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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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exists_has_deriv_within_at_eq_of_le_of_ge
(hab : a ≤ b) (hf : ∀ x ∈ (Icc a b), has_deriv_within_at f (f' x) (Icc a b) x)
{m : ℝ} (hma : f' a ≤ m) (hmb : m ≤ f' b) :
m ∈ f' '' Icc a b | (ord_connected_Icc.image_has_deriv_within_at hf).out
(mem_image_of_mem _ (left_mem_Icc.2 hab)) (mem_image_of_mem _ (right_mem_Icc.2 hab)) ⟨hma, hmb⟩ | theorem | exists_has_deriv_within_at_eq_of_le_of_ge | analysis.calculus | src/analysis/calculus/darboux.lean | [
"analysis.calculus.local_extr"
] | [
"has_deriv_within_at"
] | **Darboux's theorem**: if `a ≤ b` and `f' b ≤ m ≤ f' a`, then `f' c = m` for some
`c ∈ [a, b]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_within_at_forall_lt_or_forall_gt_of_forall_ne {s : set ℝ} (hs : convex ℝ s)
(hf : ∀ x ∈ s, has_deriv_within_at f (f' x) s x) {m : ℝ} (hf' : ∀ x ∈ s, f' x ≠ m) :
(∀ x ∈ s, f' x < m) ∨ (∀ x ∈ s, m < f' x) | begin
contrapose! hf',
rcases hf' with ⟨⟨b, hb, hmb⟩, ⟨a, ha, hma⟩⟩,
exact (hs.ord_connected.image_has_deriv_within_at hf).out (mem_image_of_mem f' ha)
(mem_image_of_mem f' hb) ⟨hma, hmb⟩
end | theorem | has_deriv_within_at_forall_lt_or_forall_gt_of_forall_ne | analysis.calculus | src/analysis/calculus/darboux.lean | [
"analysis.calculus.local_extr"
] | [
"convex",
"has_deriv_within_at"
] | If the derivative of a function is never equal to `m`, then either
it is always greater than `m`, or it is always less than `m`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
diff_cont_on_cl (f : E → F) (s : set E) : Prop | (differentiable_on : differentiable_on 𝕜 f s)
(continuous_on : continuous_on f (closure s)) | structure | diff_cont_on_cl | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"closure",
"continuous_on",
"differentiable_on"
] | A predicate saying that a function is differentiable on a set and is continuous on its
closure. This is a common assumption in complex analysis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_on.diff_cont_on_cl (h : differentiable_on 𝕜 f (closure s)) :
diff_cont_on_cl 𝕜 f s | ⟨h.mono subset_closure, h.continuous_on⟩ | lemma | differentiable_on.diff_cont_on_cl | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"closure",
"diff_cont_on_cl",
"differentiable_on",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.diff_cont_on_cl (h : differentiable 𝕜 f) : diff_cont_on_cl 𝕜 f s | ⟨h.differentiable_on, h.continuous.continuous_on⟩ | lemma | differentiable.diff_cont_on_cl | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl",
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.diff_cont_on_cl_iff (hs : is_closed s) :
diff_cont_on_cl 𝕜 f s ↔ differentiable_on 𝕜 f s | ⟨λ h, h.differentiable_on, λ h, ⟨h, hs.closure_eq.symm ▸ h.continuous_on⟩⟩ | lemma | is_closed.diff_cont_on_cl_iff | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl",
"differentiable_on",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diff_cont_on_cl_univ : diff_cont_on_cl 𝕜 f univ ↔ differentiable 𝕜 f | is_closed_univ.diff_cont_on_cl_iff.trans differentiable_on_univ | lemma | diff_cont_on_cl_univ | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl",
"differentiable",
"differentiable_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diff_cont_on_cl_const {c : F} :
diff_cont_on_cl 𝕜 (λ x : E, c) s | ⟨differentiable_on_const c, continuous_on_const⟩ | lemma | diff_cont_on_cl_const | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp {g : G → E} {t : set G} (hf : diff_cont_on_cl 𝕜 f s) (hg : diff_cont_on_cl 𝕜 g t)
(h : maps_to g t s) :
diff_cont_on_cl 𝕜 (f ∘ g) t | ⟨hf.1.comp hg.1 h, hf.2.comp hg.2 $ h.closure_of_continuous_on hg.2⟩ | lemma | diff_cont_on_cl.comp | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_ball [normed_space ℝ E] {x : E} {r : ℝ} (h : diff_cont_on_cl 𝕜 f (ball x r)) :
continuous_on f (closed_ball x r) | begin
rcases eq_or_ne r 0 with rfl|hr,
{ rw closed_ball_zero,
exact continuous_on_singleton f x },
{ rw ← closure_ball x hr,
exact h.continuous_on }
end | lemma | diff_cont_on_cl.continuous_on_ball | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"closure_ball",
"continuous_on",
"continuous_on_singleton",
"diff_cont_on_cl",
"eq_or_ne",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_ball {x : E} {r : ℝ} (hd : differentiable_on 𝕜 f (ball x r))
(hc : continuous_on f (closed_ball x r)) : diff_cont_on_cl 𝕜 f (ball x r) | ⟨hd, hc.mono $ closure_ball_subset_closed_ball⟩ | lemma | diff_cont_on_cl.mk_ball | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"continuous_on",
"diff_cont_on_cl",
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at (h : diff_cont_on_cl 𝕜 f s) (hs : is_open s) (hx : x ∈ s) :
differentiable_at 𝕜 f x | h.differentiable_on.differentiable_at $ hs.mem_nhds hx | lemma | diff_cont_on_cl.differentiable_at | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl",
"differentiable_at",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at' (h : diff_cont_on_cl 𝕜 f s) (hx : s ∈ 𝓝 x) :
differentiable_at 𝕜 f x | h.differentiable_on.differentiable_at hx | lemma | diff_cont_on_cl.differentiable_at' | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mono (h : diff_cont_on_cl 𝕜 f s) (ht : t ⊆ s) : diff_cont_on_cl 𝕜 f t | ⟨h.differentiable_on.mono ht, h.continuous_on.mono (closure_mono ht)⟩ | lemma | diff_cont_on_cl.mono | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"closure_mono",
"diff_cont_on_cl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add (hf : diff_cont_on_cl 𝕜 f s) (hg : diff_cont_on_cl 𝕜 g s) :
diff_cont_on_cl 𝕜 (f + g) s | ⟨hf.1.add hg.1, hf.2.add hg.2⟩ | lemma | diff_cont_on_cl.add | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_const (hf : diff_cont_on_cl 𝕜 f s) (c : F) :
diff_cont_on_cl 𝕜 (λ x, f x + c) s | hf.add diff_cont_on_cl_const | lemma | diff_cont_on_cl.add_const | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl",
"diff_cont_on_cl_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_add (hf : diff_cont_on_cl 𝕜 f s) (c : F) :
diff_cont_on_cl 𝕜 (λ x, c + f x) s | diff_cont_on_cl_const.add hf | lemma | diff_cont_on_cl.const_add | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg (hf : diff_cont_on_cl 𝕜 f s) : diff_cont_on_cl 𝕜 (-f) s | ⟨hf.1.neg, hf.2.neg⟩ | lemma | diff_cont_on_cl.neg | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub (hf : diff_cont_on_cl 𝕜 f s) (hg : diff_cont_on_cl 𝕜 g s) :
diff_cont_on_cl 𝕜 (f - g) s | ⟨hf.1.sub hg.1, hf.2.sub hg.2⟩ | lemma | diff_cont_on_cl.sub | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_const (hf : diff_cont_on_cl 𝕜 f s) (c : F) : diff_cont_on_cl 𝕜 (λ x, f x - c) s | hf.sub diff_cont_on_cl_const | lemma | diff_cont_on_cl.sub_const | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl",
"diff_cont_on_cl_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_sub (hf : diff_cont_on_cl 𝕜 f s) (c : F) : diff_cont_on_cl 𝕜 (λ x, c - f x) s | diff_cont_on_cl_const.sub hf | lemma | diff_cont_on_cl.const_sub | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
const_smul {R : Type*} [semiring R] [module R F] [smul_comm_class 𝕜 R F]
[has_continuous_const_smul R F] (hf : diff_cont_on_cl 𝕜 f s) (c : R) :
diff_cont_on_cl 𝕜 (c • f) s | ⟨hf.1.const_smul c, hf.2.const_smul c⟩ | lemma | diff_cont_on_cl.const_smul | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl",
"has_continuous_const_smul",
"module",
"semiring",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul {𝕜' : Type*} [nontrivially_normed_field 𝕜'] [normed_algebra 𝕜 𝕜']
[normed_space 𝕜' F] [is_scalar_tower 𝕜 𝕜' F] {c : E → 𝕜'} {f : E → F} {s : set E}
(hc : diff_cont_on_cl 𝕜 c s) (hf : diff_cont_on_cl 𝕜 f s) :
diff_cont_on_cl 𝕜 (λ x, c x • f x) s | ⟨hc.1.smul hf.1, hc.2.smul hf.2⟩ | lemma | diff_cont_on_cl.smul | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl",
"is_scalar_tower",
"nontrivially_normed_field",
"normed_algebra",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
smul_const {𝕜' : Type*} [nontrivially_normed_field 𝕜'] [normed_algebra 𝕜 𝕜']
[normed_space 𝕜' F] [is_scalar_tower 𝕜 𝕜' F] {c : E → 𝕜'} {s : set E}
(hc : diff_cont_on_cl 𝕜 c s) (y : F) :
diff_cont_on_cl 𝕜 (λ x, c x • y) s | hc.smul diff_cont_on_cl_const | lemma | diff_cont_on_cl.smul_const | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl",
"diff_cont_on_cl_const",
"is_scalar_tower",
"nontrivially_normed_field",
"normed_algebra",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv {f : E → 𝕜} (hf : diff_cont_on_cl 𝕜 f s) (h₀ : ∀ x ∈ closure s, f x ≠ 0) :
diff_cont_on_cl 𝕜 f⁻¹ s | ⟨differentiable_on_inv.comp hf.1 $ λ x hx, h₀ _ (subset_closure hx), hf.2.inv₀ h₀⟩ | lemma | diff_cont_on_cl.inv | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"closure",
"diff_cont_on_cl",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.comp_diff_cont_on_cl {g : G → E} {t : set G}
(hf : differentiable 𝕜 f) (hg : diff_cont_on_cl 𝕜 g t) :
diff_cont_on_cl 𝕜 (f ∘ g) t | hf.diff_cont_on_cl.comp hg (maps_to_image _ _) | lemma | differentiable.comp_diff_cont_on_cl | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl",
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.diff_cont_on_cl_ball {U : set E} {c : E} {R : ℝ}
(hf : differentiable_on 𝕜 f U) (hc : closed_ball c R ⊆ U) :
diff_cont_on_cl 𝕜 f (ball c R) | diff_cont_on_cl.mk_ball (hf.mono (ball_subset_closed_ball.trans hc)) (hf.continuous_on.mono hc) | lemma | differentiable_on.diff_cont_on_cl_ball | analysis.calculus | src/analysis/calculus/diff_cont_on_cl.lean | [
"analysis.calculus.deriv.inv"
] | [
"diff_cont_on_cl",
"diff_cont_on_cl.mk_ball",
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dslope (f : 𝕜 → E) (a : 𝕜) : 𝕜 → E | update (slope f a) a (deriv f a) | def | dslope | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"deriv",
"slope",
"update"
] | `dslope f a b` is defined as `slope f a b = (b - a)⁻¹ • (f b - f a)` for `a ≠ b` and
`deriv f a` for `a = b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dslope_same (f : 𝕜 → E) (a : 𝕜) : dslope f a a = deriv f a | update_same _ _ _ | lemma | dslope_same | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"deriv",
"dslope",
"update_same"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dslope_of_ne (f : 𝕜 → E) (h : b ≠ a) : dslope f a b = slope f a b | update_noteq h _ _ | lemma | dslope_of_ne | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"dslope",
"slope",
"update_noteq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.dslope_comp {F : Type*} [normed_add_comm_group F] [normed_space 𝕜 F]
(f : E →L[𝕜] F) (g : 𝕜 → E) (a b : 𝕜) (H : a = b → differentiable_at 𝕜 g a) :
dslope (f ∘ g) a b = f (dslope g a b) | begin
rcases eq_or_ne b a with rfl|hne,
{ simp only [dslope_same],
exact (f.has_fderiv_at.comp_has_deriv_at b (H rfl).has_deriv_at).deriv },
{ simpa only [dslope_of_ne _ hne] using f.to_linear_map.slope_comp g a b }
end | lemma | continuous_linear_map.dslope_comp | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"deriv",
"differentiable_at",
"dslope",
"dslope_of_ne",
"dslope_same",
"eq_or_ne",
"has_deriv_at",
"normed_add_comm_group",
"normed_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_on_dslope_slope (f : 𝕜 → E) (a : 𝕜) : eq_on (dslope f a) (slope f a) {a}ᶜ | λ b, dslope_of_ne f | lemma | eq_on_dslope_slope | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"dslope",
"dslope_of_ne",
"slope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dslope_eventually_eq_slope_of_ne (f : 𝕜 → E) (h : b ≠ a) : dslope f a =ᶠ[𝓝 b] slope f a | (eq_on_dslope_slope f a).eventually_eq_of_mem (is_open_ne.mem_nhds h) | lemma | dslope_eventually_eq_slope_of_ne | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"dslope",
"eq_on_dslope_slope",
"slope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dslope_eventually_eq_slope_punctured_nhds (f : 𝕜 → E) : dslope f a =ᶠ[𝓝[≠] a] slope f a | (eq_on_dslope_slope f a).eventually_eq_of_mem self_mem_nhds_within | lemma | dslope_eventually_eq_slope_punctured_nhds | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"dslope",
"eq_on_dslope_slope",
"self_mem_nhds_within",
"slope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_smul_dslope (f : 𝕜 → E) (a b : 𝕜) : (b - a) • dslope f a b = f b - f a | by rcases eq_or_ne b a with rfl | hne; simp [dslope_of_ne, *] | lemma | sub_smul_dslope | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"dslope",
"dslope_of_ne",
"eq_or_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dslope_sub_smul_of_ne (f : 𝕜 → E) (h : b ≠ a) : dslope (λ x, (x - a) • f x) a b = f b | by rw [dslope_of_ne _ h, slope_sub_smul _ h.symm] | lemma | dslope_sub_smul_of_ne | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"dslope",
"dslope_of_ne",
"slope_sub_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_on_dslope_sub_smul (f : 𝕜 → E) (a : 𝕜) : eq_on (dslope (λ x, (x - a) • f x) a) f {a}ᶜ | λ b, dslope_sub_smul_of_ne f | lemma | eq_on_dslope_sub_smul | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"dslope",
"dslope_sub_smul_of_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dslope_sub_smul [decidable_eq 𝕜] (f : 𝕜 → E) (a : 𝕜) :
dslope (λ x, (x - a) • f x) a = update f a (deriv (λ x, (x - a) • f x) a) | eq_update_iff.2 ⟨dslope_same _ _, eq_on_dslope_sub_smul f a⟩ | lemma | dslope_sub_smul | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"deriv",
"dslope",
"eq_on_dslope_sub_smul",
"update"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_dslope_same : continuous_at (dslope f a) a ↔ differentiable_at 𝕜 f a | by simp only [dslope, continuous_at_update_same, ← has_deriv_at_deriv_iff,
has_deriv_at_iff_tendsto_slope] | lemma | continuous_at_dslope_same | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"continuous_at",
"continuous_at_update_same",
"differentiable_at",
"dslope",
"has_deriv_at_deriv_iff",
"has_deriv_at_iff_tendsto_slope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at.of_dslope (h : continuous_within_at (dslope f a) s b) :
continuous_within_at f s b | have continuous_within_at (λ x, (x - a) • dslope f a x + f a) s b,
from ((continuous_within_at_id.sub continuous_within_at_const).smul h).add
continuous_within_at_const,
by simpa only [sub_smul_dslope, sub_add_cancel] using this | lemma | continuous_within_at.of_dslope | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"continuous_within_at",
"continuous_within_at_const",
"dslope",
"sub_smul_dslope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at.of_dslope (h : continuous_at (dslope f a) b) : continuous_at f b | (continuous_within_at_univ _ _).1 h.continuous_within_at.of_dslope | lemma | continuous_at.of_dslope | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"continuous_at",
"continuous_within_at_univ",
"dslope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on.of_dslope (h : continuous_on (dslope f a) s) : continuous_on f s | λ x hx, (h x hx).of_dslope | lemma | continuous_on.of_dslope | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"continuous_on",
"dslope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_within_at_dslope_of_ne (h : b ≠ a) :
continuous_within_at (dslope f a) s b ↔ continuous_within_at f s b | begin
refine ⟨continuous_within_at.of_dslope, λ hc, _⟩,
simp only [dslope, continuous_within_at_update_of_ne h],
exact ((continuous_within_at_id.sub continuous_within_at_const).inv₀
(sub_ne_zero.2 h)).smul (hc.sub continuous_within_at_const)
end | lemma | continuous_within_at_dslope_of_ne | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"continuous_within_at",
"continuous_within_at_const",
"continuous_within_at_update_of_ne",
"dslope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_dslope_of_ne (h : b ≠ a) : continuous_at (dslope f a) b ↔ continuous_at f b | by simp only [← continuous_within_at_univ, continuous_within_at_dslope_of_ne h] | lemma | continuous_at_dslope_of_ne | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"continuous_at",
"continuous_within_at_dslope_of_ne",
"continuous_within_at_univ",
"dslope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_dslope (h : s ∈ 𝓝 a) :
continuous_on (dslope f a) s ↔ continuous_on f s ∧ differentiable_at 𝕜 f a | begin
refine ⟨λ hc, ⟨hc.of_dslope, continuous_at_dslope_same.1 $ hc.continuous_at h⟩, _⟩,
rintro ⟨hc, hd⟩ x hx,
rcases eq_or_ne x a with rfl | hne,
exacts [(continuous_at_dslope_same.2 hd).continuous_within_at,
(continuous_within_at_dslope_of_ne hne).2 (hc x hx)]
end | lemma | continuous_on_dslope | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"continuous_on",
"continuous_within_at",
"continuous_within_at_dslope_of_ne",
"differentiable_at",
"dslope",
"eq_or_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.of_dslope (h : differentiable_within_at 𝕜 (dslope f a) s b) :
differentiable_within_at 𝕜 f s b | by simpa only [id, sub_smul_dslope f a, sub_add_cancel]
using ((differentiable_within_at_id.sub_const a).smul h).add_const (f a) | lemma | differentiable_within_at.of_dslope | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"differentiable_within_at",
"dslope",
"sub_smul_dslope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.of_dslope (h : differentiable_at 𝕜 (dslope f a) b) :
differentiable_at 𝕜 f b | differentiable_within_at_univ.1 h.differentiable_within_at.of_dslope | lemma | differentiable_at.of_dslope | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"differentiable_at",
"dslope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.of_dslope (h : differentiable_on 𝕜 (dslope f a) s) :
differentiable_on 𝕜 f s | λ x hx, (h x hx).of_dslope | lemma | differentiable_on.of_dslope | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"differentiable_on",
"dslope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_dslope_of_ne (h : b ≠ a) :
differentiable_within_at 𝕜 (dslope f a) s b ↔ differentiable_within_at 𝕜 f s b | begin
refine ⟨differentiable_within_at.of_dslope, λ hd, _⟩,
refine (((differentiable_within_at_id.sub_const a).inv
(sub_ne_zero.2 h)).smul (hd.sub_const (f a))).congr_of_eventually_eq _ (dslope_of_ne _ h),
refine (eq_on_dslope_slope _ _).eventually_eq_of_mem _,
exact mem_nhds_within_of_mem_nhds (is_open_ne.... | lemma | differentiable_within_at_dslope_of_ne | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"differentiable_within_at",
"dslope",
"dslope_of_ne",
"eq_on_dslope_slope",
"mem_nhds_within_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_dslope_of_nmem (h : a ∉ s) :
differentiable_on 𝕜 (dslope f a) s ↔ differentiable_on 𝕜 f s | forall_congr $ λ x, forall_congr $ λ hx, differentiable_within_at_dslope_of_ne $
ne_of_mem_of_not_mem hx h | lemma | differentiable_on_dslope_of_nmem | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"differentiable_on",
"differentiable_within_at_dslope_of_ne",
"dslope",
"ne_of_mem_of_not_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_dslope_of_ne (h : b ≠ a) :
differentiable_at 𝕜 (dslope f a) b ↔ differentiable_at 𝕜 f b | by simp only [← differentiable_within_at_univ,
differentiable_within_at_dslope_of_ne h] | lemma | differentiable_at_dslope_of_ne | analysis.calculus | src/analysis/calculus/dslope.lean | [
"analysis.calculus.deriv.slope",
"analysis.calculus.deriv.inv"
] | [
"differentiable_at",
"differentiable_within_at_dslope_of_ne",
"differentiable_within_at_univ",
"dslope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_boundary_of_tendsto_fderiv {f : E → F} {s : set E} {x : E} {f' : E →L[ℝ] F}
(f_diff : differentiable_on ℝ f s) (s_conv : convex ℝ s) (s_open : is_open s)
(f_cont : ∀y ∈ closure s, continuous_within_at f s y)
(h : tendsto (λy, fderiv ℝ f y) (𝓝[s] x) (𝓝 f')) :
has_fderiv_within_at f f' (closure s)... | begin
classical,
-- one can assume without loss of generality that `x` belongs to the closure of `s`, as the
-- statement is empty otherwise
by_cases hx : x ∉ closure s,
{ rw ← closure_closure at hx, exact has_fderiv_within_at_of_not_mem_closure hx },
push_neg at hx,
rw [has_fderiv_within_at, has_fderiv_a... | theorem | has_fderiv_at_boundary_of_tendsto_fderiv | analysis.calculus | src/analysis/calculus/extend_deriv.lean | [
"analysis.calculus.mean_value"
] | [
"asymptotics.is_o_iff",
"bound",
"closure",
"closure_closure",
"closure_mono",
"closure_prod_eq",
"continuous_within_at",
"continuous_within_at.closure_le",
"continuous_within_at.mono",
"convex",
"convex_ball",
"differentiable_at",
"differentiable_at.fderiv_within",
"differentiable_on",
... | If a function `f` is differentiable in a convex open set and continuous on its closure, and its
derivative converges to a limit `f'` at a point on the boundary, then `f` is differentiable there
with derivative `f'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_interval_left_endpoint_of_tendsto_deriv {s : set ℝ} {e : E} {a : ℝ} {f : ℝ → E}
(f_diff : differentiable_on ℝ f s) (f_lim : continuous_within_at f s a)
(hs : s ∈ 𝓝[>] a)
(f_lim' : tendsto (λx, deriv f x) (𝓝[>] a) (𝓝 e)) :
has_deriv_within_at f e (Ici a) a | begin
/- This is a specialization of `has_fderiv_at_boundary_of_tendsto_fderiv`. To be in the setting of
this theorem, we need to work on an open interval with closure contained in `s ∪ {a}`, that we
call `t = (a, b)`. Then, we check all the assumptions of this theorem and we apply it. -/
obtain ⟨b, ab : a < b,... | lemma | has_deriv_at_interval_left_endpoint_of_tendsto_deriv | analysis.calculus | src/analysis/calculus/extend_deriv.lean | [
"analysis.calculus.mean_value"
] | [
"Icc_mem_nhds_within_Ici",
"closure",
"closure_Ioo",
"continuous_within_at",
"convex",
"convex_Ioo",
"deriv",
"differentiable_on",
"fderiv",
"has_deriv_within_at",
"has_deriv_within_at_iff_has_fderiv_within_at",
"has_fderiv_at_boundary_of_tendsto_fderiv",
"is_bounded_bilinear_map",
"is_bou... | If a function is differentiable on the right of a point `a : ℝ`, continuous at `a`, and
its derivative also converges at `a`, then `f` is differentiable on the right at `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_interval_right_endpoint_of_tendsto_deriv {s : set ℝ} {e : E} {a : ℝ} {f : ℝ → E}
(f_diff : differentiable_on ℝ f s) (f_lim : continuous_within_at f s a)
(hs : s ∈ 𝓝[<] a)
(f_lim' : tendsto (λx, deriv f x) (𝓝[<] a) (𝓝 e)) :
has_deriv_within_at f e (Iic a) a | begin
/- This is a specialization of `has_fderiv_at_boundary_of_differentiable`. To be in the setting of
this theorem, we need to work on an open interval with closure contained in `s ∪ {a}`, that we
call `t = (b, a)`. Then, we check all the assumptions of this theorem and we apply it. -/
obtain ⟨b, ba, sab⟩ : ... | lemma | has_deriv_at_interval_right_endpoint_of_tendsto_deriv | analysis.calculus | src/analysis/calculus/extend_deriv.lean | [
"analysis.calculus.mean_value"
] | [
"Icc_mem_nhds_within_Iic",
"closure",
"closure_Ioo",
"continuous_within_at",
"convex",
"convex_Ioo",
"deriv",
"differentiable_on",
"fderiv",
"has_deriv_within_at",
"has_deriv_within_at_iff_has_fderiv_within_at",
"has_fderiv_at_boundary_of_tendsto_fderiv",
"is_bounded_bilinear_map",
"is_bou... | If a function is differentiable on the left of a point `a : ℝ`, continuous at `a`, and
its derivative also converges at `a`, then `f` is differentiable on the left at `a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_of_has_deriv_at_of_ne {f g : ℝ → E} {x : ℝ}
(f_diff : ∀ y ≠ x, has_deriv_at f (g y) y)
(hf : continuous_at f x) (hg : continuous_at g x) :
has_deriv_at f (g x) x | begin
have A : has_deriv_within_at f (g x) (Ici x) x,
{ have diff : differentiable_on ℝ f (Ioi x) :=
λy hy, (f_diff y (ne_of_gt hy)).differentiable_at.differentiable_within_at,
-- next line is the nontrivial bit of this proof, appealing to differentiability
-- extension results.
apply has_deriv_at... | lemma | has_deriv_at_of_has_deriv_at_of_ne | analysis.calculus | src/analysis/calculus/extend_deriv.lean | [
"analysis.calculus.mean_value"
] | [
"continuous_at",
"differentiable_at.differentiable_within_at",
"differentiable_on",
"has_deriv_at",
"has_deriv_at_interval_left_endpoint_of_tendsto_deriv",
"has_deriv_at_interval_right_endpoint_of_tendsto_deriv",
"has_deriv_within_at",
"self_mem_nhds_within"
] | If a real function `f` has a derivative `g` everywhere but at a point, and `f` and `g` are
continuous at this point, then `g` is also the derivative of `f` at this point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_of_has_deriv_at_of_ne' {f g : ℝ → E} {x : ℝ}
(f_diff : ∀ y ≠ x, has_deriv_at f (g y) y)
(hf : continuous_at f x) (hg : continuous_at g x) (y : ℝ) :
has_deriv_at f (g y) y | begin
rcases eq_or_ne y x with rfl|hne,
{ exact has_deriv_at_of_has_deriv_at_of_ne f_diff hf hg },
{ exact f_diff y hne }
end | lemma | has_deriv_at_of_has_deriv_at_of_ne' | analysis.calculus | src/analysis/calculus/extend_deriv.lean | [
"analysis.calculus.mean_value"
] | [
"continuous_at",
"eq_or_ne",
"has_deriv_at",
"has_deriv_at_of_has_deriv_at_of_ne"
] | If a real function `f` has a derivative `g` everywhere but at a point, and `f` and `g` are
continuous at this point, then `g` is the derivative of `f` everywhere. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_at.has_strict_fderiv_at (h : has_fpower_series_at f p x) :
has_strict_fderiv_at f (continuous_multilinear_curry_fin1 𝕜 E F (p 1)) x | begin
refine h.is_O_image_sub_norm_mul_norm_sub.trans_is_o (is_o.of_norm_right _),
refine is_o_iff_exists_eq_mul.2 ⟨λ y, ‖y - (x, x)‖, _, eventually_eq.rfl⟩,
refine (continuous_id.sub continuous_const).norm.tendsto' _ _ _,
rw [_root_.id, sub_self, norm_zero]
end | lemma | has_fpower_series_at.has_strict_fderiv_at | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"continuous_const",
"continuous_multilinear_curry_fin1",
"has_fpower_series_at",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at.has_fderiv_at (h : has_fpower_series_at f p x) :
has_fderiv_at f (continuous_multilinear_curry_fin1 𝕜 E F (p 1)) x | h.has_strict_fderiv_at.has_fderiv_at | lemma | has_fpower_series_at.has_fderiv_at | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"continuous_multilinear_curry_fin1",
"has_fderiv_at",
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at.differentiable_at (h : has_fpower_series_at f p x) :
differentiable_at 𝕜 f x | h.has_fderiv_at.differentiable_at | lemma | has_fpower_series_at.differentiable_at | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"differentiable_at",
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
analytic_at.differentiable_at : analytic_at 𝕜 f x → differentiable_at 𝕜 f x | | ⟨p, hp⟩ := hp.differentiable_at | lemma | analytic_at.differentiable_at | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"analytic_at",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
analytic_at.differentiable_within_at (h : analytic_at 𝕜 f x) :
differentiable_within_at 𝕜 f s x | h.differentiable_at.differentiable_within_at | lemma | analytic_at.differentiable_within_at | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"analytic_at",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at.fderiv_eq (h : has_fpower_series_at f p x) :
fderiv 𝕜 f x = continuous_multilinear_curry_fin1 𝕜 E F (p 1) | h.has_fderiv_at.fderiv | lemma | has_fpower_series_at.fderiv_eq | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"continuous_multilinear_curry_fin1",
"fderiv",
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball.differentiable_on [complete_space F]
(h : has_fpower_series_on_ball f p x r) :
differentiable_on 𝕜 f (emetric.ball x r) | λ y hy, (h.analytic_at_of_mem hy).differentiable_within_at | lemma | has_fpower_series_on_ball.differentiable_on | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"complete_space",
"differentiable_on",
"differentiable_within_at",
"emetric.ball",
"has_fpower_series_on_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
analytic_on.differentiable_on (h : analytic_on 𝕜 f s) :
differentiable_on 𝕜 f s | λ y hy, (h y hy).differentiable_within_at | lemma | analytic_on.differentiable_on | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"analytic_on",
"differentiable_on",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball.has_fderiv_at [complete_space F]
(h : has_fpower_series_on_ball f p x r) {y : E} (hy : (‖y‖₊ : ℝ≥0∞) < r) :
has_fderiv_at f (continuous_multilinear_curry_fin1 𝕜 E F (p.change_origin y 1)) (x + y) | (h.change_origin hy).has_fpower_series_at.has_fderiv_at | lemma | has_fpower_series_on_ball.has_fderiv_at | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"complete_space",
"continuous_multilinear_curry_fin1",
"has_fderiv_at",
"has_fpower_series_at.has_fderiv_at",
"has_fpower_series_on_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball.fderiv_eq [complete_space F]
(h : has_fpower_series_on_ball f p x r) {y : E} (hy : (‖y‖₊ : ℝ≥0∞) < r) :
fderiv 𝕜 f (x + y) = continuous_multilinear_curry_fin1 𝕜 E F (p.change_origin y 1) | (h.has_fderiv_at hy).fderiv | lemma | has_fpower_series_on_ball.fderiv_eq | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"complete_space",
"continuous_multilinear_curry_fin1",
"fderiv",
"has_fpower_series_on_ball"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_on_ball.fderiv [complete_space F]
(h : has_fpower_series_on_ball f p x r) :
has_fpower_series_on_ball (fderiv 𝕜 f)
((continuous_multilinear_curry_fin1 𝕜 E F : (E [×1]→L[𝕜] F) →L[𝕜] (E →L[𝕜] F))
.comp_formal_multilinear_series (p.change_origin_series 1)) x r | begin
suffices A : has_fpower_series_on_ball
(λ z, continuous_multilinear_curry_fin1 𝕜 E F (p.change_origin (z - x) 1))
((continuous_multilinear_curry_fin1 𝕜 E F : (E [×1]→L[𝕜] F) →L[𝕜] (E →L[𝕜] F))
.comp_formal_multilinear_series (p.change_origin_series 1)) x r,
{ apply A.congr,
assume z... | lemma | has_fpower_series_on_ball.fderiv | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"complete_space",
"continuous_multilinear_curry_fin1",
"emetric.mem_ball",
"fderiv",
"has_fpower_series_on_ball"
] | If a function has a power series on a ball, then so does its derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
analytic_on.fderiv [complete_space F] (h : analytic_on 𝕜 f s) :
analytic_on 𝕜 (fderiv 𝕜 f) s | begin
assume y hy,
rcases h y hy with ⟨p, r, hp⟩,
exact hp.fderiv.analytic_at,
end | lemma | analytic_on.fderiv | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"analytic_on",
"complete_space",
"fderiv"
] | If a function is analytic on a set `s`, so is its Fréchet derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
analytic_on.iterated_fderiv [complete_space F] (h : analytic_on 𝕜 f s) (n : ℕ) :
analytic_on 𝕜 (iterated_fderiv 𝕜 n f) s | begin
induction n with n IH,
{ rw iterated_fderiv_zero_eq_comp,
exact ((continuous_multilinear_curry_fin0 𝕜 E F).symm : F →L[𝕜] (E [×0]→L[𝕜] F))
.comp_analytic_on h },
{ rw iterated_fderiv_succ_eq_comp_left,
apply (continuous_multilinear_curry_left_equiv 𝕜 (λ (i : fin (n + 1)), E) F)
.to_c... | lemma | analytic_on.iterated_fderiv | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"analytic_on",
"complete_space",
"continuous_multilinear_curry_fin0",
"continuous_multilinear_curry_left_equiv",
"iterated_fderiv",
"iterated_fderiv_succ_eq_comp_left",
"iterated_fderiv_zero_eq_comp"
] | If a function is analytic on a set `s`, so are its successive Fréchet derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
analytic_on.cont_diff_on [complete_space F] (h : analytic_on 𝕜 f s) {n : ℕ∞} :
cont_diff_on 𝕜 n f s | begin
let t := {x | analytic_at 𝕜 f x},
suffices : cont_diff_on 𝕜 n f t, from this.mono h,
have H : analytic_on 𝕜 f t := λ x hx, hx,
have t_open : is_open t := is_open_analytic_at 𝕜 f,
apply cont_diff_on_of_continuous_on_differentiable_on,
{ assume m hm,
apply (H.iterated_fderiv m).continuous_on.con... | lemma | analytic_on.cont_diff_on | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"analytic_at",
"analytic_on",
"complete_space",
"cont_diff_on",
"cont_diff_on_of_continuous_on_differentiable_on",
"continuous_on.congr",
"differentiable_on.congr",
"is_open",
"is_open_analytic_at",
"iterated_fderiv_within_of_is_open"
] | An analytic function is infinitely differentiable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fpower_series_at.has_strict_deriv_at (h : has_fpower_series_at f p x) :
has_strict_deriv_at f (p 1 (λ _, 1)) x | h.has_strict_fderiv_at.has_strict_deriv_at | lemma | has_fpower_series_at.has_strict_deriv_at | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"has_fpower_series_at",
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at.has_deriv_at (h : has_fpower_series_at f p x) :
has_deriv_at f (p 1 (λ _, 1)) x | h.has_strict_deriv_at.has_deriv_at | lemma | has_fpower_series_at.has_deriv_at | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"has_deriv_at",
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fpower_series_at.deriv (h : has_fpower_series_at f p x) :
deriv f x = p 1 (λ _, 1) | h.has_deriv_at.deriv | lemma | has_fpower_series_at.deriv | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"deriv",
"has_fpower_series_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
analytic_on.deriv [complete_space F] (h : analytic_on 𝕜 f s) :
analytic_on 𝕜 (deriv f) s | (continuous_linear_map.apply 𝕜 F (1 : 𝕜)).comp_analytic_on h.fderiv | lemma | analytic_on.deriv | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"analytic_on",
"complete_space",
"continuous_linear_map.apply",
"deriv"
] | If a function is analytic on a set `s`, so is its derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
analytic_on.iterated_deriv [complete_space F] (h : analytic_on 𝕜 f s) (n : ℕ) :
analytic_on 𝕜 (deriv^[n] f) s | begin
induction n with n IH,
{ exact h },
{ simpa only [function.iterate_succ', function.comp_app] using IH.deriv }
end | lemma | analytic_on.iterated_deriv | analysis.calculus | src/analysis/calculus/fderiv_analytic.lean | [
"analysis.analytic.basic",
"analysis.calculus.deriv.basic",
"analysis.calculus.cont_diff_def"
] | [
"analytic_on",
"complete_space",
"deriv",
"function.iterate_succ'"
] | If a function is analytic on a set `s`, so are its successive derivatives. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
measurable_apply₂ [measurable_space E] [opens_measurable_space E]
[second_countable_topology E] [second_countable_topology (E →L[𝕜] F)]
[measurable_space F] [borel_space F] :
measurable (λ p : (E →L[𝕜] F) × E, p.1 p.2) | is_bounded_bilinear_map_apply.continuous.measurable | lemma | continuous_linear_map.measurable_apply₂ | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"borel_space",
"measurable",
"measurable_space",
"opens_measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
A (f : E → F) (L : E →L[𝕜] F) (r ε : ℝ) : set E | {x | ∃ r' ∈ Ioc (r/2) r, ∀ y z ∈ ball x r', ‖f z - f y - L (z-y)‖ ≤ ε * r} | def | fderiv_measurable_aux.A | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [] | The set `A f L r ε` is the set of points `x` around which the function `f` is well approximated
at scale `r` by the linear map `L`, up to an error `ε`. We tweak the definition to make sure that
this is an open set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
B (f : E → F) (K : set (E →L[𝕜] F)) (r s ε : ℝ) : set E | ⋃ (L ∈ K), (A f L r ε) ∩ (A f L s ε) | def | fderiv_measurable_aux.B | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [] | The set `B f K r s ε` is the set of points `x` around which there exists a continuous linear map
`L` belonging to `K` (a given set of continuous linear maps) that approximates well the
function `f` (up to an error `ε`), simultaneously at scales `r` and `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
D (f : E → F) (K : set (E →L[𝕜] F)) : set E | ⋂ (e : ℕ), ⋃ (n : ℕ), ⋂ (p ≥ n) (q ≥ n), B f K ((1/2) ^ p) ((1/2) ^ q) ((1/2) ^ e) | def | fderiv_measurable_aux.D | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [] | The set `D f K` is a complicated set constructed using countable intersections and unions. Its
main use is that, when `K` is complete, it is exactly the set of points where `f` is differentiable,
with a derivative in `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_A (L : E →L[𝕜] F) (r ε : ℝ) : is_open (A f L r ε) | begin
rw metric.is_open_iff,
rintros x ⟨r', r'_mem, hr'⟩,
obtain ⟨s, s_gt, s_lt⟩ : ∃ (s : ℝ), r / 2 < s ∧ s < r' := exists_between r'_mem.1,
have : s ∈ Ioc (r/2) r := ⟨s_gt, le_of_lt (s_lt.trans_le r'_mem.2)⟩,
refine ⟨r' - s, by linarith, λ x' hx', ⟨s, this, _⟩⟩,
have B : ball x' s ⊆ ball x r' := ball_subse... | lemma | fderiv_measurable_aux.is_open_A | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"exists_between",
"is_open",
"metric.is_open_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_B {K : set (E →L[𝕜] F)} {r s ε : ℝ} : is_open (B f K r s ε) | by simp [B, is_open_Union, is_open.inter, is_open_A] | lemma | fderiv_measurable_aux.is_open_B | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"is_open",
"is_open.inter",
"is_open_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
A_mono (L : E →L[𝕜] F) (r : ℝ) {ε δ : ℝ} (h : ε ≤ δ) :
A f L r ε ⊆ A f L r δ | begin
rintros x ⟨r', r'r, hr'⟩,
refine ⟨r', r'r, λ y hy z hz, (hr' y hy z hz).trans (mul_le_mul_of_nonneg_right h _)⟩,
linarith [mem_ball.1 hy, r'r.2, @dist_nonneg _ _ y x],
end | lemma | fderiv_measurable_aux.A_mono | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"dist_nonneg",
"mul_le_mul_of_nonneg_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_of_mem_A {r ε : ℝ} {L : E →L[𝕜] F} {x : E} (hx : x ∈ A f L r ε)
{y z : E} (hy : y ∈ closed_ball x (r/2)) (hz : z ∈ closed_ball x (r/2)) :
‖f z - f y - L (z-y)‖ ≤ ε * r | begin
rcases hx with ⟨r', r'mem, hr'⟩,
exact hr' _ ((mem_closed_ball.1 hy).trans_lt r'mem.1) _ ((mem_closed_ball.1 hz).trans_lt r'mem.1)
end | lemma | fderiv_measurable_aux.le_of_mem_A | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_A_of_differentiable {ε : ℝ} (hε : 0 < ε) {x : E} (hx : differentiable_at 𝕜 f x) :
∃ R > 0, ∀ r ∈ Ioo (0 : ℝ) R, x ∈ A f (fderiv 𝕜 f x) r ε | begin
have := hx.has_fderiv_at,
simp only [has_fderiv_at, has_fderiv_at_filter, is_o_iff] at this,
rcases eventually_nhds_iff_ball.1 (this (half_pos hε)) with ⟨R, R_pos, hR⟩,
refine ⟨R, R_pos, λ r hr, _⟩,
have : r ∈ Ioc (r/2) r := ⟨half_lt_self hr.1, le_rfl⟩,
refine ⟨r, this, λ y hy z hz, _⟩,
calc ‖f z -... | lemma | fderiv_measurable_aux.mem_A_of_differentiable | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"continuous_linear_map.map_sub",
"differentiable_at",
"fderiv",
"half_pos",
"has_fderiv_at",
"has_fderiv_at_filter",
"mul_le_mul_of_nonneg_left",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sub_le_of_mem_A {c : 𝕜} (hc : 1 < ‖c‖)
{r ε : ℝ} (hε : 0 < ε) (hr : 0 < r) {x : E} {L₁ L₂ : E →L[𝕜] F}
(h₁ : x ∈ A f L₁ r ε) (h₂ : x ∈ A f L₂ r ε) : ‖L₁ - L₂‖ ≤ 4 * ‖c‖ * ε | begin
have : 0 ≤ 4 * ‖c‖ * ε :=
mul_nonneg (mul_nonneg (by norm_num : (0 : ℝ) ≤ 4) (norm_nonneg _)) hε.le,
refine op_norm_le_of_shell (half_pos hr) this hc _,
assume y ley ylt,
rw [div_div,
div_le_iff' (mul_pos (by norm_num : (0 : ℝ) < 2) (zero_lt_one.trans hc))] at ley,
calc ‖(L₁ - L₂) y‖
=... | lemma | fderiv_measurable_aux.norm_sub_le_of_mem_A | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"dist_self",
"div_div",
"div_le_iff'",
"half_pos",
"mul_le_mul_of_nonneg_left",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_set_subset_D : {x | differentiable_at 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} ⊆ D f K | begin
assume x hx,
rw [D, mem_Inter],
assume e,
have : (0 : ℝ) < (1/2) ^ e := pow_pos (by norm_num) _,
rcases mem_A_of_differentiable this hx.1 with ⟨R, R_pos, hR⟩,
obtain ⟨n, hn⟩ : ∃ (n : ℕ), (1/2) ^ n < R :=
exists_pow_lt_of_lt_one R_pos (by norm_num : (1 : ℝ)/2 < 1),
simp only [mem_Union, mem_Inter... | lemma | fderiv_measurable_aux.differentiable_set_subset_D | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"differentiable_at",
"exists_pow_lt_of_lt_one",
"fderiv",
"pow_le_pow_of_le_one",
"pow_pos"
] | Easy inclusion: a differentiability point with derivative in `K` belongs to `D f K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
D_subset_differentiable_set {K : set (E →L[𝕜] F)} (hK : is_complete K) :
D f K ⊆ {x | differentiable_at 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} | begin
have P : ∀ {n : ℕ}, (0 : ℝ) < (1/2) ^ n := pow_pos (by norm_num),
rcases normed_field.exists_one_lt_norm 𝕜 with ⟨c, hc⟩,
have cpos : 0 < ‖c‖ := lt_trans zero_lt_one hc,
assume x hx,
have : ∀ (e : ℕ), ∃ (n : ℕ), ∀ p q, n ≤ p → n ≤ q → ∃ L ∈ K,
x ∈ A f L ((1/2) ^ p) ((1/2) ^ e) ∩ A f L ((1/2) ^ q) ((... | lemma | fderiv_measurable_aux.D_subset_differentiable_set | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"cauchy_seq",
"cauchy_seq_tendsto_of_is_complete",
"differentiable_at",
"dist_comm",
"dist_self",
"div_nonneg",
"div_pos",
"exists_nat_pow_near_of_lt_one",
"exists_pow_lt_of_lt_one",
"fderiv",
"ge_iff_le",
"has_fderiv_at",
"has_fderiv_at_iff_is_o_nhds_zero",
"is_complete",
"le_of_tendsto... | Harder inclusion: at a point in `D f K`, the function `f` has a derivative, in `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
differentiable_set_eq_D (hK : is_complete K) :
{x | differentiable_at 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} = D f K | subset.antisymm (differentiable_set_subset_D _) (D_subset_differentiable_set hK) | theorem | fderiv_measurable_aux.differentiable_set_eq_D | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"differentiable_at",
"fderiv",
"is_complete"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_set_of_differentiable_at_of_is_complete
{K : set (E →L[𝕜] F)} (hK : is_complete K) :
measurable_set {x | differentiable_at 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} | by simp [differentiable_set_eq_D K hK, D, is_open_B.measurable_set, measurable_set.Inter,
measurable_set.Union] | theorem | measurable_set_of_differentiable_at_of_is_complete | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"differentiable_at",
"fderiv",
"is_complete",
"measurable_set",
"measurable_set.Inter",
"measurable_set.Union"
] | The set of differentiability points of a function, with derivative in a given complete set,
is Borel-measurable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
measurable_set_of_differentiable_at :
measurable_set {x | differentiable_at 𝕜 f x} | begin
have : is_complete (univ : set (E →L[𝕜] F)) := complete_univ,
convert measurable_set_of_differentiable_at_of_is_complete 𝕜 f this,
simp
end | theorem | measurable_set_of_differentiable_at | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"complete_univ",
"differentiable_at",
"is_complete",
"measurable_set",
"measurable_set_of_differentiable_at_of_is_complete"
] | The set of differentiability points of a function taking values in a complete space is
Borel-measurable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
measurable_fderiv : measurable (fderiv 𝕜 f) | begin
refine measurable_of_is_closed (λ s hs, _),
have : fderiv 𝕜 f ⁻¹' s = {x | differentiable_at 𝕜 f x ∧ fderiv 𝕜 f x ∈ s} ∪
({x | ¬differentiable_at 𝕜 f x} ∩ {x | (0 : E →L[𝕜] F) ∈ s}) :=
set.ext (λ x, mem_preimage.trans fderiv_mem_iff),
rw this,
exact (measurable_set_of_differentiable_at_of_is_... | lemma | measurable_fderiv | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"differentiable_at",
"fderiv",
"fderiv_mem_iff",
"measurable",
"measurable_of_is_closed",
"measurable_set.const",
"measurable_set_of_differentiable_at",
"measurable_set_of_differentiable_at_of_is_complete",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_fderiv_apply_const [measurable_space F] [borel_space F] (y : E) :
measurable (λ x, fderiv 𝕜 f x y) | (continuous_linear_map.measurable_apply y).comp (measurable_fderiv 𝕜 f) | lemma | measurable_fderiv_apply_const | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"borel_space",
"continuous_linear_map.measurable_apply",
"fderiv",
"measurable",
"measurable_fderiv",
"measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
measurable_deriv [measurable_space 𝕜] [opens_measurable_space 𝕜]
[measurable_space F] [borel_space F] (f : 𝕜 → F) : measurable (deriv f) | by simpa only [fderiv_deriv] using measurable_fderiv_apply_const 𝕜 f 1 | lemma | measurable_deriv | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"borel_space",
"deriv",
"fderiv_deriv",
"measurable",
"measurable_fderiv_apply_const",
"measurable_space",
"opens_measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strongly_measurable_deriv [measurable_space 𝕜] [opens_measurable_space 𝕜]
[second_countable_topology F] (f : 𝕜 → F) :
strongly_measurable (deriv f) | by { borelize F, exact (measurable_deriv f).strongly_measurable } | lemma | strongly_measurable_deriv | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"deriv",
"measurable_deriv",
"measurable_space",
"opens_measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_measurable_deriv [measurable_space 𝕜] [opens_measurable_space 𝕜] [measurable_space F]
[borel_space F] (f : 𝕜 → F) (μ : measure 𝕜) : ae_measurable (deriv f) μ | (measurable_deriv f).ae_measurable | lemma | ae_measurable_deriv | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"ae_measurable",
"borel_space",
"deriv",
"measurable_deriv",
"measurable_space",
"opens_measurable_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ae_strongly_measurable_deriv [measurable_space 𝕜] [opens_measurable_space 𝕜]
[second_countable_topology F] (f : 𝕜 → F) (μ : measure 𝕜) :
ae_strongly_measurable (deriv f) μ | (strongly_measurable_deriv f).ae_strongly_measurable | lemma | ae_strongly_measurable_deriv | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"deriv",
"measurable_space",
"opens_measurable_space",
"strongly_measurable_deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
A (f : ℝ → F) (L : F) (r ε : ℝ) : set ℝ | {x | ∃ r' ∈ Ioc (r/2) r, ∀ y z ∈ Icc x (x + r'), ‖f z - f y - (z-y) • L‖ ≤ ε * r} | def | right_deriv_measurable_aux.A | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [] | The set `A f L r ε` is the set of points `x` around which the function `f` is well approximated
at scale `r` by the linear map `h ↦ h • L`, up to an error `ε`. We tweak the definition to
make sure that this is open on the right. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
B (f : ℝ → F) (K : set F) (r s ε : ℝ) : set ℝ | ⋃ (L ∈ K), (A f L r ε) ∩ (A f L s ε) | def | right_deriv_measurable_aux.B | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [] | The set `B f K r s ε` is the set of points `x` around which there exists a vector
`L` belonging to `K` (a given set of vectors) such that `h • L` approximates well `f (x + h)`
(up to an error `ε`), simultaneously at scales `r` and `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
D (f : ℝ → F) (K : set F) : set ℝ | ⋂ (e : ℕ), ⋃ (n : ℕ), ⋂ (p ≥ n) (q ≥ n), B f K ((1/2) ^ p) ((1/2) ^ q) ((1/2) ^ e) | def | right_deriv_measurable_aux.D | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [] | The set `D f K` is a complicated set constructed using countable intersections and unions. Its
main use is that, when `K` is complete, it is exactly the set of points where `f` is differentiable,
with a derivative in `K`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
A_mem_nhds_within_Ioi {L : F} {r ε x : ℝ} (hx : x ∈ A f L r ε) :
A f L r ε ∈ 𝓝[>] x | begin
rcases hx with ⟨r', rr', hr'⟩,
rw mem_nhds_within_Ioi_iff_exists_Ioo_subset,
obtain ⟨s, s_gt, s_lt⟩ : ∃ (s : ℝ), r / 2 < s ∧ s < r' := exists_between rr'.1,
have : s ∈ Ioc (r/2) r := ⟨s_gt, le_of_lt (s_lt.trans_le rr'.2)⟩,
refine ⟨x + r' - s, by { simp only [mem_Ioi], linarith }, λ x' hx', ⟨s, this, _⟩⟩... | lemma | right_deriv_measurable_aux.A_mem_nhds_within_Ioi | analysis.calculus | src/analysis/calculus/fderiv_measurable.lean | [
"analysis.calculus.deriv.basic",
"measure_theory.constructions.borel_space.continuous_linear_map",
"measure_theory.function.strongly_measurable.basic"
] | [
"exists_between",
"mem_nhds_within_Ioi_iff_exists_Ioo_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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