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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deriv_add_const' (c : F) : deriv (λ y, f y + c) = deriv f | funext $ λ x, deriv_add_const c | lemma | deriv_add_const' | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"deriv_add_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter.const_add (c : F) (hf : has_deriv_at_filter f f' x L) :
has_deriv_at_filter (λ y, c + f y) f' x L | zero_add f' ▸ (has_deriv_at_filter_const x L c).add hf | theorem | has_deriv_at_filter.const_add | 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.const_add (c : F) (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ y, c + f y) f' s x | hf.const_add c | theorem | has_deriv_within_at.const_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.const_add (c : F) (hf : has_deriv_at f f' x) :
has_deriv_at (λ x, c + f x) f' x | hf.const_add c | theorem | has_deriv_at.const_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_const_add (hxs : unique_diff_within_at 𝕜 s x) (c : F) :
deriv_within (λy, c + f y) s x = deriv_within f s x | by simp only [deriv_within, fderiv_within_const_add hxs] | lemma | deriv_within_const_add | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv_within",
"fderiv_within_const_add",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_const_add (c : F) : deriv (λy, c + f y) x = deriv f x | by simp only [deriv, fderiv_const_add] | lemma | deriv_const_add | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"fderiv_const_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_const_add' (c : F) : deriv (λ y, c + f y) = deriv f | funext $ λ x, deriv_const_add c | lemma | deriv_const_add' | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"deriv_const_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter.sum (h : ∀ i ∈ u, has_deriv_at_filter (A i) (A' i) x L) :
has_deriv_at_filter (λ y, ∑ i in u, A i y) (∑ i in u, A' i) x L | by simpa [continuous_linear_map.sum_apply] using (has_fderiv_at_filter.sum h).has_deriv_at_filter | theorem | has_deriv_at_filter.sum | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"continuous_linear_map.sum_apply",
"has_deriv_at_filter",
"has_fderiv_at_filter.sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.sum (h : ∀ i ∈ u, has_strict_deriv_at (A i) (A' i) x) :
has_strict_deriv_at (λ y, ∑ i in u, A i y) (∑ i in u, A' i) x | by simpa [continuous_linear_map.sum_apply] using (has_strict_fderiv_at.sum h).has_strict_deriv_at | theorem | has_strict_deriv_at.sum | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"continuous_linear_map.sum_apply",
"has_strict_deriv_at",
"has_strict_fderiv_at.sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.sum (h : ∀ i ∈ u, has_deriv_within_at (A i) (A' i) s x) :
has_deriv_within_at (λ y, ∑ i in u, A i y) (∑ i in u, A' i) s x | has_deriv_at_filter.sum h | theorem | has_deriv_within_at.sum | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at_filter.sum",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.sum (h : ∀ i ∈ u, has_deriv_at (A i) (A' i) x) :
has_deriv_at (λ y, ∑ i in u, A i y) (∑ i in u, A' i) x | has_deriv_at_filter.sum h | theorem | has_deriv_at.sum | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at",
"has_deriv_at_filter.sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_sum (hxs : unique_diff_within_at 𝕜 s x)
(h : ∀ i ∈ u, differentiable_within_at 𝕜 (A i) s x) :
deriv_within (λ y, ∑ i in u, A i y) s x = ∑ i in u, deriv_within (A i) s x | (has_deriv_within_at.sum (λ i hi, (h i hi).has_deriv_within_at)).deriv_within hxs | lemma | deriv_within_sum | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv_within",
"differentiable_within_at",
"has_deriv_within_at",
"has_deriv_within_at.sum",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_sum (h : ∀ i ∈ u, differentiable_at 𝕜 (A i) x) :
deriv (λ y, ∑ i in u, A i y) x = ∑ i in u, deriv (A i) x | (has_deriv_at.sum (λ i hi, (h i hi).has_deriv_at)).deriv | lemma | deriv_sum | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"differentiable_at",
"has_deriv_at",
"has_deriv_at.sum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter.neg (h : has_deriv_at_filter f f' x L) :
has_deriv_at_filter (λ x, -f x) (-f') x L | by simpa using h.neg.has_deriv_at_filter | theorem | has_deriv_at_filter.neg | 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_deriv_within_at.neg (h : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ x, -f x) (-f') s x | h.neg | theorem | has_deriv_within_at.neg | 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.neg (h : has_deriv_at f f' x) : has_deriv_at (λ x, -f x) (-f') x | h.neg | theorem | has_deriv_at.neg | 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 | |
has_strict_deriv_at.neg (h : has_strict_deriv_at f f' x) :
has_strict_deriv_at (λ x, -f x) (-f') x | by simpa using h.neg.has_strict_deriv_at | theorem | has_strict_deriv_at.neg | 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 | |
deriv_within.neg (hxs : unique_diff_within_at 𝕜 s x) :
deriv_within (λy, -f y) s x = - deriv_within f s x | by simp only [deriv_within, fderiv_within_neg hxs, continuous_linear_map.neg_apply] | lemma | deriv_within.neg | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"continuous_linear_map.neg_apply",
"deriv_within",
"fderiv_within_neg",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv.neg : deriv (λy, -f y) x = - deriv f x | by simp only [deriv, fderiv_neg, continuous_linear_map.neg_apply] | lemma | deriv.neg | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"continuous_linear_map.neg_apply",
"deriv",
"fderiv_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv.neg' : deriv (λy, -f y) = (λ x, - deriv f x) | funext $ λ x, deriv.neg | lemma | deriv.neg' | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"deriv.neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter_neg : has_deriv_at_filter has_neg.neg (-1) x L | has_deriv_at_filter.neg $ has_deriv_at_filter_id _ _ | theorem | has_deriv_at_filter_neg | 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.neg",
"has_deriv_at_filter_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_neg : has_deriv_within_at has_neg.neg (-1) s x | has_deriv_at_filter_neg _ _ | theorem | has_deriv_within_at_neg | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at_filter_neg",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_neg : has_deriv_at has_neg.neg (-1) x | has_deriv_at_filter_neg _ _ | theorem | has_deriv_at_neg | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at",
"has_deriv_at_filter_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_neg' : has_deriv_at (λ x, -x) (-1) x | has_deriv_at_filter_neg _ _ | theorem | has_deriv_at_neg' | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"has_deriv_at",
"has_deriv_at_filter_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_neg : has_strict_deriv_at has_neg.neg (-1) x | has_strict_deriv_at.neg $ has_strict_deriv_at_id _ | theorem | has_strict_deriv_at_neg | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"has_strict_deriv_at",
"has_strict_deriv_at.neg",
"has_strict_deriv_at_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_neg : deriv has_neg.neg x = -1 | has_deriv_at.deriv (has_deriv_at_neg x) | lemma | deriv_neg | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"has_deriv_at.deriv",
"has_deriv_at_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_neg' : deriv (has_neg.neg : 𝕜 → 𝕜) = λ _, -1 | funext deriv_neg | lemma | deriv_neg' | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"deriv_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_neg'' : deriv (λ x : 𝕜, -x) x = -1 | deriv_neg x | lemma | deriv_neg'' | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"deriv_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_neg (hxs : unique_diff_within_at 𝕜 s x) : deriv_within has_neg.neg s x = -1 | (has_deriv_within_at_neg x s).deriv_within hxs | lemma | deriv_within_neg | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv_within",
"has_deriv_within_at_neg",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_neg : differentiable 𝕜 (has_neg.neg : 𝕜 → 𝕜) | differentiable.neg differentiable_id | lemma | differentiable_neg | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"differentiable",
"differentiable.neg",
"differentiable_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_neg : differentiable_on 𝕜 (has_neg.neg : 𝕜 → 𝕜) s | differentiable_on.neg differentiable_on_id | lemma | differentiable_on_neg | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"differentiable_on",
"differentiable_on.neg",
"differentiable_on_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter.sub
(hf : has_deriv_at_filter f f' x L) (hg : has_deriv_at_filter g g' x L) :
has_deriv_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_deriv_at_filter.sub | 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_deriv_within_at.sub
(hf : has_deriv_within_at f f' s x) (hg : has_deriv_within_at g g' s x) :
has_deriv_within_at (λ x, f x - g x) (f' - g') s x | hf.sub hg | theorem | has_deriv_within_at.sub | 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.sub
(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.sub hg | theorem | has_deriv_at.sub | 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 | |
has_strict_deriv_at.sub
(hf : has_strict_deriv_at f f' x) (hg : has_strict_deriv_at g g' x) :
has_strict_deriv_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_deriv_at.sub | 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 | |
deriv_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) :
deriv_within (λy, f y - g y) s x = deriv_within f s x - deriv_within g s x | (hf.has_deriv_within_at.sub hg.has_deriv_within_at).deriv_within hxs | lemma | deriv_within_sub | 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_sub
(hf : differentiable_at 𝕜 f x) (hg : differentiable_at 𝕜 g x) :
deriv (λ y, f y - g y) x = deriv f x - deriv g x | (hf.has_deriv_at.sub hg.has_deriv_at).deriv | lemma | deriv_sub | 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.sub_const
(hf : has_deriv_at_filter f f' x L) (c : F) :
has_deriv_at_filter (λ x, f x - c) f' x L | by simpa only [sub_eq_add_neg] using hf.add_const (-c) | theorem | has_deriv_at_filter.sub_const | 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_deriv_within_at.sub_const
(hf : has_deriv_within_at f f' s x) (c : F) :
has_deriv_within_at (λ x, f x - c) f' s x | hf.sub_const c | theorem | has_deriv_within_at.sub_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.sub_const
(hf : has_deriv_at f f' x) (c : F) :
has_deriv_at (λ x, f x - c) f' x | hf.sub_const c | theorem | has_deriv_at.sub_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_sub_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_sub_const hxs] | lemma | deriv_within_sub_const | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv_within",
"fderiv_within_sub_const",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_sub_const (c : F) : deriv (λ y, f y - c) x = deriv f x | by simp only [deriv, fderiv_sub_const] | lemma | deriv_sub_const | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"fderiv_sub_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter.const_sub (c : F) (hf : has_deriv_at_filter f f' x L) :
has_deriv_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_deriv_at_filter.const_sub | 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_deriv_within_at.const_sub (c : F) (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ x, c - f x) (-f') s x | hf.const_sub c | theorem | has_deriv_within_at.const_sub | 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_strict_deriv_at.const_sub (c : F) (hf : has_strict_deriv_at f f' x) :
has_strict_deriv_at (λ x, c - f x) (-f') x | by simpa only [sub_eq_add_neg] using hf.neg.const_add c | theorem | has_strict_deriv_at.const_sub | 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_at.const_sub (c : F) (hf : has_deriv_at f f' x) :
has_deriv_at (λ x, c - f x) (-f') x | hf.const_sub c | theorem | has_deriv_at.const_sub | 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_const_sub (hxs : unique_diff_within_at 𝕜 s x) (c : F) :
deriv_within (λy, c - f y) s x = -deriv_within f s x | by simp [deriv_within, fderiv_within_const_sub hxs] | lemma | deriv_within_const_sub | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv_within",
"fderiv_within_const_sub",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_const_sub (c : F) : deriv (λ y, c - f y) x = -deriv f x | by simp only [← deriv_within_univ,
deriv_within_const_sub (unique_diff_within_at_univ : unique_diff_within_at 𝕜 _ _)] | lemma | deriv_const_sub | analysis.calculus.deriv | src/analysis/calculus/deriv/add.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.add"
] | [
"deriv",
"deriv_within_const_sub",
"deriv_within_univ",
"unique_diff_within_at",
"unique_diff_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter (f : 𝕜 → F) (f' : F) (x : 𝕜) (L : filter 𝕜) | has_fderiv_at_filter f (smul_right (1 : 𝕜 →L[𝕜] 𝕜) f') x L | def | has_deriv_at_filter | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"filter",
"has_fderiv_at_filter"
] | `f` has the derivative `f'` at the point `x` as `x` goes along the filter `L`.
That is, `f x' = f x + (x' - x) • f' + o(x' - x)` where `x'` converges along the filter `L`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_within_at (f : 𝕜 → F) (f' : F) (s : set 𝕜) (x : 𝕜) | has_deriv_at_filter f f' x (𝓝[s] x) | def | has_deriv_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at_filter"
] | `f` has the derivative `f'` at the point `x` within the subset `s`.
That is, `f x' = f x + (x' - x) • f' + o(x' - x)` where `x'` converges to `x` inside `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at (f : 𝕜 → F) (f' : F) (x : 𝕜) | has_deriv_at_filter f f' x (𝓝 x) | def | has_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at_filter"
] | `f` has the derivative `f'` at the point `x`.
That is, `f x' = f x + (x' - x) • f' + o(x' - x)` where `x'` converges to `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_deriv_at (f : 𝕜 → F) (f' : F) (x : 𝕜) | has_strict_fderiv_at f (smul_right (1 : 𝕜 →L[𝕜] 𝕜) f') x | def | has_strict_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_strict_fderiv_at"
] | `f` has the derivative `f'` at the point `x` in the sense of strict differentiability.
That is, `f y - f z = (y - z) • f' + o(y - z)` as `y, z → x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
deriv_within (f : 𝕜 → F) (s : set 𝕜) (x : 𝕜) | fderiv_within 𝕜 f s x 1 | def | deriv_within | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"fderiv_within"
] | Derivative of `f` at the point `x` within the set `s`, if it exists. Zero otherwise.
If the derivative exists (i.e., `∃ f', has_deriv_within_at f f' s x`), then
`f x' = f x + (x' - x) • deriv_within f s x + o(x' - x)` where `x'` converges to `x` inside `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
deriv (f : 𝕜 → F) (x : 𝕜) | fderiv 𝕜 f x 1 | def | deriv | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"fderiv"
] | Derivative of `f` at the point `x`, if it exists. Zero otherwise.
If the derivative exists (i.e., `∃ f', has_deriv_at f f' x`), then
`f x' = f x + (x' - x) • deriv f x + o(x' - x)` where `x'` converges to `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_filter_iff_has_deriv_at_filter {f' : 𝕜 →L[𝕜] F} :
has_fderiv_at_filter f f' x L ↔ has_deriv_at_filter f (f' 1) x L | by simp [has_deriv_at_filter] | lemma | has_fderiv_at_filter_iff_has_deriv_at_filter | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at_filter",
"has_fderiv_at_filter"
] | Expressing `has_fderiv_at_filter f f' x L` in terms of `has_deriv_at_filter` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_filter.has_deriv_at_filter {f' : 𝕜 →L[𝕜] F} :
has_fderiv_at_filter f f' x L → has_deriv_at_filter f (f' 1) x L | has_fderiv_at_filter_iff_has_deriv_at_filter.mp | lemma | has_fderiv_at_filter.has_deriv_at_filter | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at_filter",
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_iff_has_deriv_within_at {f' : 𝕜 →L[𝕜] F} :
has_fderiv_within_at f f' s x ↔ has_deriv_within_at f (f' 1) s x | has_fderiv_at_filter_iff_has_deriv_at_filter | lemma | has_fderiv_within_at_iff_has_deriv_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at",
"has_fderiv_at_filter_iff_has_deriv_at_filter",
"has_fderiv_within_at"
] | Expressing `has_fderiv_within_at f f' s x` in terms of `has_deriv_within_at` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_within_at_iff_has_fderiv_within_at {f' : F} :
has_deriv_within_at f f' s x ↔
has_fderiv_within_at f (smul_right (1 : 𝕜 →L[𝕜] 𝕜) f') s x | iff.rfl | lemma | has_deriv_within_at_iff_has_fderiv_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at",
"has_fderiv_within_at"
] | Expressing `has_deriv_within_at f f' s x` in terms of `has_fderiv_within_at` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_within_at.has_deriv_within_at {f' : 𝕜 →L[𝕜] F} :
has_fderiv_within_at f f' s x → has_deriv_within_at f (f' 1) s x | has_fderiv_within_at_iff_has_deriv_within_at.mp | lemma | has_fderiv_within_at.has_deriv_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.has_fderiv_within_at {f' : F} :
has_deriv_within_at f f' s x → has_fderiv_within_at f (smul_right (1 : 𝕜 →L[𝕜] 𝕜) f') s x | has_deriv_within_at_iff_has_fderiv_within_at.mp | lemma | has_deriv_within_at.has_fderiv_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_iff_has_deriv_at {f' : 𝕜 →L[𝕜] F} :
has_fderiv_at f f' x ↔ has_deriv_at f (f' 1) x | has_fderiv_at_filter_iff_has_deriv_at_filter | lemma | has_fderiv_at_iff_has_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at",
"has_fderiv_at",
"has_fderiv_at_filter_iff_has_deriv_at_filter"
] | Expressing `has_fderiv_at f f' x` in terms of `has_deriv_at` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at.has_deriv_at {f' : 𝕜 →L[𝕜] F} :
has_fderiv_at f f' x → has_deriv_at f (f' 1) x | has_fderiv_at_iff_has_deriv_at.mp | lemma | has_fderiv_at.has_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at_iff_has_strict_deriv_at {f' : 𝕜 →L[𝕜] F} :
has_strict_fderiv_at f f' x ↔ has_strict_deriv_at f (f' 1) x | by simp [has_strict_deriv_at, has_strict_fderiv_at] | lemma | has_strict_fderiv_at_iff_has_strict_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_strict_deriv_at",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.has_strict_deriv_at {f' : 𝕜 →L[𝕜] F} :
has_strict_fderiv_at f f' x → has_strict_deriv_at f (f' 1) x | has_strict_fderiv_at_iff_has_strict_deriv_at.mp | lemma | has_strict_fderiv_at.has_strict_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_strict_deriv_at",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_iff_has_strict_fderiv_at :
has_strict_deriv_at f f' x ↔ has_strict_fderiv_at f (smul_right (1 : 𝕜 →L[𝕜] 𝕜) f') x | iff.rfl | lemma | has_strict_deriv_at_iff_has_strict_fderiv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_strict_deriv_at",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_iff_has_fderiv_at {f' : F} :
has_deriv_at f f' x ↔
has_fderiv_at f (smul_right (1 : 𝕜 →L[𝕜] 𝕜) f') x | iff.rfl | lemma | has_deriv_at_iff_has_fderiv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at",
"has_fderiv_at"
] | Expressing `has_deriv_at f f' x` in terms of `has_fderiv_at` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
deriv_within_zero_of_not_differentiable_within_at
(h : ¬ differentiable_within_at 𝕜 f s x) : deriv_within f s x = 0 | by { unfold deriv_within, rw fderiv_within_zero_of_not_differentiable_within_at, simp, assumption } | lemma | deriv_within_zero_of_not_differentiable_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv_within",
"differentiable_within_at",
"fderiv_within_zero_of_not_differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_of_deriv_within_ne_zero (h : deriv_within f s x ≠ 0) :
differentiable_within_at 𝕜 f s x | not_imp_comm.1 deriv_within_zero_of_not_differentiable_within_at h | lemma | differentiable_within_at_of_deriv_within_ne_zero | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv_within",
"deriv_within_zero_of_not_differentiable_within_at",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_zero_of_not_differentiable_at (h : ¬ differentiable_at 𝕜 f x) : deriv f x = 0 | by { unfold deriv, rw fderiv_zero_of_not_differentiable_at, simp, assumption } | lemma | deriv_zero_of_not_differentiable_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"differentiable_at",
"fderiv_zero_of_not_differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_of_deriv_ne_zero (h : deriv f x ≠ 0) : differentiable_at 𝕜 f x | not_imp_comm.1 deriv_zero_of_not_differentiable_at h | lemma | differentiable_at_of_deriv_ne_zero | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"deriv_zero_of_not_differentiable_at",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
unique_diff_within_at.eq_deriv (s : set 𝕜) (H : unique_diff_within_at 𝕜 s x)
(h : has_deriv_within_at f f' s x) (h₁ : has_deriv_within_at f f₁' s x) : f' = f₁' | smul_right_one_eq_iff.mp $ unique_diff_within_at.eq H h h₁ | theorem | unique_diff_within_at.eq_deriv | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at",
"unique_diff_within_at",
"unique_diff_within_at.eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter_iff_is_o :
has_deriv_at_filter f f' x L ↔ (λ x' : 𝕜, f x' - f x - (x' - x) • f') =o[L] (λ x', x' - x) | iff.rfl | theorem | has_deriv_at_filter_iff_is_o | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter_iff_tendsto :
has_deriv_at_filter f f' x L ↔
tendsto (λ x' : 𝕜, ‖x' - x‖⁻¹ * ‖f x' - f x - (x' - x) • f'‖) L (𝓝 0) | has_fderiv_at_filter_iff_tendsto | theorem | has_deriv_at_filter_iff_tendsto | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at_filter",
"has_fderiv_at_filter_iff_tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_iff_is_o :
has_deriv_within_at f f' s x
↔ (λ x' : 𝕜, f x' - f x - (x' - x) • f') =o[𝓝[s] x] (λ x', x' - x) | iff.rfl | theorem | has_deriv_within_at_iff_is_o | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_iff_tendsto : has_deriv_within_at f f' s x ↔
tendsto (λ x', ‖x' - x‖⁻¹ * ‖f x' - f x - (x' - x) • f'‖) (𝓝[s] x) (𝓝 0) | has_fderiv_at_filter_iff_tendsto | theorem | has_deriv_within_at_iff_tendsto | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at",
"has_fderiv_at_filter_iff_tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_iff_is_o :
has_deriv_at f f' x ↔ (λ x' : 𝕜, f x' - f x - (x' - x) • f') =o[𝓝 x] (λ x', x' - x) | iff.rfl | theorem | has_deriv_at_iff_is_o | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_iff_tendsto : has_deriv_at f f' x ↔
tendsto (λ x', ‖x' - x‖⁻¹ * ‖f x' - f x - (x' - x) • f'‖) (𝓝 x) (𝓝 0) | has_fderiv_at_filter_iff_tendsto | theorem | has_deriv_at_iff_tendsto | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at",
"has_fderiv_at_filter_iff_tendsto"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter.is_O_sub (h : has_deriv_at_filter f f' x L) :
(λ x', f x' - f x) =O[L] (λ x', x' - x) | has_fderiv_at_filter.is_O_sub h | theorem | has_deriv_at_filter.is_O_sub | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at_filter",
"has_fderiv_at_filter.is_O_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter.is_O_sub_rev (hf : has_deriv_at_filter f f' x L) (hf' : f' ≠ 0) :
(λ x', x' - x) =O[L] (λ x', f x' - f x) | suffices antilipschitz_with ‖f'‖₊⁻¹ (smul_right (1 : 𝕜 →L[𝕜] 𝕜) f'), from hf.is_O_sub_rev this,
add_monoid_hom_class.antilipschitz_of_bound (smul_right (1 : 𝕜 →L[𝕜] 𝕜) f') $
λ x, by simp [norm_smul, ← div_eq_inv_mul, mul_div_cancel _ (mt norm_eq_zero.1 hf')] | theorem | has_deriv_at_filter.is_O_sub_rev | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"antilipschitz_with",
"div_eq_inv_mul",
"has_deriv_at_filter",
"mul_div_cancel",
"norm_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.has_deriv_at (h : has_strict_deriv_at f f' x) :
has_deriv_at f f' x | h.has_fderiv_at | theorem | has_strict_deriv_at.has_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at",
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_congr_set' {s t : set 𝕜} (y : 𝕜) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
has_deriv_within_at f f' s x ↔ has_deriv_within_at f f' t x | has_fderiv_within_at_congr_set' y h | theorem | has_deriv_within_at_congr_set' | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at",
"has_fderiv_within_at_congr_set'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_congr_set {s t : set 𝕜} (h : s =ᶠ[𝓝 x] t) :
has_deriv_within_at f f' s x ↔ has_deriv_within_at f f' t x | has_fderiv_within_at_congr_set h | theorem | has_deriv_within_at_congr_set | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at",
"has_fderiv_within_at_congr_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_diff_singleton :
has_deriv_within_at f f' (s \ {x}) x ↔ has_deriv_within_at f f' s x | has_fderiv_within_at_diff_singleton _ | lemma | has_deriv_within_at_diff_singleton | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at",
"has_fderiv_within_at_diff_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_Ioi_iff_Ici [partial_order 𝕜] :
has_deriv_within_at f f' (Ioi x) x ↔ has_deriv_within_at f f' (Ici x) x | by rw [← Ici_diff_left, has_deriv_within_at_diff_singleton] | lemma | has_deriv_within_at_Ioi_iff_Ici | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at",
"has_deriv_within_at_diff_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_Iio_iff_Iic [partial_order 𝕜] :
has_deriv_within_at f f' (Iio x) x ↔ has_deriv_within_at f f' (Iic x) x | by rw [← Iic_diff_right, has_deriv_within_at_diff_singleton] | lemma | has_deriv_within_at_Iio_iff_Iic | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at",
"has_deriv_within_at_diff_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.Ioi_iff_Ioo [linear_order 𝕜] [order_closed_topology 𝕜] {x y : 𝕜}
(h : x < y) :
has_deriv_within_at f f' (Ioo x y) x ↔ has_deriv_within_at f f' (Ioi x) x | has_fderiv_within_at_inter $ Iio_mem_nhds h | theorem | has_deriv_within_at.Ioi_iff_Ioo | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"Iio_mem_nhds",
"has_deriv_within_at",
"has_fderiv_within_at_inter",
"order_closed_topology"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_iff_is_o_nhds_zero : has_deriv_at f f' x ↔
(λh, f (x + h) - f x - h • f') =o[𝓝 0] (λh, h) | has_fderiv_at_iff_is_o_nhds_zero | theorem | has_deriv_at_iff_is_o_nhds_zero | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at",
"has_fderiv_at_iff_is_o_nhds_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter.mono (h : has_deriv_at_filter f f' x L₂) (hst : L₁ ≤ L₂) :
has_deriv_at_filter f f' x L₁ | has_fderiv_at_filter.mono h hst | theorem | has_deriv_at_filter.mono | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at_filter",
"has_fderiv_at_filter.mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.mono (h : has_deriv_within_at f f' t x) (hst : s ⊆ t) :
has_deriv_within_at f f' s x | has_fderiv_within_at.mono h hst | theorem | has_deriv_within_at.mono | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at",
"has_fderiv_within_at.mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.mono_of_mem (h : has_deriv_within_at f f' t x) (hst : t ∈ 𝓝[s] x) :
has_deriv_within_at f f' s x | has_fderiv_within_at.mono_of_mem h hst | theorem | has_deriv_within_at.mono_of_mem | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at",
"has_fderiv_within_at.mono_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.has_deriv_at_filter (h : has_deriv_at f f' x) (hL : L ≤ 𝓝 x) :
has_deriv_at_filter f f' x L | has_fderiv_at.has_fderiv_at_filter h hL | theorem | has_deriv_at.has_deriv_at_filter | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at",
"has_deriv_at_filter",
"has_fderiv_at.has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.has_deriv_within_at
(h : has_deriv_at f f' x) : has_deriv_within_at f f' s x | has_fderiv_at.has_fderiv_within_at h | theorem | has_deriv_at.has_deriv_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at",
"has_deriv_within_at",
"has_fderiv_at.has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.differentiable_within_at (h : has_deriv_within_at f f' s x) :
differentiable_within_at 𝕜 f s x | has_fderiv_within_at.differentiable_within_at h | lemma | has_deriv_within_at.differentiable_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_within_at",
"has_deriv_within_at",
"has_fderiv_within_at.differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.differentiable_at (h : has_deriv_at f f' x) : differentiable_at 𝕜 f x | has_fderiv_at.differentiable_at h | lemma | has_deriv_at.differentiable_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_at",
"has_deriv_at",
"has_fderiv_at.differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_univ : has_deriv_within_at f f' univ x ↔ has_deriv_at f f' x | has_fderiv_within_at_univ | lemma | has_deriv_within_at_univ | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at",
"has_deriv_within_at",
"has_fderiv_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.unique
(h₀ : has_deriv_at f f₀' x) (h₁ : has_deriv_at f f₁' x) : f₀' = f₁' | smul_right_one_eq_iff.mp $ h₀.has_fderiv_at.unique h₁ | theorem | has_deriv_at.unique | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_inter' (h : t ∈ 𝓝[s] x) :
has_deriv_within_at f f' (s ∩ t) x ↔ has_deriv_within_at f f' s x | has_fderiv_within_at_inter' h | lemma | has_deriv_within_at_inter' | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at",
"has_fderiv_within_at_inter'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_inter (h : t ∈ 𝓝 x) :
has_deriv_within_at f f' (s ∩ t) x ↔ has_deriv_within_at f f' s x | has_fderiv_within_at_inter h | lemma | has_deriv_within_at_inter | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at",
"has_fderiv_within_at_inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.union (hs : has_deriv_within_at f f' s x)
(ht : has_deriv_within_at f f' t x) :
has_deriv_within_at f f' (s ∪ t) x | hs.has_fderiv_within_at.union ht.has_fderiv_within_at | lemma | has_deriv_within_at.union | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.nhds_within (h : has_deriv_within_at f f' s x)
(ht : s ∈ 𝓝[t] x) : has_deriv_within_at f f' t x | (has_deriv_within_at_inter' ht).1 (h.mono (inter_subset_right _ _)) | lemma | has_deriv_within_at.nhds_within | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at",
"has_deriv_within_at_inter'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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