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
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
differentiable_on.add
(hf : differentiable_on 𝕜 f s) (hg : differentiable_on 𝕜 g s) :
differentiable_on 𝕜 (λy, f y + g y) s | λx hx, (hf x hx).add (hg x hx) | lemma | differentiable_on.add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.add
(hf : differentiable 𝕜 f) (hg : differentiable 𝕜 g) :
differentiable 𝕜 (λy, f y + g y) | λx, (hf x).add (hg x) | lemma | differentiable.add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_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) :
fderiv_within 𝕜 (λy, f y + g y) s x = fderiv_within 𝕜 f s x + fderiv_within 𝕜 g s x | (hf.has_fderiv_within_at.add hg.has_fderiv_within_at).fderiv_within hxs | lemma | fderiv_within_add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_add
(hf : differentiable_at 𝕜 f x) (hg : differentiable_at 𝕜 g x) :
fderiv 𝕜 (λy, f y + g y) x = fderiv 𝕜 f x + fderiv 𝕜 g x | (hf.has_fderiv_at.add hg.has_fderiv_at).fderiv | lemma | fderiv_add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.add_const (hf : has_strict_fderiv_at f f' x) (c : F) :
has_strict_fderiv_at (λ y, f y + c) f' x | add_zero f' ▸ hf.add (has_strict_fderiv_at_const _ _) | theorem | has_strict_fderiv_at.add_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_strict_fderiv_at",
"has_strict_fderiv_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.add_const
(hf : has_fderiv_at_filter f f' x L) (c : F) :
has_fderiv_at_filter (λ y, f y + c) f' x L | add_zero f' ▸ hf.add (has_fderiv_at_filter_const _ _ _) | theorem | has_fderiv_at_filter.add_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at_filter",
"has_fderiv_at_filter_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.add_const
(hf : has_fderiv_within_at f f' s x) (c : F) :
has_fderiv_within_at (λ y, f y + c) f' s x | hf.add_const c | theorem | has_fderiv_within_at.add_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.add_const (hf : has_fderiv_at f f' x) (c : F):
has_fderiv_at (λ x, f x + c) f' x | hf.add_const c | theorem | has_fderiv_at.add_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.add_const
(hf : differentiable_within_at 𝕜 f s x) (c : F) :
differentiable_within_at 𝕜 (λ y, f y + c) s x | (hf.has_fderiv_within_at.add_const c).differentiable_within_at | lemma | differentiable_within_at.add_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_add_const_iff (c : F) :
differentiable_within_at 𝕜 (λ y, f y + c) s x ↔ differentiable_within_at 𝕜 f s x | ⟨λ h, by simpa using h.add_const (-c), λ h, h.add_const c⟩ | lemma | differentiable_within_at_add_const_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.add_const
(hf : differentiable_at 𝕜 f x) (c : F) :
differentiable_at 𝕜 (λ y, f y + c) x | (hf.has_fderiv_at.add_const c).differentiable_at | lemma | differentiable_at.add_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_add_const_iff (c : F) :
differentiable_at 𝕜 (λ y, f y + c) x ↔ differentiable_at 𝕜 f x | ⟨λ h, by simpa using h.add_const (-c), λ h, h.add_const c⟩ | lemma | differentiable_at_add_const_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.add_const
(hf : differentiable_on 𝕜 f s) (c : F) :
differentiable_on 𝕜 (λy, f y + c) s | λx hx, (hf x hx).add_const c | lemma | differentiable_on.add_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_add_const_iff (c : F) :
differentiable_on 𝕜 (λ y, f y + c) s ↔ differentiable_on 𝕜 f s | ⟨λ h, by simpa using h.add_const (-c), λ h, h.add_const c⟩ | lemma | differentiable_on_add_const_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.add_const
(hf : differentiable 𝕜 f) (c : F) :
differentiable 𝕜 (λy, f y + c) | λx, (hf x).add_const c | lemma | differentiable.add_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_add_const_iff (c : F) :
differentiable 𝕜 (λ y, f y + c) ↔ differentiable 𝕜 f | ⟨λ h, by simpa using h.add_const (-c), λ h, h.add_const c⟩ | lemma | differentiable_add_const_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_add_const (hxs : unique_diff_within_at 𝕜 s x) (c : F) :
fderiv_within 𝕜 (λy, f y + c) s x = fderiv_within 𝕜 f s x | if hf : differentiable_within_at 𝕜 f s x
then (hf.has_fderiv_within_at.add_const c).fderiv_within hxs
else by { rw [fderiv_within_zero_of_not_differentiable_within_at hf,
fderiv_within_zero_of_not_differentiable_within_at], simpa } | lemma | fderiv_within_add_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at",
"fderiv_within",
"fderiv_within_zero_of_not_differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_add_const (c : F) : fderiv 𝕜 (λy, f y + c) x = fderiv 𝕜 f x | by simp only [← fderiv_within_univ, fderiv_within_add_const unique_diff_within_at_univ] | lemma | fderiv_add_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv",
"fderiv_within_add_const",
"fderiv_within_univ",
"unique_diff_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.const_add (hf : has_strict_fderiv_at f f' x) (c : F) :
has_strict_fderiv_at (λ y, c + f y) f' x | zero_add f' ▸ (has_strict_fderiv_at_const _ _).add hf | theorem | has_strict_fderiv_at.const_add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_strict_fderiv_at",
"has_strict_fderiv_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.const_add
(hf : has_fderiv_at_filter f f' x L) (c : F) :
has_fderiv_at_filter (λ y, c + f y) f' x L | zero_add f' ▸ (has_fderiv_at_filter_const _ _ _).add hf | theorem | has_fderiv_at_filter.const_add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at_filter",
"has_fderiv_at_filter_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.const_add
(hf : has_fderiv_within_at f f' s x) (c : F) :
has_fderiv_within_at (λ y, c + f y) f' s x | hf.const_add c | theorem | has_fderiv_within_at.const_add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.const_add
(hf : has_fderiv_at f f' x) (c : F):
has_fderiv_at (λ x, c + f x) f' x | hf.const_add c | theorem | has_fderiv_at.const_add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.const_add
(hf : differentiable_within_at 𝕜 f s x) (c : F) :
differentiable_within_at 𝕜 (λ y, c + f y) s x | (hf.has_fderiv_within_at.const_add c).differentiable_within_at | lemma | differentiable_within_at.const_add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_const_add_iff (c : F) :
differentiable_within_at 𝕜 (λ y, c + f y) s x ↔ differentiable_within_at 𝕜 f s x | ⟨λ h, by simpa using h.const_add (-c), λ h, h.const_add c⟩ | lemma | differentiable_within_at_const_add_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.const_add
(hf : differentiable_at 𝕜 f x) (c : F) :
differentiable_at 𝕜 (λ y, c + f y) x | (hf.has_fderiv_at.const_add c).differentiable_at | lemma | differentiable_at.const_add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_const_add_iff (c : F) :
differentiable_at 𝕜 (λ y, c + f y) x ↔ differentiable_at 𝕜 f x | ⟨λ h, by simpa using h.const_add (-c), λ h, h.const_add c⟩ | lemma | differentiable_at_const_add_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.const_add (hf : differentiable_on 𝕜 f s) (c : F) :
differentiable_on 𝕜 (λy, c + f y) s | λx hx, (hf x hx).const_add c | lemma | differentiable_on.const_add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_const_add_iff (c : F) :
differentiable_on 𝕜 (λ y, c + f y) s ↔ differentiable_on 𝕜 f s | ⟨λ h, by simpa using h.const_add (-c), λ h, h.const_add c⟩ | lemma | differentiable_on_const_add_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.const_add (hf : differentiable 𝕜 f) (c : F) :
differentiable 𝕜 (λy, c + f y) | λx, (hf x).const_add c | lemma | differentiable.const_add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_const_add_iff (c : F) :
differentiable 𝕜 (λ y, c + f y) ↔ differentiable 𝕜 f | ⟨λ h, by simpa using h.const_add (-c), λ h, h.const_add c⟩ | lemma | differentiable_const_add_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_const_add (hxs : unique_diff_within_at 𝕜 s x) (c : F) :
fderiv_within 𝕜 (λy, c + f y) s x = fderiv_within 𝕜 f s x | by simpa only [add_comm] using fderiv_within_add_const hxs c | lemma | fderiv_within_const_add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv_within",
"fderiv_within_add_const",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_const_add (c : F) : fderiv 𝕜 (λy, c + f y) x = fderiv 𝕜 f x | by simp only [add_comm c, fderiv_add_const] | lemma | fderiv_const_add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv",
"fderiv_add_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.sum (h : ∀ i ∈ u, has_strict_fderiv_at (A i) (A' i) x) :
has_strict_fderiv_at (λ y, ∑ i in u, A i y) (∑ i in u, A' i) x | begin
dsimp [has_strict_fderiv_at] at *,
convert is_o.sum h,
simp [finset.sum_sub_distrib, continuous_linear_map.sum_apply]
end | theorem | has_strict_fderiv_at.sum | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"continuous_linear_map.sum_apply",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.sum (h : ∀ i ∈ u, has_fderiv_at_filter (A i) (A' i) x L) :
has_fderiv_at_filter (λ y, ∑ i in u, A i y) (∑ i in u, A' i) x L | begin
dsimp [has_fderiv_at_filter] at *,
convert is_o.sum h,
simp [continuous_linear_map.sum_apply]
end | theorem | has_fderiv_at_filter.sum | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"continuous_linear_map.sum_apply",
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.sum (h : ∀ i ∈ u, has_fderiv_within_at (A i) (A' i) s x) :
has_fderiv_within_at (λ y, ∑ i in u, A i y) (∑ i in u, A' i) s x | has_fderiv_at_filter.sum h | theorem | has_fderiv_within_at.sum | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at_filter.sum",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.sum (h : ∀ i ∈ u, has_fderiv_at (A i) (A' i) x) :
has_fderiv_at (λ y, ∑ i in u, A i y) (∑ i in u, A' i) x | has_fderiv_at_filter.sum h | theorem | has_fderiv_at.sum | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at",
"has_fderiv_at_filter.sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.sum (h : ∀ i ∈ u, differentiable_within_at 𝕜 (A i) s x) :
differentiable_within_at 𝕜 (λ y, ∑ i in u, A i y) s x | has_fderiv_within_at.differentiable_within_at $ has_fderiv_within_at.sum $
λ i hi, (h i hi).has_fderiv_within_at | theorem | differentiable_within_at.sum | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at",
"has_fderiv_within_at",
"has_fderiv_within_at.differentiable_within_at",
"has_fderiv_within_at.sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.sum (h : ∀ i ∈ u, differentiable_at 𝕜 (A i) x) :
differentiable_at 𝕜 (λ y, ∑ i in u, A i y) x | has_fderiv_at.differentiable_at $ has_fderiv_at.sum $ λ i hi, (h i hi).has_fderiv_at | theorem | differentiable_at.sum | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at",
"has_fderiv_at",
"has_fderiv_at.differentiable_at",
"has_fderiv_at.sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.sum (h : ∀ i ∈ u, differentiable_on 𝕜 (A i) s) :
differentiable_on 𝕜 (λ y, ∑ i in u, A i y) s | λ x hx, differentiable_within_at.sum $ λ i hi, h i hi x hx | theorem | differentiable_on.sum | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on",
"differentiable_within_at.sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.sum (h : ∀ i ∈ u, differentiable 𝕜 (A i)) :
differentiable 𝕜 (λ y, ∑ i in u, A i y) | λ x, differentiable_at.sum $ λ i hi, h i hi x | theorem | differentiable.sum | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable",
"differentiable_at.sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_sum (hxs : unique_diff_within_at 𝕜 s x)
(h : ∀ i ∈ u, differentiable_within_at 𝕜 (A i) s x) :
fderiv_within 𝕜 (λ y, ∑ i in u, A i y) s x = (∑ i in u, fderiv_within 𝕜 (A i) s x) | (has_fderiv_within_at.sum (λ i hi, (h i hi).has_fderiv_within_at)).fderiv_within hxs | theorem | fderiv_within_sum | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at",
"fderiv_within",
"has_fderiv_within_at",
"has_fderiv_within_at.sum",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_sum (h : ∀ i ∈ u, differentiable_at 𝕜 (A i) x) :
fderiv 𝕜 (λ y, ∑ i in u, A i y) x = (∑ i in u, fderiv 𝕜 (A i) x) | (has_fderiv_at.sum (λ i hi, (h i hi).has_fderiv_at)).fderiv | theorem | fderiv_sum | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at",
"fderiv",
"has_fderiv_at",
"has_fderiv_at.sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.neg (h : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ x, -f x) (-f') x | (-1 : F →L[𝕜] F).has_strict_fderiv_at.comp x h | theorem | has_strict_fderiv_at.neg | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_strict_fderiv_at",
"has_strict_fderiv_at.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.neg (h : has_fderiv_at_filter f f' x L) :
has_fderiv_at_filter (λ x, -f x) (-f') x L | (-1 : F →L[𝕜] F).has_fderiv_at_filter.comp x h tendsto_map | theorem | has_fderiv_at_filter.neg | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at_filter",
"has_fderiv_at_filter.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.neg (h : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ x, -f x) (-f') s x | h.neg | theorem | has_fderiv_within_at.neg | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.neg (h : has_fderiv_at f f' x) :
has_fderiv_at (λ x, -f x) (-f') x | h.neg | theorem | has_fderiv_at.neg | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.neg (h : differentiable_within_at 𝕜 f s x) :
differentiable_within_at 𝕜 (λy, -f y) s x | h.has_fderiv_within_at.neg.differentiable_within_at | lemma | differentiable_within_at.neg | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_neg_iff :
differentiable_within_at 𝕜 (λy, -f y) s x ↔ differentiable_within_at 𝕜 f s x | ⟨λ h, by simpa only [neg_neg] using h.neg, λ h, h.neg⟩ | lemma | differentiable_within_at_neg_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.neg (h : differentiable_at 𝕜 f x) :
differentiable_at 𝕜 (λy, -f y) x | h.has_fderiv_at.neg.differentiable_at | lemma | differentiable_at.neg | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_neg_iff :
differentiable_at 𝕜 (λy, -f y) x ↔ differentiable_at 𝕜 f x | ⟨λ h, by simpa only [neg_neg] using h.neg, λ h, h.neg⟩ | lemma | differentiable_at_neg_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.neg (h : differentiable_on 𝕜 f s) :
differentiable_on 𝕜 (λy, -f y) s | λx hx, (h x hx).neg | lemma | differentiable_on.neg | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_neg_iff :
differentiable_on 𝕜 (λy, -f y) s ↔ differentiable_on 𝕜 f s | ⟨λ h, by simpa only [neg_neg] using h.neg, λ h, h.neg⟩ | lemma | differentiable_on_neg_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.neg (h : differentiable 𝕜 f) :
differentiable 𝕜 (λy, -f y) | λx, (h x).neg | lemma | differentiable.neg | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_neg_iff : differentiable 𝕜 (λy, -f y) ↔ differentiable 𝕜 f | ⟨λ h, by simpa only [neg_neg] using h.neg, λ h, h.neg⟩ | lemma | differentiable_neg_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_neg (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (λy, -f y) s x = - fderiv_within 𝕜 f s x | if h : differentiable_within_at 𝕜 f s x
then h.has_fderiv_within_at.neg.fderiv_within hxs
else by { rw [fderiv_within_zero_of_not_differentiable_within_at h,
fderiv_within_zero_of_not_differentiable_within_at, neg_zero], simpa } | lemma | fderiv_within_neg | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at",
"fderiv_within",
"fderiv_within_zero_of_not_differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_neg : fderiv 𝕜 (λy, -f y) x = - fderiv 𝕜 f x | by simp only [← fderiv_within_univ, fderiv_within_neg unique_diff_within_at_univ] | lemma | fderiv_neg | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv",
"fderiv_within_neg",
"fderiv_within_univ",
"unique_diff_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.sub
(hf : has_strict_fderiv_at f f' x) (hg : has_strict_fderiv_at g g' x) :
has_strict_fderiv_at (λ x, f x - g x) (f' - g') x | by simpa only [sub_eq_add_neg] using hf.add hg.neg | theorem | has_strict_fderiv_at.sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.sub
(hf : has_fderiv_at_filter f f' x L) (hg : has_fderiv_at_filter g g' x L) :
has_fderiv_at_filter (λ x, f x - g x) (f' - g') x L | by simpa only [sub_eq_add_neg] using hf.add hg.neg | theorem | has_fderiv_at_filter.sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.sub
(hf : has_fderiv_within_at f f' s x) (hg : has_fderiv_within_at g g' s x) :
has_fderiv_within_at (λ x, f x - g x) (f' - g') s x | hf.sub hg | theorem | has_fderiv_within_at.sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.sub
(hf : has_fderiv_at f f' x) (hg : has_fderiv_at g g' x) :
has_fderiv_at (λ x, f x - g x) (f' - g') x | hf.sub hg | theorem | has_fderiv_at.sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.sub
(hf : differentiable_within_at 𝕜 f s x) (hg : differentiable_within_at 𝕜 g s x) :
differentiable_within_at 𝕜 (λ y, f y - g y) s x | (hf.has_fderiv_within_at.sub hg.has_fderiv_within_at).differentiable_within_at | lemma | differentiable_within_at.sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.sub
(hf : differentiable_at 𝕜 f x) (hg : differentiable_at 𝕜 g x) :
differentiable_at 𝕜 (λ y, f y - g y) x | (hf.has_fderiv_at.sub hg.has_fderiv_at).differentiable_at | lemma | differentiable_at.sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.sub
(hf : differentiable_on 𝕜 f s) (hg : differentiable_on 𝕜 g s) :
differentiable_on 𝕜 (λy, f y - g y) s | λx hx, (hf x hx).sub (hg x hx) | lemma | differentiable_on.sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.sub
(hf : differentiable 𝕜 f) (hg : differentiable 𝕜 g) :
differentiable 𝕜 (λy, f y - g y) | λx, (hf x).sub (hg x) | lemma | differentiable.sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_sub (hxs : unique_diff_within_at 𝕜 s x)
(hf : differentiable_within_at 𝕜 f s x) (hg : differentiable_within_at 𝕜 g s x) :
fderiv_within 𝕜 (λy, f y - g y) s x = fderiv_within 𝕜 f s x - fderiv_within 𝕜 g s x | (hf.has_fderiv_within_at.sub hg.has_fderiv_within_at).fderiv_within hxs | lemma | fderiv_within_sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_sub
(hf : differentiable_at 𝕜 f x) (hg : differentiable_at 𝕜 g x) :
fderiv 𝕜 (λy, f y - g y) x = fderiv 𝕜 f x - fderiv 𝕜 g x | (hf.has_fderiv_at.sub hg.has_fderiv_at).fderiv | lemma | fderiv_sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.sub_const
(hf : has_strict_fderiv_at f f' x) (c : F) :
has_strict_fderiv_at (λ x, f x - c) f' x | by simpa only [sub_eq_add_neg] using hf.add_const (-c) | theorem | has_strict_fderiv_at.sub_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.sub_const
(hf : has_fderiv_at_filter f f' x L) (c : F) :
has_fderiv_at_filter (λ x, f x - c) f' x L | by simpa only [sub_eq_add_neg] using hf.add_const (-c) | theorem | has_fderiv_at_filter.sub_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.sub_const
(hf : has_fderiv_within_at f f' s x) (c : F) :
has_fderiv_within_at (λ x, f x - c) f' s x | hf.sub_const c | theorem | has_fderiv_within_at.sub_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.sub_const
(hf : has_fderiv_at f f' x) (c : F) :
has_fderiv_at (λ x, f x - c) f' x | hf.sub_const c | theorem | has_fderiv_at.sub_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.sub_const
(hf : differentiable_within_at 𝕜 f s x) (c : F) :
differentiable_within_at 𝕜 (λ y, f y - c) s x | (hf.has_fderiv_within_at.sub_const c).differentiable_within_at | lemma | differentiable_within_at.sub_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_sub_const_iff (c : F) :
differentiable_within_at 𝕜 (λ y, f y - c) s x ↔ differentiable_within_at 𝕜 f s x | by simp only [sub_eq_add_neg, differentiable_within_at_add_const_iff] | lemma | differentiable_within_at_sub_const_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at",
"differentiable_within_at_add_const_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.sub_const (hf : differentiable_at 𝕜 f x) (c : F) :
differentiable_at 𝕜 (λ y, f y - c) x | (hf.has_fderiv_at.sub_const c).differentiable_at | lemma | differentiable_at.sub_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_sub_const_iff (c : F) :
differentiable_at 𝕜 (λ y, f y - c) x ↔ differentiable_at 𝕜 f x | by simp only [sub_eq_add_neg, differentiable_at_add_const_iff] | lemma | differentiable_at_sub_const_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at",
"differentiable_at_add_const_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.sub_const (hf : differentiable_on 𝕜 f s) (c : F) :
differentiable_on 𝕜 (λy, f y - c) s | λx hx, (hf x hx).sub_const c | lemma | differentiable_on.sub_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_sub_const_iff (c : F) :
differentiable_on 𝕜 (λ y, f y - c) s ↔ differentiable_on 𝕜 f s | by simp only [sub_eq_add_neg, differentiable_on_add_const_iff] | lemma | differentiable_on_sub_const_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on",
"differentiable_on_add_const_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.sub_const (hf : differentiable 𝕜 f) (c : F) :
differentiable 𝕜 (λy, f y - c) | λx, (hf x).sub_const c | lemma | differentiable.sub_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_sub_const_iff (c : F) :
differentiable 𝕜 (λ y, f y - c) ↔ differentiable 𝕜 f | by simp only [sub_eq_add_neg, differentiable_add_const_iff] | lemma | differentiable_sub_const_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable",
"differentiable_add_const_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_sub_const (hxs : unique_diff_within_at 𝕜 s x) (c : F) :
fderiv_within 𝕜 (λy, f y - c) s x = fderiv_within 𝕜 f s x | by simp only [sub_eq_add_neg, fderiv_within_add_const hxs] | lemma | fderiv_within_sub_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv_within",
"fderiv_within_add_const",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_sub_const (c : F) : fderiv 𝕜 (λy, f y - c) x = fderiv 𝕜 f x | by simp only [sub_eq_add_neg, fderiv_add_const] | lemma | fderiv_sub_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv",
"fderiv_add_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.const_sub
(hf : has_strict_fderiv_at f f' x) (c : F) :
has_strict_fderiv_at (λ x, c - f x) (-f') x | by simpa only [sub_eq_add_neg] using hf.neg.const_add c | theorem | has_strict_fderiv_at.const_sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.const_sub
(hf : has_fderiv_at_filter f f' x L) (c : F) :
has_fderiv_at_filter (λ x, c - f x) (-f') x L | by simpa only [sub_eq_add_neg] using hf.neg.const_add c | theorem | has_fderiv_at_filter.const_sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.const_sub
(hf : has_fderiv_within_at f f' s x) (c : F) :
has_fderiv_within_at (λ x, c - f x) (-f') s x | hf.const_sub c | theorem | has_fderiv_within_at.const_sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.const_sub
(hf : has_fderiv_at f f' x) (c : F) :
has_fderiv_at (λ x, c - f x) (-f') x | hf.const_sub c | theorem | has_fderiv_at.const_sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.const_sub
(hf : differentiable_within_at 𝕜 f s x) (c : F) :
differentiable_within_at 𝕜 (λ y, c - f y) s x | (hf.has_fderiv_within_at.const_sub c).differentiable_within_at | lemma | differentiable_within_at.const_sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_const_sub_iff (c : F) :
differentiable_within_at 𝕜 (λ y, c - f y) s x ↔ differentiable_within_at 𝕜 f s x | by simp [sub_eq_add_neg] | lemma | differentiable_within_at_const_sub_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.const_sub
(hf : differentiable_at 𝕜 f x) (c : F) :
differentiable_at 𝕜 (λ y, c - f y) x | (hf.has_fderiv_at.const_sub c).differentiable_at | lemma | differentiable_at.const_sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_const_sub_iff (c : F) :
differentiable_at 𝕜 (λ y, c - f y) x ↔ differentiable_at 𝕜 f x | by simp [sub_eq_add_neg] | lemma | differentiable_at_const_sub_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.const_sub (hf : differentiable_on 𝕜 f s) (c : F) :
differentiable_on 𝕜 (λy, c - f y) s | λx hx, (hf x hx).const_sub c | lemma | differentiable_on.const_sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_const_sub_iff (c : F) :
differentiable_on 𝕜 (λ y, c - f y) s ↔ differentiable_on 𝕜 f s | by simp [sub_eq_add_neg] | lemma | differentiable_on_const_sub_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.const_sub (hf : differentiable 𝕜 f) (c : F) :
differentiable 𝕜 (λy, c - f y) | λx, (hf x).const_sub c | lemma | differentiable.const_sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_const_sub_iff (c : F) :
differentiable 𝕜 (λ y, c - f y) ↔ differentiable 𝕜 f | by simp [sub_eq_add_neg] | lemma | differentiable_const_sub_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_const_sub (hxs : unique_diff_within_at 𝕜 s x) (c : F) :
fderiv_within 𝕜 (λy, c - f y) s x = -fderiv_within 𝕜 f s x | by simp only [sub_eq_add_neg, fderiv_within_const_add, fderiv_within_neg, hxs] | lemma | fderiv_within_const_sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv_within",
"fderiv_within_const_add",
"fderiv_within_neg",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_const_sub (c : F) : fderiv 𝕜 (λy, c - f y) x = -fderiv 𝕜 f x | by simp only [← fderiv_within_univ, fderiv_within_const_sub unique_diff_within_at_univ] | lemma | fderiv_const_sub | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"fderiv",
"fderiv_within_const_sub",
"fderiv_within_univ",
"unique_diff_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter (f : E → F) (f' : E →L[𝕜] F) (x : E) (L : filter E) | (λ x', f x' - f x - f' (x' - x)) =o[L] (λ x', x' - x) | def | has_fderiv_at_filter | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"filter"
] | A function `f` has the continuous linear map `f'` as derivative along the filter `L` if
`f x' = f x + f' (x' - x) + o (x' - x)` when `x'` converges along the filter `L`. This definition
is designed to be specialized for `L = 𝓝 x` (in `has_fderiv_at`), giving rise to the usual notion
of Fréchet derivative, and for `L =... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_within_at (f : E → F) (f' : E →L[𝕜] F) (s : set E) (x : E) | has_fderiv_at_filter f f' x (𝓝[s] x) | def | has_fderiv_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at_filter"
] | A function `f` has the continuous linear map `f'` as derivative at `x` within a set `s` if
`f x' = f x + f' (x' - x) + o (x' - x)` when `x'` tends to `x` inside `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at (f : E → F) (f' : E →L[𝕜] F) (x : E) | has_fderiv_at_filter f f' x (𝓝 x) | def | has_fderiv_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at_filter"
] | A function `f` has the continuous linear map `f'` as derivative at `x` if
`f x' = f x + f' (x' - x) + o (x' - x)` when `x'` tends to `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_fderiv_at (f : E → F) (f' : E →L[𝕜] F) (x : E) | (λ p : E × E, f p.1 - f p.2 - f' (p.1 - p.2)) =o[𝓝 (x, x)] (λ p : E × E, p.1 - p.2) | def | has_strict_fderiv_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [] | A function `f` has derivative `f'` at `a` in the sense of *strict differentiability*
if `f x - f y - f' (x - y) = o(x - y)` as `x, y → a`. This form of differentiability is required,
e.g., by the inverse function theorem. Any `C^1` function on a vector space over `ℝ` is strictly
differentiable but this definition works... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_within_at (f : E → F) (s : set E) (x : E) | ∃f' : E →L[𝕜] F, has_fderiv_within_at f f' s x | def | differentiable_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at"
] | A function `f` is differentiable at a point `x` within a set `s` if it admits a derivative
there (possibly non-unique). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_at (f : E → F) (x : E) | ∃f' : E →L[𝕜] F, has_fderiv_at f f' x | def | differentiable_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at"
] | A function `f` is differentiable at a point `x` if it admits a derivative there (possibly
non-unique). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.