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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has_deriv_within_at.has_deriv_at (h : has_deriv_within_at f f' s x) (hs : s ∈ 𝓝 x) :
has_deriv_at f f' x | has_fderiv_within_at.has_fderiv_at h hs | lemma | has_deriv_within_at.has_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at",
"has_deriv_within_at",
"has_fderiv_within_at.has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.has_deriv_within_at (h : differentiable_within_at 𝕜 f s x) :
has_deriv_within_at f (deriv_within f s x) s x | h.has_fderiv_within_at.has_deriv_within_at | lemma | differentiable_within_at.has_deriv_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv_within",
"differentiable_within_at",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.has_deriv_at (h : differentiable_at 𝕜 f x) : has_deriv_at f (deriv f x) x | h.has_fderiv_at.has_deriv_at | lemma | differentiable_at.has_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"differentiable_at",
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_deriv_iff : has_deriv_at f (deriv f x) x ↔ differentiable_at 𝕜 f x | ⟨λ h, h.differentiable_at, λ h, h.has_deriv_at⟩ | lemma | has_deriv_at_deriv_iff | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"differentiable_at",
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_deriv_within_iff :
has_deriv_within_at f (deriv_within f s x) s x ↔ differentiable_within_at 𝕜 f s x | ⟨λ h, h.differentiable_within_at, λ h, h.has_deriv_within_at⟩ | lemma | has_deriv_within_at_deriv_within_iff | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv_within",
"differentiable_within_at",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.has_deriv_at (h : differentiable_on 𝕜 f s) (hs : s ∈ 𝓝 x) :
has_deriv_at f (deriv f x) x | (h.has_fderiv_at hs).has_deriv_at | lemma | differentiable_on.has_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"differentiable_on",
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.deriv (h : has_deriv_at f f' x) : deriv f x = f' | h.differentiable_at.has_deriv_at.unique h | lemma | has_deriv_at.deriv | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_eq {f' : 𝕜 → F} (h : ∀ x, has_deriv_at f (f' x) x) : deriv f = f' | funext $ λ x, (h x).deriv | lemma | deriv_eq | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.deriv_within
(h : has_deriv_within_at f f' s x) (hxs : unique_diff_within_at 𝕜 s x) :
deriv_within f s x = f' | hxs.eq_deriv _ h.differentiable_within_at.has_deriv_within_at h | lemma | has_deriv_within_at.deriv_within | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv_within",
"has_deriv_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_deriv_within : (fderiv_within 𝕜 f s x : 𝕜 → F) 1 = deriv_within f s x | rfl | lemma | fderiv_within_deriv_within | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv_within",
"fderiv_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_fderiv_within :
smul_right (1 : 𝕜 →L[𝕜] 𝕜) (deriv_within f s x) = fderiv_within 𝕜 f s x | by simp [deriv_within] | lemma | deriv_within_fderiv_within | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv_within",
"fderiv_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_deriv : (fderiv 𝕜 f x : 𝕜 → F) 1 = deriv f x | rfl | lemma | fderiv_deriv | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_fderiv :
smul_right (1 : 𝕜 →L[𝕜] 𝕜) (deriv f x) = fderiv 𝕜 f x | by simp [deriv] | lemma | deriv_fderiv | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.deriv_within (h : differentiable_at 𝕜 f x)
(hxs : unique_diff_within_at 𝕜 s x) : deriv_within f s x = deriv f x | by { unfold deriv_within deriv, rw h.fderiv_within hxs } | lemma | differentiable_at.deriv_within | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"deriv_within",
"differentiable_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.deriv_eq_zero (hd : has_deriv_within_at f 0 s x)
(H : unique_diff_within_at 𝕜 s x) : deriv f x = 0 | (em' (differentiable_at 𝕜 f x)).elim deriv_zero_of_not_differentiable_at $
λ h, H.eq_deriv _ h.has_deriv_at.has_deriv_within_at hd | theorem | has_deriv_within_at.deriv_eq_zero | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"deriv_zero_of_not_differentiable_at",
"differentiable_at",
"em'",
"has_deriv_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_of_mem (st : t ∈ 𝓝[s] x) (ht : unique_diff_within_at 𝕜 s x)
(h : differentiable_within_at 𝕜 f t x) :
deriv_within f s x = deriv_within f t x | ((differentiable_within_at.has_deriv_within_at h).mono_of_mem st).deriv_within ht | lemma | deriv_within_of_mem | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv_within",
"differentiable_within_at",
"differentiable_within_at.has_deriv_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_subset (st : s ⊆ t) (ht : unique_diff_within_at 𝕜 s x)
(h : differentiable_within_at 𝕜 f t x) :
deriv_within f s x = deriv_within f t x | ((differentiable_within_at.has_deriv_within_at h).mono st).deriv_within ht | lemma | deriv_within_subset | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv_within",
"differentiable_within_at",
"differentiable_within_at.has_deriv_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_congr_set' (y : 𝕜) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) :
deriv_within f s x = deriv_within f t x | by simp only [deriv_within, fderiv_within_congr_set' y h] | lemma | deriv_within_congr_set' | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv_within",
"fderiv_within_congr_set'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_congr_set (h : s =ᶠ[𝓝 x] t) : deriv_within f s x = deriv_within f t x | by simp only [deriv_within, fderiv_within_congr_set h] | lemma | deriv_within_congr_set | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv_within",
"fderiv_within_congr_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_univ : deriv_within f univ = deriv f | by { ext, unfold deriv_within deriv, rw fderiv_within_univ } | lemma | deriv_within_univ | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"deriv_within",
"fderiv_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_inter (ht : t ∈ 𝓝 x) :
deriv_within f (s ∩ t) x = deriv_within f s x | by { unfold deriv_within, rw fderiv_within_inter ht } | lemma | deriv_within_inter | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv_within",
"fderiv_within_inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_of_open (hs : is_open s) (hx : x ∈ s) :
deriv_within f s x = deriv f x | by { unfold deriv_within, rw fderiv_within_of_open hs hx, refl } | lemma | deriv_within_of_open | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"deriv_within",
"fderiv_within_of_open",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_mem_iff {f : 𝕜 → F} {s : set F} {x : 𝕜} :
deriv f x ∈ s ↔ (differentiable_at 𝕜 f x ∧ deriv f x ∈ s) ∨
(¬differentiable_at 𝕜 f x ∧ (0 : F) ∈ s) | by by_cases hx : differentiable_at 𝕜 f x; simp [deriv_zero_of_not_differentiable_at, *] | lemma | deriv_mem_iff | 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 | |
deriv_within_mem_iff {f : 𝕜 → F} {t : set 𝕜} {s : set F} {x : 𝕜} :
deriv_within f t x ∈ s ↔ (differentiable_within_at 𝕜 f t x ∧ deriv_within f t x ∈ s) ∨
(¬differentiable_within_at 𝕜 f t x ∧ (0 : F) ∈ s) | by by_cases hx : differentiable_within_at 𝕜 f t x;
simp [deriv_within_zero_of_not_differentiable_within_at, *] | lemma | deriv_within_mem_iff | 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 | |
differentiable_within_at_Ioi_iff_Ici [partial_order 𝕜] :
differentiable_within_at 𝕜 f (Ioi x) x ↔ differentiable_within_at 𝕜 f (Ici x) x | ⟨λ h, h.has_deriv_within_at.Ici_of_Ioi.differentiable_within_at,
λ h, h.has_deriv_within_at.Ioi_of_Ici.differentiable_within_at⟩ | lemma | differentiable_within_at_Ioi_iff_Ici | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_Ioi_eq_Ici {E : Type*} [normed_add_comm_group E] [normed_space ℝ E] (f : ℝ → E)
(x : ℝ) :
deriv_within f (Ioi x) x = deriv_within f (Ici x) x | begin
by_cases H : differentiable_within_at ℝ f (Ioi x) x,
{ have A := H.has_deriv_within_at.Ici_of_Ioi,
have B := (differentiable_within_at_Ioi_iff_Ici.1 H).has_deriv_within_at,
simpa using (unique_diff_on_Ici x).eq le_rfl A B },
{ rw [deriv_within_zero_of_not_differentiable_within_at H,
deriv_with... | lemma | deriv_within_Ioi_eq_Ici | 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",
"differentiable_within_at_Ioi_iff_Ici",
"has_deriv_within_at",
"le_rfl",
"normed_add_comm_group",
"normed_space",
"unique_diff_on_Ici"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.has_deriv_at_filter_iff
(h₀ : f₀ =ᶠ[L] f₁) (hx : f₀ x = f₁ x) (h₁ : f₀' = f₁') :
has_deriv_at_filter f₀ f₀' x L ↔ has_deriv_at_filter f₁ f₁' x L | h₀.has_fderiv_at_filter_iff hx (by simp [h₁]) | theorem | filter.eventually_eq.has_deriv_at_filter_iff | 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.congr_of_eventually_eq (h : has_deriv_at_filter f f' x L)
(hL : f₁ =ᶠ[L] f) (hx : f₁ x = f x) : has_deriv_at_filter f₁ f' x L | by rwa hL.has_deriv_at_filter_iff hx rfl | lemma | has_deriv_at_filter.congr_of_eventually_eq | 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_within_at.congr_mono (h : has_deriv_within_at f f' s x) (ht : ∀x ∈ t, f₁ x = f x)
(hx : f₁ x = f x) (h₁ : t ⊆ s) : has_deriv_within_at f₁ f' t x | has_fderiv_within_at.congr_mono h ht hx h₁ | lemma | has_deriv_within_at.congr_mono | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_within_at",
"has_fderiv_within_at.congr_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.congr (h : has_deriv_within_at f f' s x) (hs : ∀x ∈ s, f₁ x = f x)
(hx : f₁ x = f x) : has_deriv_within_at f₁ f' s x | h.congr_mono hs hx (subset.refl _) | lemma | has_deriv_within_at.congr | 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.congr_of_mem (h : has_deriv_within_at f f' s x) (hs : ∀x ∈ s, f₁ x = f x)
(hx : x ∈ s) : has_deriv_within_at f₁ f' s x | h.congr hs (hs _ hx) | lemma | has_deriv_within_at.congr_of_mem | 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.congr_of_eventually_eq (h : has_deriv_within_at f f' s x)
(h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) : has_deriv_within_at f₁ f' s x | has_deriv_at_filter.congr_of_eventually_eq h h₁ hx | lemma | has_deriv_within_at.congr_of_eventually_eq | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at_filter.congr_of_eventually_eq",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.congr_of_eventually_eq_of_mem (h : has_deriv_within_at f f' s x)
(h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : x ∈ s) : has_deriv_within_at f₁ f' s x | h.congr_of_eventually_eq h₁ (h₁.eq_of_nhds_within hx) | lemma | has_deriv_within_at.congr_of_eventually_eq_of_mem | 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_at.congr_of_eventually_eq (h : has_deriv_at f f' x)
(h₁ : f₁ =ᶠ[𝓝 x] f) : has_deriv_at f₁ f' x | has_deriv_at_filter.congr_of_eventually_eq h h₁ (mem_of_mem_nhds h₁ : _) | lemma | has_deriv_at.congr_of_eventually_eq | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at",
"has_deriv_at_filter.congr_of_eventually_eq",
"mem_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.deriv_within_eq (hL : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
deriv_within f₁ s x = deriv_within f s x | by { unfold deriv_within, rw hL.fderiv_within_eq hx } | lemma | filter.eventually_eq.deriv_within_eq | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_congr (hs : eq_on f₁ f s) (hx : f₁ x = f x) :
deriv_within f₁ s x = deriv_within f s x | by { unfold deriv_within, rw fderiv_within_congr hs hx } | lemma | deriv_within_congr | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv_within",
"fderiv_within_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.deriv_eq (hL : f₁ =ᶠ[𝓝 x] f) : deriv f₁ x = deriv f x | by { unfold deriv, rwa filter.eventually_eq.fderiv_eq } | lemma | filter.eventually_eq.deriv_eq | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"filter.eventually_eq.fderiv_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.deriv (h : f₁ =ᶠ[𝓝 x] f) : deriv f₁ =ᶠ[𝓝 x] deriv f | h.eventually_eq_nhds.mono $ λ x h, h.deriv_eq | lemma | filter.eventually_eq.deriv | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter_id : has_deriv_at_filter id 1 x L | (has_fderiv_at_filter_id x L).has_deriv_at_filter | theorem | has_deriv_at_filter_id | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at_filter",
"has_fderiv_at_filter_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_id : has_deriv_within_at id 1 s x | has_deriv_at_filter_id _ _ | theorem | has_deriv_within_at_id | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at_filter_id",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_id : has_deriv_at id 1 x | has_deriv_at_filter_id _ _ | theorem | has_deriv_at_id | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at",
"has_deriv_at_filter_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_id' : has_deriv_at (λ (x : 𝕜), x) 1 x | has_deriv_at_filter_id _ _ | theorem | has_deriv_at_id' | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at",
"has_deriv_at_filter_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_id : has_strict_deriv_at id 1 x | (has_strict_fderiv_at_id x).has_strict_deriv_at | theorem | has_strict_deriv_at_id | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_strict_deriv_at",
"has_strict_fderiv_at_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_id : deriv id x = 1 | has_deriv_at.deriv (has_deriv_at_id x) | lemma | deriv_id | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"has_deriv_at.deriv",
"has_deriv_at_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_id' : deriv (@id 𝕜) = λ _, 1 | funext deriv_id | lemma | deriv_id' | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"deriv_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_id'' : deriv (λ x : 𝕜, x) = λ _, 1 | deriv_id' | lemma | deriv_id'' | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"deriv_id'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_id (hxs : unique_diff_within_at 𝕜 s x) : deriv_within id s x = 1 | (has_deriv_within_at_id x s).deriv_within hxs | lemma | deriv_within_id | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv_within",
"has_deriv_within_at_id",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter_const : has_deriv_at_filter (λ x, c) 0 x L | (has_fderiv_at_filter_const c x L).has_deriv_at_filter | theorem | has_deriv_at_filter_const | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at_filter",
"has_fderiv_at_filter_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_const : has_strict_deriv_at (λ x, c) 0 x | (has_strict_fderiv_at_const c x).has_strict_deriv_at | theorem | has_strict_deriv_at_const | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_strict_deriv_at",
"has_strict_fderiv_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_const : has_deriv_within_at (λ x, c) 0 s x | has_deriv_at_filter_const _ _ _ | theorem | has_deriv_within_at_const | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at_filter_const",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_const : has_deriv_at (λ x, c) 0 x | has_deriv_at_filter_const _ _ _ | theorem | has_deriv_at_const | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_deriv_at",
"has_deriv_at_filter_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_const : deriv (λ x, c) x = 0 | has_deriv_at.deriv (has_deriv_at_const x c) | lemma | deriv_const | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"has_deriv_at.deriv",
"has_deriv_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_const' : deriv (λ x:𝕜, c) = λ x, 0 | funext (λ x, deriv_const x c) | lemma | deriv_const' | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv",
"deriv_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_const (hxs : unique_diff_within_at 𝕜 s x) : deriv_within (λ x, c) s x = 0 | (has_deriv_within_at_const _ _ _).deriv_within hxs | lemma | deriv_within_const | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"deriv_within",
"has_deriv_within_at_const",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter.tendsto_nhds
(hL : L ≤ 𝓝 x) (h : has_deriv_at_filter f f' x L) :
tendsto f L (𝓝 (f x)) | h.tendsto_nhds hL | theorem | has_deriv_at_filter.tendsto_nhds | 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_within_at.continuous_within_at
(h : has_deriv_within_at f f' s x) : continuous_within_at f s x | has_deriv_at_filter.tendsto_nhds inf_le_left h | theorem | has_deriv_within_at.continuous_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"continuous_within_at",
"has_deriv_at_filter.tendsto_nhds",
"has_deriv_within_at",
"inf_le_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.continuous_at (h : has_deriv_at f f' x) : continuous_at f x | has_deriv_at_filter.tendsto_nhds le_rfl h | theorem | has_deriv_at.continuous_at | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"continuous_at",
"has_deriv_at",
"has_deriv_at_filter.tendsto_nhds",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.continuous_on {f f' : 𝕜 → F}
(hderiv : ∀ x ∈ s, has_deriv_at f (f' x) x) : continuous_on f s | λ x hx, (hderiv x hx).continuous_at.continuous_within_at | theorem | has_deriv_at.continuous_on | analysis.calculus.deriv | src/analysis/calculus/deriv/basic.lean | [
"analysis.calculus.fderiv.basic"
] | [
"continuous_at.continuous_within_at",
"continuous_on",
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter.scomp
(hg : has_deriv_at_filter g₁ g₁' (h x) L')
(hh : has_deriv_at_filter h h' x L) (hL : tendsto h L L'):
has_deriv_at_filter (g₁ ∘ h) (h' • g₁') x L | by simpa using ((hg.restrict_scalars 𝕜).comp x hh hL).has_deriv_at_filter | theorem | has_deriv_at_filter.scomp | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_deriv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.scomp_has_deriv_at
(hg : has_deriv_within_at g₁ g₁' s' (h x))
(hh : has_deriv_at h h' x) (hs : ∀ x, h x ∈ s') :
has_deriv_at (g₁ ∘ h) (h' • g₁') x | hg.scomp x hh $ tendsto_inf.2 ⟨hh.continuous_at, tendsto_principal.2 $ eventually_of_forall hs⟩ | theorem | has_deriv_within_at.scomp_has_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_deriv_at",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.scomp
(hg : has_deriv_within_at g₁ g₁' t' (h x))
(hh : has_deriv_within_at h h' s x) (hst : maps_to h s t') :
has_deriv_within_at (g₁ ∘ h) (h' • g₁') s x | hg.scomp x hh $ hh.continuous_within_at.tendsto_nhds_within hst | theorem | has_deriv_within_at.scomp | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.scomp
(hg : has_deriv_at g₁ g₁' (h x)) (hh : has_deriv_at h h' x) :
has_deriv_at (g₁ ∘ h) (h' • g₁') x | hg.scomp x hh hh.continuous_at | theorem | has_deriv_at.scomp | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_deriv_at"
] | The chain rule. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_deriv_at.scomp
(hg : has_strict_deriv_at g₁ g₁' (h x)) (hh : has_strict_deriv_at h h' x) :
has_strict_deriv_at (g₁ ∘ h) (h' • g₁') x | by simpa using ((hg.restrict_scalars 𝕜).comp x hh).has_strict_deriv_at | theorem | has_strict_deriv_at.scomp | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.scomp_has_deriv_within_at
(hg : has_deriv_at g₁ g₁' (h x)) (hh : has_deriv_within_at h h' s x) :
has_deriv_within_at (g₁ ∘ h) (h' • g₁') s x | has_deriv_within_at.scomp x hg.has_deriv_within_at hh (maps_to_univ _ _) | theorem | has_deriv_at.scomp_has_deriv_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_deriv_at",
"has_deriv_within_at",
"has_deriv_within_at.scomp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within.scomp
(hg : differentiable_within_at 𝕜' g₁ t' (h x)) (hh : differentiable_within_at 𝕜 h s x)
(hs : maps_to h s t') (hxs : unique_diff_within_at 𝕜 s x) :
deriv_within (g₁ ∘ h) s x = deriv_within h s x • deriv_within g₁ t' (h x) | (has_deriv_within_at.scomp x hg.has_deriv_within_at hh.has_deriv_within_at hs).deriv_within hxs | lemma | deriv_within.scomp | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"deriv_within",
"differentiable_within_at",
"has_deriv_within_at.scomp",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv.scomp
(hg : differentiable_at 𝕜' g₁ (h x)) (hh : differentiable_at 𝕜 h x) :
deriv (g₁ ∘ h) x = deriv h x • deriv g₁ (h x) | (has_deriv_at.scomp x hg.has_deriv_at hh.has_deriv_at).deriv | lemma | deriv.scomp | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"deriv",
"differentiable_at",
"has_deriv_at.scomp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter.comp_has_fderiv_at_filter {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} (x)
{L'' : filter E} (hh₂ : has_deriv_at_filter h₂ h₂' (f x) L')
(hf : has_fderiv_at_filter f f' x L'') (hL : tendsto f L'' L') :
has_fderiv_at_filter (h₂ ∘ f) (h₂' • f') x L'' | by { convert (hh₂.restrict_scalars 𝕜).comp x hf hL, ext x, simp [mul_comm] } | theorem | has_deriv_at_filter.comp_has_fderiv_at_filter | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"filter",
"has_deriv_at_filter",
"has_fderiv_at_filter",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.comp_has_strict_fderiv_at {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} (x)
(hh : has_strict_deriv_at h₂ h₂' (f x)) (hf : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (h₂ ∘ f) (h₂' • f') x | begin
rw has_strict_deriv_at at hh,
convert (hh.restrict_scalars 𝕜).comp x hf,
ext x,
simp [mul_comm]
end | theorem | has_strict_deriv_at.comp_has_strict_fderiv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_strict_deriv_at",
"has_strict_fderiv_at",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.comp_has_fderiv_at {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} (x)
(hh : has_deriv_at h₂ h₂' (f x)) (hf : has_fderiv_at f f' x) :
has_fderiv_at (h₂ ∘ f) (h₂' • f') x | hh.comp_has_fderiv_at_filter x hf hf.continuous_at | theorem | has_deriv_at.comp_has_fderiv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_deriv_at",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.comp_has_fderiv_within_at {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} {s} (x)
(hh : has_deriv_at h₂ h₂' (f x)) (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (h₂ ∘ f) (h₂' • f') s x | hh.comp_has_fderiv_at_filter x hf hf.continuous_within_at | theorem | has_deriv_at.comp_has_fderiv_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_deriv_at",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.comp_has_fderiv_within_at {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} {s t} (x)
(hh : has_deriv_within_at h₂ h₂' t (f x)) (hf : has_fderiv_within_at f f' s x)
(hst : maps_to f s t) :
has_fderiv_within_at (h₂ ∘ f) (h₂' • f') s x | hh.comp_has_fderiv_at_filter x hf $ hf.continuous_within_at.tendsto_nhds_within hst | theorem | has_deriv_within_at.comp_has_fderiv_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_deriv_within_at",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter.comp
(hh₂ : has_deriv_at_filter h₂ h₂' (h x) L')
(hh : has_deriv_at_filter h h' x L) (hL : tendsto h L L') :
has_deriv_at_filter (h₂ ∘ h) (h₂' * h') x L | by { rw mul_comm, exact hh₂.scomp x hh hL } | theorem | has_deriv_at_filter.comp | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_deriv_at_filter",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.comp
(hh₂ : has_deriv_within_at h₂ h₂' s' (h x))
(hh : has_deriv_within_at h h' s x) (hst : maps_to h s s') :
has_deriv_within_at (h₂ ∘ h) (h₂' * h') s x | by { rw mul_comm, exact hh₂.scomp x hh hst, } | theorem | has_deriv_within_at.comp | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_deriv_within_at",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.comp
(hh₂ : has_deriv_at h₂ h₂' (h x)) (hh : has_deriv_at h h' x) :
has_deriv_at (h₂ ∘ h) (h₂' * h') x | hh₂.comp x hh hh.continuous_at | theorem | has_deriv_at.comp | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_deriv_at"
] | The chain rule. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_deriv_at.comp
(hh₂ : has_strict_deriv_at h₂ h₂' (h x)) (hh : has_strict_deriv_at h h' x) :
has_strict_deriv_at (h₂ ∘ h) (h₂' * h') x | by { rw mul_comm, exact hh₂.scomp x hh } | theorem | has_strict_deriv_at.comp | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_strict_deriv_at",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.comp_has_deriv_within_at
(hh₂ : has_deriv_at h₂ h₂' (h x)) (hh : has_deriv_within_at h h' s x) :
has_deriv_within_at (h₂ ∘ h) (h₂' * h') s x | hh₂.has_deriv_within_at.comp x hh (maps_to_univ _ _) | theorem | has_deriv_at.comp_has_deriv_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_deriv_at",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within.comp
(hh₂ : differentiable_within_at 𝕜' h₂ s' (h x)) (hh : differentiable_within_at 𝕜 h s x)
(hs : maps_to h s s') (hxs : unique_diff_within_at 𝕜 s x) :
deriv_within (h₂ ∘ h) s x = deriv_within h₂ s' (h x) * deriv_within h s x | (hh₂.has_deriv_within_at.comp x hh.has_deriv_within_at hs).deriv_within hxs | lemma | deriv_within.comp | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv.comp
(hh₂ : differentiable_at 𝕜' h₂ (h x)) (hh : differentiable_at 𝕜 h x) :
deriv (h₂ ∘ h) x = deriv h₂ (h x) * deriv h x | (hh₂.has_deriv_at.comp x hh.has_deriv_at).deriv | lemma | deriv.comp | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter.iterate {f : 𝕜 → 𝕜} {f' : 𝕜}
(hf : has_deriv_at_filter f f' x L) (hL : tendsto f L L) (hx : f x = x) (n : ℕ) :
has_deriv_at_filter (f^[n]) (f'^n) x L | begin
have := hf.iterate hL hx n,
rwa [continuous_linear_map.smul_right_one_pow] at this
end | lemma | has_deriv_at_filter.iterate | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"continuous_linear_map.smul_right_one_pow",
"has_deriv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.iterate {f : 𝕜 → 𝕜} {f' : 𝕜}
(hf : has_deriv_at f f' x) (hx : f x = x) (n : ℕ) :
has_deriv_at (f^[n]) (f'^n) x | begin
have := has_fderiv_at.iterate hf hx n,
rwa [continuous_linear_map.smul_right_one_pow] at this
end | lemma | has_deriv_at.iterate | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"continuous_linear_map.smul_right_one_pow",
"has_deriv_at",
"has_fderiv_at.iterate"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.iterate {f : 𝕜 → 𝕜} {f' : 𝕜}
(hf : has_deriv_within_at f f' s x) (hx : f x = x) (hs : maps_to f s s) (n : ℕ) :
has_deriv_within_at (f^[n]) (f'^n) s x | begin
have := has_fderiv_within_at.iterate hf hx hs n,
rwa [continuous_linear_map.smul_right_one_pow] at this
end | lemma | has_deriv_within_at.iterate | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"continuous_linear_map.smul_right_one_pow",
"has_deriv_within_at",
"has_fderiv_within_at.iterate"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.iterate {f : 𝕜 → 𝕜} {f' : 𝕜}
(hf : has_strict_deriv_at f f' x) (hx : f x = x) (n : ℕ) :
has_strict_deriv_at (f^[n]) (f'^n) x | begin
have := hf.iterate hx n,
rwa [continuous_linear_map.smul_right_one_pow] at this
end | lemma | has_strict_deriv_at.iterate | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"continuous_linear_map.smul_right_one_pow",
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.comp_has_deriv_within_at {t : set F}
(hl : has_fderiv_within_at l l' t (f x)) (hf : has_deriv_within_at f f' s x)
(hst : maps_to f s t) :
has_deriv_within_at (l ∘ f) (l' f') s x | by simpa only [one_apply, one_smul, smul_right_apply, coe_comp', (∘)]
using (hl.comp x hf.has_fderiv_within_at hst).has_deriv_within_at | theorem | has_fderiv_within_at.comp_has_deriv_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_deriv_within_at",
"has_fderiv_within_at",
"one_smul"
] | The composition `l ∘ f` where `l : F → E` and `f : 𝕜 → F`, has a derivative within a set
equal to the Fréchet derivative of `l` applied to the derivative of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at.comp_has_deriv_within_at
(hl : has_fderiv_at l l' (f x)) (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (l ∘ f) (l' f') s x | hl.has_fderiv_within_at.comp_has_deriv_within_at x hf (maps_to_univ _ _) | theorem | has_fderiv_at.comp_has_deriv_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_deriv_within_at",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.comp_has_deriv_at (hl : has_fderiv_at l l' (f x)) (hf : has_deriv_at f f' x) :
has_deriv_at (l ∘ f) (l' f') x | has_deriv_within_at_univ.mp $ hl.comp_has_deriv_within_at x hf.has_deriv_within_at | theorem | has_fderiv_at.comp_has_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_deriv_at",
"has_fderiv_at"
] | The composition `l ∘ f` where `l : F → E` and `f : 𝕜 → F`, has a derivative equal to the
Fréchet derivative of `l` applied to the derivative of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_fderiv_at.comp_has_strict_deriv_at
(hl : has_strict_fderiv_at l l' (f x)) (hf : has_strict_deriv_at f f' x) :
has_strict_deriv_at (l ∘ f) (l' f') x | by simpa only [one_apply, one_smul, smul_right_apply, coe_comp', (∘)]
using (hl.comp x hf.has_strict_fderiv_at).has_strict_deriv_at | theorem | has_strict_fderiv_at.comp_has_strict_deriv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"has_strict_deriv_at",
"has_strict_fderiv_at",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within.comp_deriv_within {t : set F}
(hl : differentiable_within_at 𝕜 l t (f x)) (hf : differentiable_within_at 𝕜 f s x)
(hs : maps_to f s t) (hxs : unique_diff_within_at 𝕜 s x) :
deriv_within (l ∘ f) s x = (fderiv_within 𝕜 l t (f x) : F → E) (deriv_within f s x) | (hl.has_fderiv_within_at.comp_has_deriv_within_at x hf.has_deriv_within_at hs).deriv_within hxs | lemma | fderiv_within.comp_deriv_within | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"deriv_within",
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv.comp_deriv
(hl : differentiable_at 𝕜 l (f x)) (hf : differentiable_at 𝕜 f x) :
deriv (l ∘ f) x = (fderiv 𝕜 l (f x) : F → E) (deriv f x) | (hl.has_fderiv_at.comp_has_deriv_at x hf.has_deriv_at).deriv | lemma | fderiv.comp_deriv | analysis.calculus.deriv | src/analysis/calculus/deriv/comp.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.comp",
"analysis.calculus.fderiv.restrict_scalars"
] | [
"deriv",
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_inv (hx : x ≠ 0) : has_strict_deriv_at has_inv.inv (-(x^2)⁻¹) x | begin
suffices : (λ p : 𝕜 × 𝕜, (p.1 - p.2) * ((x * x)⁻¹ - (p.1 * p.2)⁻¹)) =o[𝓝 (x, x)]
(λ p, (p.1 - p.2) * 1),
{ refine this.congr' _ (eventually_of_forall $ λ _, mul_one _),
refine eventually.mono ((is_open_ne.prod is_open_ne).mem_nhds ⟨hx, hx⟩) _,
rintro ⟨y, z⟩ ⟨hy, hz⟩,
simp only [mem_set_of_e... | theorem | has_strict_deriv_at_inv | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"has_strict_deriv_at",
"is_open_ne",
"mul_ne_zero",
"mul_one",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_inv (x_ne_zero : x ≠ 0) :
has_deriv_at (λy, y⁻¹) (-(x^2)⁻¹) x | (has_strict_deriv_at_inv x_ne_zero).has_deriv_at | theorem | has_deriv_at_inv | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"has_deriv_at",
"has_strict_deriv_at_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_inv (x_ne_zero : x ≠ 0) (s : set 𝕜) :
has_deriv_within_at (λx, x⁻¹) (-(x^2)⁻¹) s x | (has_deriv_at_inv x_ne_zero).has_deriv_within_at | theorem | has_deriv_within_at_inv | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"has_deriv_at_inv",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_inv :
differentiable_at 𝕜 (λx, x⁻¹) x ↔ x ≠ 0 | ⟨λ H, normed_field.continuous_at_inv.1 H.continuous_at,
λ H, (has_deriv_at_inv H).differentiable_at⟩ | lemma | differentiable_at_inv | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"differentiable_at",
"has_deriv_at_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_inv (x_ne_zero : x ≠ 0) :
differentiable_within_at 𝕜 (λx, x⁻¹) s x | (differentiable_at_inv.2 x_ne_zero).differentiable_within_at | lemma | differentiable_within_at_inv | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_inv : differentiable_on 𝕜 (λx:𝕜, x⁻¹) {x | x ≠ 0} | λx hx, differentiable_within_at_inv hx | lemma | differentiable_on_inv | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"differentiable_on",
"differentiable_within_at_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_inv : deriv (λx, x⁻¹) x = -(x^2)⁻¹ | begin
rcases eq_or_ne x 0 with rfl|hne,
{ simp [deriv_zero_of_not_differentiable_at (mt differentiable_at_inv.1 (not_not.2 rfl))] },
{ exact (has_deriv_at_inv hne).deriv }
end | lemma | deriv_inv | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"deriv",
"deriv_zero_of_not_differentiable_at",
"eq_or_ne",
"has_deriv_at_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_inv' : deriv (λ x : 𝕜, x⁻¹) = λ x, -(x ^ 2)⁻¹ | funext (λ x, deriv_inv) | lemma | deriv_inv' | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"deriv",
"deriv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_inv (x_ne_zero : x ≠ 0) (hxs : unique_diff_within_at 𝕜 s x) :
deriv_within (λx, x⁻¹) s x = -(x^2)⁻¹ | begin
rw differentiable_at.deriv_within (differentiable_at_inv.2 x_ne_zero) hxs,
exact deriv_inv
end | lemma | deriv_within_inv | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"deriv_inv",
"deriv_within",
"differentiable_at.deriv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_inv (x_ne_zero : x ≠ 0) :
has_fderiv_at (λx, x⁻¹) (smul_right (1 : 𝕜 →L[𝕜] 𝕜) (-(x^2)⁻¹) : 𝕜 →L[𝕜] 𝕜) x | has_deriv_at_inv x_ne_zero | lemma | has_fderiv_at_inv | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"has_deriv_at_inv",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_inv (x_ne_zero : x ≠ 0) :
has_fderiv_within_at (λx, x⁻¹) (smul_right (1 : 𝕜 →L[𝕜] 𝕜) (-(x^2)⁻¹) : 𝕜 →L[𝕜] 𝕜) s x | (has_fderiv_at_inv x_ne_zero).has_fderiv_within_at | lemma | has_fderiv_within_at_inv | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"has_fderiv_at_inv",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_inv :
fderiv 𝕜 (λx, x⁻¹) x = smul_right (1 : 𝕜 →L[𝕜] 𝕜) (-(x^2)⁻¹) | by rw [← deriv_fderiv, deriv_inv] | lemma | fderiv_inv | analysis.calculus.deriv | src/analysis/calculus/deriv/inv.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"deriv_fderiv",
"deriv_inv",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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