"is_conformal_map"
] | A function is a conformal map if and only if its differential is a conformal linear map | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_at {f : X → Y} {x : X} (h : conformal_at f x) :
differentiable_at ℝ f x | let ⟨_, h₁, _⟩ := h in h₁.differentiable_at | lemma | conformal_at.differentiable_at | analysis.calculus.conformal | src/analysis/calculus/conformal/normed_space.lean | [
"analysis.normed_space.conformal_linear_map",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"conformal_at",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
congr {f g : X → Y} {x : X} {u : set X} (hx : x ∈ u) (hu : is_open u)
(hf : conformal_at f x) (h : ∀ (x : X), x ∈ u → g x = f x) :
conformal_at g x | let ⟨f', hfderiv, hf'⟩ := hf in
⟨f', hfderiv.congr_of_eventually_eq ((hu.eventually_mem hx).mono h), hf'⟩ | lemma | conformal_at.congr | analysis.calculus.conformal | src/analysis/calculus/conformal/normed_space.lean | [
"analysis.normed_space.conformal_linear_map",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"conformal_at",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {f : X → Y} {g : Y → Z} (x : X)
(hg : conformal_at g (f x)) (hf : conformal_at f x) : conformal_at (g ∘ f) x | begin
rcases hf with ⟨f', hf₁, cf⟩,
rcases hg with ⟨g', hg₁, cg⟩,
exact ⟨g'.comp f', hg₁.comp x hf₁, cg.comp cf⟩,
end | lemma | conformal_at.comp | analysis.calculus.conformal | src/analysis/calculus/conformal/normed_space.lean | [
"analysis.normed_space.conformal_linear_map",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"conformal_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_smul {f : X → Y} {x : X} {c : ℝ} (hc : c ≠ 0) (hf : conformal_at f x) :
conformal_at (c • f) x | (conformal_at_const_smul hc $ f x).comp x hf | lemma | conformal_at.const_smul | analysis.calculus.conformal | src/analysis/calculus/conformal/normed_space.lean | [
"analysis.normed_space.conformal_linear_map",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"conformal_at",
"conformal_at_const_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conformal (f : X → Y) | ∀ (x : X), conformal_at f x | def | conformal | analysis.calculus.conformal | src/analysis/calculus/conformal/normed_space.lean | [
"analysis.normed_space.conformal_linear_map",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"conformal_at"
] | A map `f` is conformal if it's conformal at every point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
conformal_id : conformal (id : X → X) | λ x, conformal_at_id x | lemma | conformal_id | analysis.calculus.conformal | src/analysis/calculus/conformal/normed_space.lean | [
"analysis.normed_space.conformal_linear_map",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"conformal",
"conformal_at_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conformal_const_smul {c : ℝ} (h : c ≠ 0) : conformal (λ (x : X), c • x) | λ x, conformal_at_const_smul h x | lemma | conformal_const_smul | analysis.calculus.conformal | src/analysis/calculus/conformal/normed_space.lean | [
"analysis.normed_space.conformal_linear_map",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"conformal",
"conformal_at_const_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
conformal_at {f : X → Y} (h : conformal f) (x : X) : conformal_at f x | h x | lemma | conformal.conformal_at | analysis.calculus.conformal | src/analysis/calculus/conformal/normed_space.lean | [
"analysis.normed_space.conformal_linear_map",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"conformal",
"conformal_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable {f : X → Y} (h : conformal f) : differentiable ℝ f | λ x, (h x).differentiable_at | lemma | conformal.differentiable | analysis.calculus.conformal | src/analysis/calculus/conformal/normed_space.lean | [
"analysis.normed_space.conformal_linear_map",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"conformal",
"differentiable",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {f : X → Y} {g : Y → Z} (hf : conformal f) (hg : conformal g) : conformal (g ∘ f) | λ x, (hg $ f x).comp x (hf x) | lemma | conformal.comp | analysis.calculus.conformal | src/analysis/calculus/conformal/normed_space.lean | [
"analysis.normed_space.conformal_linear_map",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"conformal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_smul {f : X → Y} (hf : conformal f) {c : ℝ} (hc : c ≠ 0) : conformal (c • f) | λ x, (hf x).const_smul hc | lemma | conformal.const_smul | analysis.calculus.conformal | src/analysis/calculus/conformal/normed_space.lean | [
"analysis.normed_space.conformal_linear_map",
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"conformal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter.add
(hf : has_deriv_at_filter f f' x L) (hg : has_deriv_at_filter g g' x L) :
has_deriv_at_filter (λ y, f y + g y) (f' + g') x L | by simpa using (hf.add hg).has_deriv_at_filter | theorem | has_deriv_at_filter.add | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.add
(hf : has_strict_deriv_at f f' x) (hg : has_strict_deriv_at g g' x) :
has_strict_deriv_at (λ y, f y + g y) (f' + g') x | by simpa using (hf.add hg).has_strict_deriv_at | theorem | has_strict_deriv_at.add | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.add
(hf : has_deriv_within_at f f' s x) (hg : has_deriv_within_at g g' s x) :
has_deriv_within_at (λ y, f y + g y) (f' + g') s x | hf.add hg | theorem | has_deriv_within_at.add | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.add
(hf : has_deriv_at f f' x) (hg : has_deriv_at g g' x) :
has_deriv_at (λ x, f x + g x) (f' + g') x | hf.add hg | theorem | has_deriv_at.add | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_add (hxs : unique_diff_within_at 𝕜 s x)
(hf : differentiable_within_at 𝕜 f s x) (hg : differentiable_within_at 𝕜 g s x) :
deriv_within (λy, f y + g y) s x = deriv_within f s x + deriv_within g s x | (hf.has_deriv_within_at.add hg.has_deriv_within_at).deriv_within hxs | lemma | deriv_within_add | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_add
(hf : differentiable_at 𝕜 f x) (hg : differentiable_at 𝕜 g x) :
deriv (λy, f y + g y) x = deriv f x + deriv g x | (hf.has_deriv_at.add hg.has_deriv_at).deriv | lemma | deriv_add | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter.add_const
(hf : has_deriv_at_filter f f' x L) (c : F) :
has_deriv_at_filter (λ y, f y + c) f' x L | add_zero f' ▸ hf.add (has_deriv_at_filter_const x L c) | theorem | has_deriv_at_filter.add_const | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at_filter",
"has_deriv_at_filter_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.add_const
(hf : has_deriv_within_at f f' s x) (c : F) :
has_deriv_within_at (λ y, f y + c) f' s x | hf.add_const c | theorem | has_deriv_within_at.add_const | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.add_const
(hf : has_deriv_at f f' x) (c : F) :
has_deriv_at (λ x, f x + c) f' x | hf.add_const c | theorem | has_deriv_at.add_const | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_add_const (hxs : unique_diff_within_at 𝕜 s x) (c : F) :
deriv_within (λy, f y + c) s x = deriv_within f s x | by simp only [deriv_within, fderiv_within_add_const hxs] | lemma | deriv_within_add_const | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv_within",
"fderiv_within_add_const",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_add_const (c : F) : deriv (λy, f y + c) x = deriv f x | by simp only [deriv, fderiv_add_const] | lemma | deriv_add_const | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"fderiv_add_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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