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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filter.eventually_eq.has_fderiv_within_at_iff (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : f₀ x = f₁ x) :
has_fderiv_within_at f₀ f' s x ↔ has_fderiv_within_at f₁ f' s x | h.has_fderiv_at_filter_iff hx (λ _, rfl) | theorem | filter.eventually_eq.has_fderiv_within_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.has_fderiv_within_at_iff_of_mem (h : f₀ =ᶠ[𝓝[s] x] f₁) (hx : x ∈ s) :
has_fderiv_within_at f₀ f' s x ↔ has_fderiv_within_at f₁ f' s x | h.has_fderiv_within_at_iff (h.eq_of_nhds_within hx) | theorem | filter.eventually_eq.has_fderiv_within_at_iff_of_mem | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.differentiable_within_at_iff (h : f₀ =ᶠ[𝓝[s] x] f₁)
(hx : f₀ x = f₁ x) :
differentiable_within_at 𝕜 f₀ s x ↔ differentiable_within_at 𝕜 f₁ s x | exists_congr $ λ f', h.has_fderiv_within_at_iff hx | theorem | filter.eventually_eq.differentiable_within_at_iff | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.differentiable_within_at_iff_of_mem (h : f₀ =ᶠ[𝓝[s] x] f₁)
(hx : x ∈ s) :
differentiable_within_at 𝕜 f₀ s x ↔ differentiable_within_at 𝕜 f₁ s x | h.differentiable_within_at_iff (h.eq_of_nhds_within hx) | theorem | filter.eventually_eq.differentiable_within_at_iff_of_mem | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.congr_mono (h : has_fderiv_within_at f f' s x) (ht : eq_on f₁ f t)
(hx : f₁ x = f x) (h₁ : t ⊆ s) : has_fderiv_within_at f₁ f' t x | has_fderiv_at_filter.congr_of_eventually_eq (h.mono h₁) (filter.mem_inf_of_right ht) hx | lemma | has_fderiv_within_at.congr_mono | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"filter.mem_inf_of_right",
"has_fderiv_at_filter.congr_of_eventually_eq",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.congr (h : has_fderiv_within_at f f' s x) (hs : eq_on f₁ f s)
(hx : f₁ x = f x) : has_fderiv_within_at f₁ f' s x | h.congr_mono hs hx (subset.refl _) | lemma | has_fderiv_within_at.congr | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.congr' (h : has_fderiv_within_at f f' s x) (hs : eq_on f₁ f s)
(hx : x ∈ s) : has_fderiv_within_at f₁ f' s x | h.congr hs (hs hx) | lemma | has_fderiv_within_at.congr' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.congr_of_eventually_eq (h : has_fderiv_within_at f f' s x)
(h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) : has_fderiv_within_at f₁ f' s x | has_fderiv_at_filter.congr_of_eventually_eq h h₁ hx | lemma | has_fderiv_within_at.congr_of_eventually_eq | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at_filter.congr_of_eventually_eq",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.congr_of_eventually_eq (h : has_fderiv_at f f' x)
(h₁ : f₁ =ᶠ[𝓝 x] f) : has_fderiv_at f₁ f' x | has_fderiv_at_filter.congr_of_eventually_eq h h₁ (mem_of_mem_nhds h₁ : _) | lemma | has_fderiv_at.congr_of_eventually_eq | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at",
"has_fderiv_at_filter.congr_of_eventually_eq",
"mem_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.congr_mono (h : differentiable_within_at 𝕜 f s x)
(ht : eq_on f₁ f t) (hx : f₁ x = f x) (h₁ : t ⊆ s) : differentiable_within_at 𝕜 f₁ t x | (h.has_fderiv_within_at.congr_mono ht hx h₁).differentiable_within_at | lemma | differentiable_within_at.congr_mono | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.congr (h : differentiable_within_at 𝕜 f s x)
(ht : ∀x ∈ s, f₁ x = f x) (hx : f₁ x = f x) : differentiable_within_at 𝕜 f₁ s x | differentiable_within_at.congr_mono h ht hx (subset.refl _) | lemma | differentiable_within_at.congr | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at",
"differentiable_within_at.congr_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.congr_of_eventually_eq
(h : differentiable_within_at 𝕜 f s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f)
(hx : f₁ x = f x) : differentiable_within_at 𝕜 f₁ s x | (h.has_fderiv_within_at.congr_of_eventually_eq h₁ hx).differentiable_within_at | lemma | differentiable_within_at.congr_of_eventually_eq | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.congr_mono (h : differentiable_on 𝕜 f s) (h' : ∀x ∈ t, f₁ x = f x)
(h₁ : t ⊆ s) : differentiable_on 𝕜 f₁ t | λ x hx, (h x (h₁ hx)).congr_mono h' (h' x hx) h₁ | lemma | differentiable_on.congr_mono | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.congr (h : differentiable_on 𝕜 f s) (h' : ∀x ∈ s, f₁ x = f x) :
differentiable_on 𝕜 f₁ s | λ x hx, (h x hx).congr h' (h' x hx) | lemma | differentiable_on.congr | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_congr (h' : ∀x ∈ s, f₁ x = f x) :
differentiable_on 𝕜 f₁ s ↔ differentiable_on 𝕜 f s | ⟨λ h, differentiable_on.congr h (λy hy, (h' y hy).symm),
λ h, differentiable_on.congr h h'⟩ | lemma | differentiable_on_congr | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_on",
"differentiable_on.congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.congr_of_eventually_eq (h : differentiable_at 𝕜 f x) (hL : f₁ =ᶠ[𝓝 x] f) :
differentiable_at 𝕜 f₁ x | hL.differentiable_at_iff.2 h | lemma | differentiable_at.congr_of_eventually_eq | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.fderiv_within_congr_mono (h : differentiable_within_at 𝕜 f s x)
(hs : eq_on f₁ f t) (hx : f₁ x = f x) (hxt : unique_diff_within_at 𝕜 t x) (h₁ : t ⊆ s) :
fderiv_within 𝕜 f₁ t x = fderiv_within 𝕜 f s x | (has_fderiv_within_at.congr_mono h.has_fderiv_within_at hs hx h₁).fderiv_within hxt | lemma | differentiable_within_at.fderiv_within_congr_mono | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at",
"fderiv_within",
"has_fderiv_within_at.congr_mono",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.fderiv_within_eq (hs : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
fderiv_within 𝕜 f₁ s x = fderiv_within 𝕜 f s x | by simp only [fderiv_within, hs.has_fderiv_within_at_iff hx] | lemma | filter.eventually_eq.fderiv_within_eq | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.fderiv_within' (hs : f₁ =ᶠ[𝓝[s] x] f) (ht : t ⊆ s) :
fderiv_within 𝕜 f₁ t =ᶠ[𝓝[s] x] fderiv_within 𝕜 f t | (eventually_nhds_within_nhds_within.2 hs).mp $ eventually_mem_nhds_within.mono $ λ y hys hs,
filter.eventually_eq.fderiv_within_eq (hs.filter_mono $ nhds_within_mono _ ht)
(hs.self_of_nhds_within hys) | lemma | filter.eventually_eq.fderiv_within' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv_within",
"filter.eventually_eq.fderiv_within_eq",
"nhds_within_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.fderiv_within (hs : f₁ =ᶠ[𝓝[s] x] f) :
fderiv_within 𝕜 f₁ s =ᶠ[𝓝[s] x] fderiv_within 𝕜 f s | hs.fderiv_within' subset.rfl | lemma | filter.eventually_eq.fderiv_within | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.fderiv_within_eq_nhds (h : f₁ =ᶠ[𝓝 x] f) :
fderiv_within 𝕜 f₁ s x = fderiv_within 𝕜 f s x | (h.filter_mono nhds_within_le_nhds).fderiv_within_eq h.self_of_nhds | lemma | filter.eventually_eq.fderiv_within_eq_nhds | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv_within",
"nhds_within_le_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_congr (hs : eq_on f₁ f s) (hx : f₁ x = f x) :
fderiv_within 𝕜 f₁ s x = fderiv_within 𝕜 f s x | (hs.eventually_eq.filter_mono inf_le_right).fderiv_within_eq hx | lemma | fderiv_within_congr | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv_within",
"inf_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_congr' (hs : eq_on f₁ f s) (hx : x ∈ s) :
fderiv_within 𝕜 f₁ s x = fderiv_within 𝕜 f s x | fderiv_within_congr hs (hs hx) | lemma | fderiv_within_congr' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv_within",
"fderiv_within_congr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.fderiv_eq (h : f₁ =ᶠ[𝓝 x] f) :
fderiv 𝕜 f₁ x = fderiv 𝕜 f x | by rw [← fderiv_within_univ, ← fderiv_within_univ, h.fderiv_within_eq_nhds] | lemma | filter.eventually_eq.fderiv_eq | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv",
"fderiv_within_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.fderiv (h : f₁ =ᶠ[𝓝 x] f) :
fderiv 𝕜 f₁ =ᶠ[𝓝 x] fderiv 𝕜 f | h.eventually_eq_nhds.mono $ λ x h, h.fderiv_eq | lemma | filter.eventually_eq.fderiv | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at_id (x : E) :
has_strict_fderiv_at id (id 𝕜 E) x | (is_o_zero _ _).congr_left $ by simp | theorem | has_strict_fderiv_at_id | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter_id (x : E) (L : filter E) :
has_fderiv_at_filter id (id 𝕜 E) x L | (is_o_zero _ _).congr_left $ by simp | theorem | has_fderiv_at_filter_id | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"filter",
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_id (x : E) (s : set E) :
has_fderiv_within_at id (id 𝕜 E) s x | has_fderiv_at_filter_id _ _ | theorem | has_fderiv_within_at_id | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at_filter_id",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_id (x : E) : has_fderiv_at id (id 𝕜 E) x | has_fderiv_at_filter_id _ _ | theorem | has_fderiv_at_id | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at",
"has_fderiv_at_filter_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_id : differentiable_at 𝕜 id x | (has_fderiv_at_id x).differentiable_at | lemma | differentiable_at_id | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at",
"has_fderiv_at_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_id' : differentiable_at 𝕜 (λ x, x) x | (has_fderiv_at_id x).differentiable_at | lemma | differentiable_at_id' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at",
"has_fderiv_at_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_id : differentiable_within_at 𝕜 id s x | differentiable_at_id.differentiable_within_at | lemma | differentiable_within_at_id | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_id : differentiable 𝕜 (id : E → E) | λx, differentiable_at_id | lemma | differentiable_id | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable",
"differentiable_at_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_id' : differentiable 𝕜 (λ (x : E), x) | λx, differentiable_at_id | lemma | differentiable_id' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable",
"differentiable_at_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_id : differentiable_on 𝕜 id s | differentiable_id.differentiable_on | lemma | differentiable_on_id | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_id : fderiv 𝕜 id x = id 𝕜 E | has_fderiv_at.fderiv (has_fderiv_at_id x) | lemma | fderiv_id | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv",
"has_fderiv_at.fderiv",
"has_fderiv_at_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_id' : fderiv 𝕜 (λ (x : E), x) x = continuous_linear_map.id 𝕜 E | fderiv_id | lemma | fderiv_id' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"continuous_linear_map.id",
"fderiv",
"fderiv_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_id (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 id s x = id 𝕜 E | begin
rw differentiable_at.fderiv_within (differentiable_at_id) hxs,
exact fderiv_id
end | lemma | fderiv_within_id | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at.fderiv_within",
"differentiable_at_id",
"fderiv_id",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_id' (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (λ (x : E), x) s x = continuous_linear_map.id 𝕜 E | fderiv_within_id hxs | lemma | fderiv_within_id' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"continuous_linear_map.id",
"fderiv_within",
"fderiv_within_id",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at_const (c : F) (x : E) :
has_strict_fderiv_at (λ _, c) (0 : E →L[𝕜] F) x | (is_o_zero _ _).congr_left $ λ _, by simp only [zero_apply, sub_self] | theorem | has_strict_fderiv_at_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter_const (c : F) (x : E) (L : filter E) :
has_fderiv_at_filter (λ x, c) (0 : E →L[𝕜] F) x L | (is_o_zero _ _).congr_left $ λ _, by simp only [zero_apply, sub_self] | theorem | has_fderiv_at_filter_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"filter",
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_const (c : F) (x : E) (s : set E) :
has_fderiv_within_at (λ x, c) (0 : E →L[𝕜] F) s x | has_fderiv_at_filter_const _ _ _ | theorem | has_fderiv_within_at_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at_filter_const",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_const (c : F) (x : E) :
has_fderiv_at (λ x, c) (0 : E →L[𝕜] F) x | has_fderiv_at_filter_const _ _ _ | theorem | has_fderiv_at_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at",
"has_fderiv_at_filter_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_const (c : F) : differentiable_at 𝕜 (λx, c) x | ⟨0, has_fderiv_at_const c x⟩ | lemma | differentiable_at_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at",
"has_fderiv_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_const (c : F) : differentiable_within_at 𝕜 (λx, c) s x | differentiable_at.differentiable_within_at (differentiable_at_const _) | lemma | differentiable_within_at_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at.differentiable_within_at",
"differentiable_at_const",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_const_apply (c : F) : fderiv 𝕜 (λy, c) x = 0 | has_fderiv_at.fderiv (has_fderiv_at_const c x) | lemma | fderiv_const_apply | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv",
"has_fderiv_at.fderiv",
"has_fderiv_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_const (c : F) : fderiv 𝕜 (λ (y : E), c) = 0 | by { ext m, rw fderiv_const_apply, refl } | lemma | fderiv_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv",
"fderiv_const_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_const_apply (c : F) (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (λy, c) s x = 0 | begin
rw differentiable_at.fderiv_within (differentiable_at_const _) hxs,
exact fderiv_const_apply _
end | lemma | fderiv_within_const_apply | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_at.fderiv_within",
"differentiable_at_const",
"fderiv_const_apply",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_const (c : F) : differentiable 𝕜 (λx : E, c) | λx, differentiable_at_const _ | lemma | differentiable_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable",
"differentiable_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_const (c : F) : differentiable_on 𝕜 (λx, c) s | (differentiable_const _).differentiable_on | lemma | differentiable_on_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_const",
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_singleton (f : E → F) (x : E) :
has_fderiv_within_at f (0 : E →L[𝕜] F) {x} x | by simp only [has_fderiv_within_at, nhds_within_singleton, has_fderiv_at_filter, is_o_pure,
continuous_linear_map.zero_apply, sub_self] | lemma | has_fderiv_within_at_singleton | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"continuous_linear_map.zero_apply",
"has_fderiv_at_filter",
"has_fderiv_within_at",
"nhds_within_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_of_subsingleton [h : subsingleton E] (f : E → F) (x : E) :
has_fderiv_at f (0 : E →L[𝕜] F) x | begin
rw [← has_fderiv_within_at_univ, subsingleton_univ.eq_singleton_of_mem (mem_univ x)],
exact has_fderiv_within_at_singleton f x
end | lemma | has_fderiv_at_of_subsingleton | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at",
"has_fderiv_within_at_singleton",
"has_fderiv_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_empty : differentiable_on 𝕜 f ∅ | λ x, false.elim | lemma | differentiable_on_empty | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_singleton : differentiable_on 𝕜 f {x} | forall_eq.2 (has_fderiv_within_at_singleton f x).differentiable_within_at | lemma | differentiable_on_singleton | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_on",
"differentiable_within_at",
"has_fderiv_within_at_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
set.subsingleton.differentiable_on (hs : s.subsingleton) : differentiable_on 𝕜 f s | hs.induction_on differentiable_on_empty (λ x, differentiable_on_singleton) | lemma | set.subsingleton.differentiable_on | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"differentiable_on",
"differentiable_on_empty",
"differentiable_on_singleton"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_zero_of_eventually_const
(c : F) (hf : f =ᶠ[𝓝 x] (λ y, c)) :
has_fderiv_at f (0 : E →L[𝕜] F) x | (has_fderiv_at_const _ _).congr_of_eventually_eq hf | lemma | has_fderiv_at_zero_of_eventually_const | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"has_fderiv_at",
"has_fderiv_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_fderiv_subset : support (fderiv 𝕜 f) ⊆ tsupport f | begin
intros x,
rw [← not_imp_not, not_mem_tsupport_iff_eventually_eq, nmem_support],
exact λ hx, (hx.fderiv_eq.trans $ fderiv_const_apply 0),
end | lemma | support_fderiv_subset | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv",
"fderiv_const_apply",
"not_imp_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsupport_fderiv_subset : tsupport (fderiv 𝕜 f) ⊆ tsupport f | closure_minimal (support_fderiv_subset 𝕜) is_closed_closure | lemma | tsupport_fderiv_subset | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"closure_minimal",
"fderiv",
"is_closed_closure",
"support_fderiv_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.fderiv (hf : has_compact_support f) : has_compact_support (fderiv 𝕜 f) | hf.mono' $ support_fderiv_subset 𝕜 | lemma | has_compact_support.fderiv | analysis.calculus.fderiv | src/analysis/calculus/fderiv/basic.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.calculus.tangent_cone",
"analysis.normed_space.bounded_linear_maps"
] | [
"fderiv",
"support_fderiv_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map.has_strict_fderiv_at (h : is_bounded_bilinear_map 𝕜 b) (p : E × F) :
has_strict_fderiv_at b (h.deriv p) p | begin
rw has_strict_fderiv_at,
set T := (E × F) × (E × F),
have : (λ q : T, b (q.1 - q.2)) =o[𝓝 (p, p)] (λ q : T, ‖q.1 - q.2‖ * 1),
{ refine (h.is_O'.comp_tendsto le_top).trans_is_o _,
simp only [(∘)],
refine (is_O_refl (λ q : T, ‖q.1 - q.2‖) _).mul_is_o (is_o.norm_left $ (is_o_one_iff _).2 _),
rw ... | lemma | is_bounded_bilinear_map.has_strict_fderiv_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/bilinear.lean | [
"analysis.calculus.fderiv.prod"
] | [
"continuous_at_snd",
"has_strict_fderiv_at",
"is_bounded_bilinear_map",
"is_bounded_bilinear_map_deriv_coe",
"le_top",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map.has_fderiv_at (h : is_bounded_bilinear_map 𝕜 b) (p : E × F) :
has_fderiv_at b (h.deriv p) p | (h.has_strict_fderiv_at p).has_fderiv_at | lemma | is_bounded_bilinear_map.has_fderiv_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/bilinear.lean | [
"analysis.calculus.fderiv.prod"
] | [
"has_fderiv_at",
"is_bounded_bilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map.has_fderiv_within_at (h : is_bounded_bilinear_map 𝕜 b) (p : E × F) :
has_fderiv_within_at b (h.deriv p) u p | (h.has_fderiv_at p).has_fderiv_within_at | lemma | is_bounded_bilinear_map.has_fderiv_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/bilinear.lean | [
"analysis.calculus.fderiv.prod"
] | [
"has_fderiv_within_at",
"is_bounded_bilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map.differentiable_at (h : is_bounded_bilinear_map 𝕜 b) (p : E × F) :
differentiable_at 𝕜 b p | (h.has_fderiv_at p).differentiable_at | lemma | is_bounded_bilinear_map.differentiable_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/bilinear.lean | [
"analysis.calculus.fderiv.prod"
] | [
"differentiable_at",
"is_bounded_bilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map.differentiable_within_at (h : is_bounded_bilinear_map 𝕜 b)
(p : E × F) :
differentiable_within_at 𝕜 b u p | (h.differentiable_at p).differentiable_within_at | lemma | is_bounded_bilinear_map.differentiable_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/bilinear.lean | [
"analysis.calculus.fderiv.prod"
] | [
"differentiable_within_at",
"is_bounded_bilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map.fderiv (h : is_bounded_bilinear_map 𝕜 b) (p : E × F) :
fderiv 𝕜 b p = h.deriv p | has_fderiv_at.fderiv (h.has_fderiv_at p) | lemma | is_bounded_bilinear_map.fderiv | analysis.calculus.fderiv | src/analysis/calculus/fderiv/bilinear.lean | [
"analysis.calculus.fderiv.prod"
] | [
"fderiv",
"has_fderiv_at.fderiv",
"is_bounded_bilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map.fderiv_within (h : is_bounded_bilinear_map 𝕜 b) (p : E × F)
(hxs : unique_diff_within_at 𝕜 u p) : fderiv_within 𝕜 b u p = h.deriv p | begin
rw differentiable_at.fderiv_within (h.differentiable_at p) hxs,
exact h.fderiv p
end | lemma | is_bounded_bilinear_map.fderiv_within | analysis.calculus.fderiv | src/analysis/calculus/fderiv/bilinear.lean | [
"analysis.calculus.fderiv.prod"
] | [
"differentiable_at.fderiv_within",
"fderiv_within",
"is_bounded_bilinear_map",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map.differentiable (h : is_bounded_bilinear_map 𝕜 b) :
differentiable 𝕜 b | λx, h.differentiable_at x | lemma | is_bounded_bilinear_map.differentiable | analysis.calculus.fderiv | src/analysis/calculus/fderiv/bilinear.lean | [
"analysis.calculus.fderiv.prod"
] | [
"differentiable",
"is_bounded_bilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_bilinear_map.differentiable_on (h : is_bounded_bilinear_map 𝕜 b) :
differentiable_on 𝕜 b u | h.differentiable.differentiable_on | lemma | is_bounded_bilinear_map.differentiable_on | analysis.calculus.fderiv | src/analysis/calculus/fderiv/bilinear.lean | [
"analysis.calculus.fderiv.prod"
] | [
"differentiable_on",
"is_bounded_bilinear_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.has_fderiv_within_at_of_bilinear
{f : G' → E} {g : G' → F} {f' : G' →L[𝕜] E} {g' : G' →L[𝕜] F} {x : G'} {s : set G'}
(hf : has_fderiv_within_at f f' s x) (hg : has_fderiv_within_at g g' s x) :
has_fderiv_within_at (λ y, B (f y) (g y)) (B.precompR G' (f x) g' + B.precompL G' f' (g x)) s x | (B.is_bounded_bilinear_map.has_fderiv_at (f x, g x)).comp_has_fderiv_within_at x (hf.prod hg) | lemma | continuous_linear_map.has_fderiv_within_at_of_bilinear | analysis.calculus.fderiv | src/analysis/calculus/fderiv/bilinear.lean | [
"analysis.calculus.fderiv.prod"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.has_fderiv_at_of_bilinear
{f : G' → E} {g : G' → F} {f' : G' →L[𝕜] E} {g' : G' →L[𝕜] F} {x : G'}
(hf : has_fderiv_at f f' x) (hg : has_fderiv_at g g' x) :
has_fderiv_at (λ y, B (f y) (g y)) (B.precompR G' (f x) g' + B.precompL G' f' (g x)) x | (B.is_bounded_bilinear_map.has_fderiv_at (f x, g x)).comp x (hf.prod hg) | lemma | continuous_linear_map.has_fderiv_at_of_bilinear | analysis.calculus.fderiv | src/analysis/calculus/fderiv/bilinear.lean | [
"analysis.calculus.fderiv.prod"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.fderiv_within_of_bilinear
{f : G' → E} {g : G' → F} {x : G'} {s : set G'}
(hf : differentiable_within_at 𝕜 f s x) (hg : differentiable_within_at 𝕜 g s x)
(hs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (λ y, B (f y) (g y)) s x =
(B.precompR G' (f x) (fderiv_within 𝕜 g s x) + ... | (B.has_fderiv_within_at_of_bilinear hf.has_fderiv_within_at hg.has_fderiv_within_at).fderiv_within
hs | lemma | continuous_linear_map.fderiv_within_of_bilinear | analysis.calculus.fderiv | src/analysis/calculus/fderiv/bilinear.lean | [
"analysis.calculus.fderiv.prod"
] | [
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.fderiv_of_bilinear {f : G' → E} {g : G' → F} {x : G'}
(hf : differentiable_at 𝕜 f x) (hg : differentiable_at 𝕜 g x) :
fderiv 𝕜 (λ y, B (f y) (g y)) x =
(B.precompR G' (f x) (fderiv 𝕜 g x) + B.precompL G' (fderiv 𝕜 f x) (g x)) | (B.has_fderiv_at_of_bilinear hf.has_fderiv_at hg.has_fderiv_at).fderiv | lemma | continuous_linear_map.fderiv_of_bilinear | analysis.calculus.fderiv | src/analysis/calculus/fderiv/bilinear.lean | [
"analysis.calculus.fderiv.prod"
] | [
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.comp {g : F → G} {g' : F →L[𝕜] G} {L' : filter F}
(hg : has_fderiv_at_filter g g' (f x) L')
(hf : has_fderiv_at_filter f f' x L) (hL : tendsto f L L') :
has_fderiv_at_filter (g ∘ f) (g'.comp f') x L | let eq₁ := (g'.is_O_comp _ _).trans_is_o hf in
let eq₂ := (hg.comp_tendsto hL).trans_is_O hf.is_O_sub in
by { refine eq₂.triangle (eq₁.congr_left (λ x', _)), simp } | theorem | has_fderiv_at_filter.comp | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"filter",
"has_fderiv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.comp {g : F → G} {g' : F →L[𝕜] G} {t : set F}
(hg : has_fderiv_within_at g g' t (f x)) (hf : has_fderiv_within_at f f' s x)
(hst : maps_to f s t) :
has_fderiv_within_at (g ∘ f) (g'.comp f') s x | hg.comp x hf $ hf.continuous_within_at.tendsto_nhds_within hst | theorem | has_fderiv_within_at.comp | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.comp_has_fderiv_within_at {g : F → G} {g' : F →L[𝕜] G}
(hg : has_fderiv_at g g' (f x)) (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (g ∘ f) (g'.comp f') s x | hg.comp x hf hf.continuous_within_at | theorem | has_fderiv_at.comp_has_fderiv_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_fderiv_at",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.comp_of_mem {g : F → G} {g' : F →L[𝕜] G} {t : set F}
(hg : has_fderiv_within_at g g' t (f x)) (hf : has_fderiv_within_at f f' s x)
(hst : tendsto f (𝓝[s] x) (𝓝[t] f x)) :
has_fderiv_within_at (g ∘ f) (g'.comp f') s x | has_fderiv_at_filter.comp x hg hf hst | theorem | has_fderiv_within_at.comp_of_mem | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_fderiv_at_filter.comp",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.comp {g : F → G} {g' : F →L[𝕜] G}
(hg : has_fderiv_at g g' (f x)) (hf : has_fderiv_at f f' x) :
has_fderiv_at (g ∘ f) (g'.comp f') x | hg.comp x hf hf.continuous_at | theorem | has_fderiv_at.comp | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_fderiv_at"
] | The chain rule. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_within_at.comp {g : F → G} {t : set F}
(hg : differentiable_within_at 𝕜 g t (f x)) (hf : differentiable_within_at 𝕜 f s x)
(h : maps_to f s t) : differentiable_within_at 𝕜 (g ∘ f) s x | (hg.has_fderiv_within_at.comp x hf.has_fderiv_within_at h).differentiable_within_at | lemma | differentiable_within_at.comp | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.comp' {g : F → G} {t : set F}
(hg : differentiable_within_at 𝕜 g t (f x)) (hf : differentiable_within_at 𝕜 f s x) :
differentiable_within_at 𝕜 (g ∘ f) (s ∩ f⁻¹' t) x | hg.comp x (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _) | lemma | differentiable_within_at.comp' | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.comp {g : F → G}
(hg : differentiable_at 𝕜 g (f x)) (hf : differentiable_at 𝕜 f x) :
differentiable_at 𝕜 (g ∘ f) x | (hg.has_fderiv_at.comp x hf.has_fderiv_at).differentiable_at | lemma | differentiable_at.comp | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.comp_differentiable_within_at {g : F → G}
(hg : differentiable_at 𝕜 g (f x)) (hf : differentiable_within_at 𝕜 f s x) :
differentiable_within_at 𝕜 (g ∘ f) s x | hg.differentiable_within_at.comp x hf (maps_to_univ _ _) | lemma | differentiable_at.comp_differentiable_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_at",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within.comp {g : F → G} {t : set F}
(hg : differentiable_within_at 𝕜 g t (f x)) (hf : differentiable_within_at 𝕜 f s x)
(h : maps_to f s t) (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (g ∘ f) s x = (fderiv_within 𝕜 g t (f x)).comp (fderiv_within 𝕜 f s x) | (hg.has_fderiv_within_at.comp x (hf.has_fderiv_within_at) h).fderiv_within hxs | lemma | fderiv_within.comp | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_fderiv_within {g : F → G} {f : E → F} {x : E} {y : F} {s : set E} {t : set F}
(hg : differentiable_within_at 𝕜 g t y) (hf : differentiable_within_at 𝕜 f s x)
(h : maps_to f s t) (hxs : unique_diff_within_at 𝕜 s x) (hy : f x = y) (v : E) :
fderiv_within 𝕜 g t y (fderiv_within 𝕜 f s x v) = fderiv... | by { subst y, rw [fderiv_within.comp x hg hf h hxs], refl } | lemma | fderiv_within_fderiv_within | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_within_at",
"fderiv_within",
"fderiv_within.comp",
"unique_diff_within_at"
] | A version of `fderiv_within.comp` that is useful to rewrite the composition of two derivatives
into a single derivative. This version always applies, but creates a new side-goal `f x = y`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fderiv_within.comp₃ {g' : G → G'} {g : F → G} {t : set F} {u : set G} {y : F} {y' : G}
(hg' : differentiable_within_at 𝕜 g' u y') (hg : differentiable_within_at 𝕜 g t y)
(hf : differentiable_within_at 𝕜 f s x)
(h2g : maps_to g t u) (h2f : maps_to f s t)
(h3g : g y = y') (h3f : f x = y) (hxs : unique_diff_wit... | begin
substs h3g h3f,
exact (hg'.has_fderiv_within_at.comp x
(hg.has_fderiv_within_at.comp x (hf.has_fderiv_within_at) h2f) $ h2g.comp h2f).fderiv_within hxs
end | lemma | fderiv_within.comp₃ | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | Ternary version of `fderiv_within.comp`, with equality assumptions of basepoints added, in
order to apply more easily as a rewrite from right-to-left. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fderiv.comp {g : F → G}
(hg : differentiable_at 𝕜 g (f x)) (hf : differentiable_at 𝕜 f x) :
fderiv 𝕜 (g ∘ f) x = (fderiv 𝕜 g (f x)).comp (fderiv 𝕜 f x) | (hg.has_fderiv_at.comp x hf.has_fderiv_at).fderiv | lemma | fderiv.comp | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv.comp_fderiv_within {g : F → G}
(hg : differentiable_at 𝕜 g (f x)) (hf : differentiable_within_at 𝕜 f s x)
(hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (g ∘ f) s x = (fderiv 𝕜 g (f x)).comp (fderiv_within 𝕜 f s x) | (hg.has_fderiv_at.comp_has_fderiv_within_at x hf.has_fderiv_within_at).fderiv_within hxs | lemma | fderiv.comp_fderiv_within | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_at",
"differentiable_within_at",
"fderiv",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.comp {g : F → G} {t : set F}
(hg : differentiable_on 𝕜 g t) (hf : differentiable_on 𝕜 f s) (st : maps_to f s t) :
differentiable_on 𝕜 (g ∘ f) s | λx hx, differentiable_within_at.comp x (hg (f x) (st hx)) (hf x hx) st | lemma | differentiable_on.comp | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_on",
"differentiable_within_at.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.comp {g : F → G} (hg : differentiable 𝕜 g) (hf : differentiable 𝕜 f) :
differentiable 𝕜 (g ∘ f) | λx, differentiable_at.comp x (hg (f x)) (hf x) | lemma | differentiable.comp | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable",
"differentiable_at.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.comp_differentiable_on {g : F → G} (hg : differentiable 𝕜 g)
(hf : differentiable_on 𝕜 f s) :
differentiable_on 𝕜 (g ∘ f) s | hg.differentiable_on.comp hf (maps_to_univ _ _) | lemma | differentiable.comp_differentiable_on | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable",
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.comp {g : F → G} {g' : F →L[𝕜] G}
(hg : has_strict_fderiv_at g g' (f x)) (hf : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ x, g (f x)) (g'.comp f') x | ((hg.comp_tendsto (hf.continuous_at.prod_map' hf.continuous_at)).trans_is_O hf.is_O_sub).triangle $
by simpa only [g'.map_sub, f'.coe_comp'] using (g'.is_O_comp _ _).trans_is_o hf | lemma | has_strict_fderiv_at.comp | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_strict_fderiv_at"
] | The chain rule for derivatives in the sense of strict differentiability. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable.iterate {f : E → E} (hf : differentiable 𝕜 f) (n : ℕ) :
differentiable 𝕜 (f^[n]) | nat.rec_on n differentiable_id (λ n ihn, ihn.comp hf) | lemma | differentiable.iterate | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable",
"differentiable_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.iterate {f : E → E} (hf : differentiable_on 𝕜 f s)
(hs : maps_to f s s) (n : ℕ) :
differentiable_on 𝕜 (f^[n]) s | nat.rec_on n differentiable_on_id (λ n ihn, ihn.comp hf hs) | lemma | differentiable_on.iterate | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_on",
"differentiable_on_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.iterate {f : E → E} {f' : E →L[𝕜] E}
(hf : has_fderiv_at_filter f f' x L) (hL : tendsto f L L) (hx : f x = x) (n : ℕ) :
has_fderiv_at_filter (f^[n]) (f'^n) x L | begin
induction n with n ihn,
{ exact has_fderiv_at_filter_id x L },
{ rw [function.iterate_succ, pow_succ'],
rw ← hx at ihn,
exact ihn.comp x hf hL }
end | lemma | has_fderiv_at_filter.iterate | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"function.iterate_succ",
"has_fderiv_at_filter",
"has_fderiv_at_filter_id",
"pow_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.iterate {f : E → E} {f' : E →L[𝕜] E}
(hf : has_fderiv_at f f' x) (hx : f x = x) (n : ℕ) :
has_fderiv_at (f^[n]) (f'^n) x | begin
refine hf.iterate _ hx n,
convert hf.continuous_at,
exact hx.symm
end | lemma | has_fderiv_at.iterate | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.iterate {f : E → E} {f' : E →L[𝕜] E}
(hf : has_fderiv_within_at f f' s x) (hx : f x = x) (hs : maps_to f s s) (n : ℕ) :
has_fderiv_within_at (f^[n]) (f'^n) s x | begin
refine hf.iterate _ hx n,
convert tendsto_inf.2 ⟨hf.continuous_within_at, _⟩,
exacts [hx.symm, (tendsto_principal_principal.2 hs).mono_left inf_le_right]
end | lemma | has_fderiv_within_at.iterate | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"has_fderiv_within_at",
"inf_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.iterate {f : E → E} {f' : E →L[𝕜] E}
(hf : has_strict_fderiv_at f f' x) (hx : f x = x) (n : ℕ) :
has_strict_fderiv_at (f^[n]) (f'^n) x | begin
induction n with n ihn,
{ exact has_strict_fderiv_at_id x },
{ rw [function.iterate_succ, pow_succ'],
rw ← hx at ihn,
exact ihn.comp x hf }
end | lemma | has_strict_fderiv_at.iterate | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"function.iterate_succ",
"has_strict_fderiv_at",
"has_strict_fderiv_at_id",
"pow_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.iterate {f : E → E} (hf : differentiable_at 𝕜 f x)
(hx : f x = x) (n : ℕ) :
differentiable_at 𝕜 (f^[n]) x | (hf.has_fderiv_at.iterate hx n).differentiable_at | lemma | differentiable_at.iterate | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.iterate {f : E → E} (hf : differentiable_within_at 𝕜 f s x)
(hx : f x = x) (hs : maps_to f s s) (n : ℕ) :
differentiable_within_at 𝕜 (f^[n]) s x | (hf.has_fderiv_within_at.iterate hx hs n).differentiable_within_at | lemma | differentiable_within_at.iterate | analysis.calculus.fderiv | src/analysis/calculus/fderiv/comp.lean | [
"analysis.calculus.fderiv.basic"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at :
has_strict_fderiv_at iso (iso : E →L[𝕜] F) x | iso.to_continuous_linear_map.has_strict_fderiv_at | lemma | continuous_linear_equiv.has_strict_fderiv_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_strict_fderiv_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at :
has_fderiv_within_at iso (iso : E →L[𝕜] F) s x | iso.to_continuous_linear_map.has_fderiv_within_at | lemma | continuous_linear_equiv.has_fderiv_within_at | analysis.calculus.fderiv | src/analysis/calculus/fderiv/equiv.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_within_at",
"iso"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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