statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
cont_diff_at.restrict_scalars (h : cont_diff_at 𝕜' n f x) :
cont_diff_at 𝕜 n f x | cont_diff_within_at_univ.1 $ h.cont_diff_within_at.restrict_scalars _ | lemma | cont_diff_at.restrict_scalars | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.restrict_scalars (h : cont_diff 𝕜' n f) :
cont_diff 𝕜 n f | cont_diff_iff_cont_diff_at.2 $ λ x, h.cont_diff_at.restrict_scalars _ | lemma | cont_diff.restrict_scalars | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_linear_map.norm_iterated_fderiv_within_le_of_bilinear_aux
{Du Eu Fu Gu : Type u}
[normed_add_comm_group Du] [normed_space 𝕜 Du]
[normed_add_comm_group Eu] [normed_space 𝕜 Eu]
[normed_add_comm_group Fu] [normed_space 𝕜 Fu]
[normed_add_comm_group Gu] [normed_space 𝕜 Gu]
(B : Eu →L[𝕜] Fu →L[𝕜]... | begin
/- We argue by induction on `n`. The bound is trivial for `n = 0`. For `n + 1`, we write
the `(n+1)`-th derivative as the `n`-th derivative of the derivative `B f g' + B f' g`, and apply
the inductive assumption to each of those two terms. For this induction to make sense,
the spaces of linear maps that a... | lemma | continuous_linear_map.norm_iterated_fderiv_within_le_of_bilinear_aux | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"algebra_map.coe_one",
"cont_diff_on",
"enat.coe_one",
"fderiv_within",
"finset.range",
"finset.range_one",
"finset.sum_choose_succ_mul",
"iterated_fderiv_within",
"iterated_fderiv_within_add_apply'",
"iterated_fderiv_within_congr",
"iterated_fderiv_within_succ_eq_comp_right",
"linear_isometry... | Bounding the norm of the iterated derivative of `B (f x) (g x)` within a set in terms of the
iterated derivatives of `f` and `g` when `B` is bilinear. This lemma is an auxiliary version
assuming all spaces live in the same universe, to enable an induction. Use instead
`continuous_linear_map.norm_iterated_fderiv_within_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.norm_iterated_fderiv_within_le_of_bilinear
(B : E →L[𝕜] F →L[𝕜] G) {f : D → E} {g : D → F} {N : ℕ∞} {s : set D} {x : D}
(hf : cont_diff_on 𝕜 N f s) (hg : cont_diff_on 𝕜 N g s) (hs : unique_diff_on 𝕜 s) (hx : x ∈ s)
{n : ℕ} (hn : (n : ℕ∞) ≤ N) :
‖iterated_fderiv_within 𝕜 n (λ y, B (f ... | begin
/- We reduce the bound to the case where all spaces live in the same universe (in which we
already have proved the result), by using linear isometries between the spaces and their `ulift`
to a common universe. These linear isometries preserve the norm of the iterated derivative. -/
let Du : Type (max uD u... | lemma | continuous_linear_map.norm_iterated_fderiv_within_le_of_bilinear | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on",
"continuous_linear_map.coe_comp'",
"continuous_linear_map.compL",
"continuous_linear_map.compL_apply",
"continuous_linear_map.flip_apply",
"continuous_linear_map.le_op_norm",
"continuous_linear_map.op_norm_le_bound",
"finset.range",
"linear_isometry_equiv.apply_symm_apply",
"linear... | Bounding the norm of the iterated derivative of `B (f x) (g x)` within a set in terms of the
iterated derivatives of `f` and `g` when `B` is bilinear:
`‖D^n (x ↦ B (f x) (g x))‖ ≤ ‖B‖ ∑_{k ≤ n} n.choose k ‖D^k f‖ ‖D^{n-k} g‖` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.norm_iterated_fderiv_le_of_bilinear
(B : E →L[𝕜] F →L[𝕜] G) {f : D → E} {g : D → F} {N : ℕ∞}
(hf : cont_diff 𝕜 N f) (hg : cont_diff 𝕜 N g) (x : D)
{n : ℕ} (hn : (n : ℕ∞) ≤ N) :
‖iterated_fderiv 𝕜 n (λ y, B (f y) (g y)) x‖
≤ ‖B‖ * ∑ i in finset.range (n+1), (n.choose i : ℝ)
*... | begin
simp_rw [← iterated_fderiv_within_univ],
exact B.norm_iterated_fderiv_within_le_of_bilinear hf.cont_diff_on hg.cont_diff_on
unique_diff_on_univ (mem_univ x) hn,
end | lemma | continuous_linear_map.norm_iterated_fderiv_le_of_bilinear | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"finset.range",
"iterated_fderiv_within_univ",
"unique_diff_on_univ"
] | Bounding the norm of the iterated derivative of `B (f x) (g x)` in terms of the
iterated derivatives of `f` and `g` when `B` is bilinear:
`‖D^n (x ↦ B (f x) (g x))‖ ≤ ‖B‖ ∑_{k ≤ n} n.choose k ‖D^k f‖ ‖D^{n-k} g‖` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.norm_iterated_fderiv_within_le_of_bilinear_of_le_one
(B : E →L[𝕜] F →L[𝕜] G) {f : D → E} {g : D → F} {N : ℕ∞} {s : set D} {x : D}
(hf : cont_diff_on 𝕜 N f s) (hg : cont_diff_on 𝕜 N g s) (hs : unique_diff_on 𝕜 s) (hx : x ∈ s)
{n : ℕ} (hn : (n : ℕ∞) ≤ N) (hB : ‖B‖ ≤ 1) :
‖iterated_fderi... | begin
apply (B.norm_iterated_fderiv_within_le_of_bilinear hf hg hs hx hn).trans,
apply mul_le_of_le_one_left (finset.sum_nonneg' (λ i, _)) hB,
positivity
end | lemma | continuous_linear_map.norm_iterated_fderiv_within_le_of_bilinear_of_le_one | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on",
"finset.range",
"mul_le_of_le_one_left",
"unique_diff_on"
] | Bounding the norm of the iterated derivative of `B (f x) (g x)` within a set in terms of the
iterated derivatives of `f` and `g` when `B` is bilinear of norm at most `1`:
`‖D^n (x ↦ B (f x) (g x))‖ ≤ ∑_{k ≤ n} n.choose k ‖D^k f‖ ‖D^{n-k} g‖` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_linear_map.norm_iterated_fderiv_le_of_bilinear_of_le_one
(B : E →L[𝕜] F →L[𝕜] G) {f : D → E} {g : D → F} {N : ℕ∞}
(hf : cont_diff 𝕜 N f) (hg : cont_diff 𝕜 N g) (x : D)
{n : ℕ} (hn : (n : ℕ∞) ≤ N) (hB : ‖B‖ ≤ 1) :
‖iterated_fderiv 𝕜 n (λ y, B (f y) (g y)) x‖
≤ ∑ i in finset.range (n+1), (n.ch... | begin
simp_rw [← iterated_fderiv_within_univ],
exact B.norm_iterated_fderiv_within_le_of_bilinear_of_le_one hf.cont_diff_on hg.cont_diff_on
unique_diff_on_univ (mem_univ x) hn hB,
end | lemma | continuous_linear_map.norm_iterated_fderiv_le_of_bilinear_of_le_one | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"finset.range",
"iterated_fderiv_within_univ",
"unique_diff_on_univ"
] | Bounding the norm of the iterated derivative of `B (f x) (g x)` in terms of the
iterated derivatives of `f` and `g` when `B` is bilinear of norm at most `1`:
`‖D^n (x ↦ B (f x) (g x))‖ ≤ ∑_{k ≤ n} n.choose k ‖D^k f‖ ‖D^{n-k} g‖` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_iterated_fderiv_within_smul_le
{f : E → 𝕜'} {g : E → F} {N : ℕ∞} (hf : cont_diff_on 𝕜 N f s) (hg : cont_diff_on 𝕜 N g s)
(hs : unique_diff_on 𝕜 s) {x : E} (hx : x ∈ s) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
‖iterated_fderiv_within 𝕜 n (λ y, f y • g y) s x‖
≤ ∑ i in finset.range (n+1), (n.choose i : ℝ)
... | (continuous_linear_map.lsmul 𝕜 𝕜' : 𝕜' →L[𝕜] F →L[𝕜] F)
.norm_iterated_fderiv_within_le_of_bilinear_of_le_one hf hg hs hx hn
continuous_linear_map.op_norm_lsmul_le | lemma | norm_iterated_fderiv_within_smul_le | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on",
"continuous_linear_map.lsmul",
"continuous_linear_map.op_norm_lsmul_le",
"finset.range",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_iterated_fderiv_smul_le
{f : E → 𝕜'} {g : E → F} {N : ℕ∞} (hf : cont_diff 𝕜 N f) (hg : cont_diff 𝕜 N g)
(x : E) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
‖iterated_fderiv 𝕜 n (λ y, f y • g y) x‖
≤ ∑ i in finset.range (n+1), (n.choose i : ℝ)
* ‖iterated_fderiv 𝕜 i f x‖ * ‖iterated_fderiv 𝕜 (n-i) g x‖ | (continuous_linear_map.lsmul 𝕜 𝕜' : 𝕜' →L[𝕜] F →L[𝕜] F)
.norm_iterated_fderiv_le_of_bilinear_of_le_one hf hg x hn
continuous_linear_map.op_norm_lsmul_le | lemma | norm_iterated_fderiv_smul_le | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"continuous_linear_map.lsmul",
"continuous_linear_map.op_norm_lsmul_le",
"finset.range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_iterated_fderiv_within_mul_le
{f : E → A} {g : E → A} {N : ℕ∞} (hf : cont_diff_on 𝕜 N f s) (hg : cont_diff_on 𝕜 N g s)
(hs : unique_diff_on 𝕜 s) {x : E} (hx : x ∈ s) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
‖iterated_fderiv_within 𝕜 n (λ y, f y * g y) s x‖
≤ ∑ i in finset.range (n+1), (n.choose i : ℝ)
* ‖... | (continuous_linear_map.mul 𝕜 A : A →L[𝕜] A →L[𝕜] A)
.norm_iterated_fderiv_within_le_of_bilinear_of_le_one hf hg hs hx hn
(continuous_linear_map.op_norm_mul_le _ _) | lemma | norm_iterated_fderiv_within_mul_le | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on",
"continuous_linear_map.mul",
"continuous_linear_map.op_norm_mul_le",
"finset.range",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_iterated_fderiv_mul_le
{f : E → A} {g : E → A} {N : ℕ∞} (hf : cont_diff 𝕜 N f) (hg : cont_diff 𝕜 N g)
(x : E) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
‖iterated_fderiv 𝕜 n (λ y, f y * g y) x‖
≤ ∑ i in finset.range (n+1), (n.choose i : ℝ)
* ‖iterated_fderiv 𝕜 i f x‖ * ‖iterated_fderiv 𝕜 (n-i) g x‖ | begin
simp_rw [← iterated_fderiv_within_univ],
exact norm_iterated_fderiv_within_mul_le hf.cont_diff_on
hg.cont_diff_on unique_diff_on_univ (mem_univ x) hn,
end | lemma | norm_iterated_fderiv_mul_le | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"finset.range",
"iterated_fderiv_within_univ",
"norm_iterated_fderiv_within_mul_le",
"unique_diff_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_iterated_fderiv_within_comp_le_aux
{Fu Gu : Type u} [normed_add_comm_group Fu] [normed_space 𝕜 Fu]
[normed_add_comm_group Gu] [normed_space 𝕜 Gu]
{g : Fu → Gu} {f : E → Fu} {n : ℕ} {s : set E} {t : set Fu} {x : E}
(hg : cont_diff_on 𝕜 n g t) (hf : cont_diff_on 𝕜 n f s)
(ht : unique_diff_on 𝕜 t) (hs ... | begin
/- We argue by induction on `n`, using that `D^(n+1) (g ∘ f) = D^n (g ' ∘ f ⬝ f')`. The successive
derivatives of `g' ∘ f` are controlled thanks to the inductive assumption, and those of `f'` are
controlled by assumption.
As composition of linear maps is a bilinear map, one may use
`continuous_linear_ma... | lemma | norm_iterated_fderiv_within_comp_le_aux | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"algebra_map.coe_one",
"bot_le",
"cont_diff_on",
"cont_diff_on.comp",
"continuous_linear_map.compL",
"continuous_linear_map.norm_compL_le",
"div_eq_inv_mul",
"enat.coe_one",
"fderiv_within",
"fderiv_within.comp",
"finset.card_range",
"finset.mem_range_succ_iff",
"finset.range",
"inv_le_one... | If the derivatives within a set of `g` at `f x` are bounded by `C`, and the `i`-th derivative
within a set of `f` at `x` is bounded by `D^i` for all `1 ≤ i ≤ n`, then the `n`-th derivative
of `g ∘ f` is bounded by `n! * C * D^n`.
This lemma proves this estimate assuming additionally that two of the spaces live in the s... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_iterated_fderiv_within_comp_le
{g : F → G} {f : E → F} {n : ℕ} {s : set E} {t : set F} {x : E} {N : ℕ∞}
(hg : cont_diff_on 𝕜 N g t) (hf : cont_diff_on 𝕜 N f s) (hn : (n : ℕ∞) ≤ N)
(ht : unique_diff_on 𝕜 t) (hs : unique_diff_on 𝕜 s)
(hst : maps_to f s t) (hx : x ∈ s)
{C : ℝ} {D : ℝ} (hC : ∀ i, i ≤ n →... | begin
/- We reduce the bound to the case where all spaces live in the same universe (in which we
already have proved the result), by using linear isometries between the spaces and their `ulift`
to a common universe. These linear isometries preserve the norm of the iterated derivative. -/
let Fu : Type (max uF u... | lemma | norm_iterated_fderiv_within_comp_le | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on",
"linear_isometry_equiv.apply_symm_apply",
"linear_isometry_equiv.map_eq_iff",
"linear_isometry_equiv.norm_iterated_fderiv_within_comp_left",
"linear_isometry_equiv.norm_iterated_fderiv_within_comp_right",
"linear_isometry_equiv.ulift",
"norm_iterated_fderiv_within_comp_le_aux",
"unique... | If the derivatives within a set of `g` at `f x` are bounded by `C`, and the `i`-th derivative
within a set of `f` at `x` is bounded by `D^i` for all `1 ≤ i ≤ n`, then the `n`-th derivative
of `g ∘ f` is bounded by `n! * C * D^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_iterated_fderiv_comp_le
{g : F → G} {f : E → F} {n : ℕ} {N : ℕ∞}
(hg : cont_diff 𝕜 N g) (hf : cont_diff 𝕜 N f) (hn : (n : ℕ∞) ≤ N) (x : E)
{C : ℝ} {D : ℝ} (hC : ∀ i, i ≤ n → ‖iterated_fderiv 𝕜 i g (f x)‖ ≤ C)
(hD : ∀ i, 1 ≤ i → i ≤ n → ‖iterated_fderiv 𝕜 i f x‖ ≤ D^i) :
‖iterated_fderiv 𝕜 n (g ∘ f) ... | begin
simp_rw [← iterated_fderiv_within_univ] at ⊢ hC hD,
exact norm_iterated_fderiv_within_comp_le hg.cont_diff_on hf.cont_diff_on hn unique_diff_on_univ
unique_diff_on_univ (maps_to_univ _ _) (mem_univ x) hC hD,
end | lemma | norm_iterated_fderiv_comp_le | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"iterated_fderiv_within_univ",
"norm_iterated_fderiv_within_comp_le",
"unique_diff_on_univ"
] | If the derivatives of `g` at `f x` are bounded by `C`, and the `i`-th derivative
of `f` at `x` is bounded by `D^i` for all `1 ≤ i ≤ n`, then the `n`-th derivative
of `g ∘ f` is bounded by `n! * C * D^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_iterated_fderiv_within_clm_apply {f : E → (F →L[𝕜] G)} {g : E → F} {s : set E} {x : E}
{N : ℕ∞} {n : ℕ} (hf : cont_diff_on 𝕜 N f s) (hg : cont_diff_on 𝕜 N g s) (hs : unique_diff_on 𝕜 s)
(hx : x ∈ s) (hn : ↑n ≤ N) :
‖iterated_fderiv_within 𝕜 n (λ y, (f y) (g y)) s x‖ ≤
(finset.range (n + 1)).sum (λ... | begin
let B : (F →L[𝕜] G) →L[𝕜] F →L[𝕜] G :=
continuous_linear_map.flip (continuous_linear_map.apply 𝕜 G),
have hB : ‖B‖ ≤ 1 :=
begin
simp only [continuous_linear_map.op_norm_flip, continuous_linear_map.apply],
refine continuous_linear_map.op_norm_le_bound _ zero_le_one (λ f, _),
simp only [cont... | lemma | norm_iterated_fderiv_within_clm_apply | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on",
"continuous_linear_map.apply",
"continuous_linear_map.coe_id'",
"continuous_linear_map.flip",
"continuous_linear_map.op_norm_flip",
"continuous_linear_map.op_norm_le_bound",
"finset.range",
"one_mul",
"unique_diff_on",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_iterated_fderiv_clm_apply {f : E → (F →L[𝕜] G)} {g : E → F}
{N : ℕ∞} {n : ℕ} (hf : cont_diff 𝕜 N f) (hg : cont_diff 𝕜 N g) (x : E) (hn : ↑n ≤ N):
‖iterated_fderiv 𝕜 n (λ (y : E), (f y) (g y)) x‖ ≤
(finset.range (n + 1)).sum (λ (i : ℕ), ↑(n.choose i) * ‖iterated_fderiv 𝕜 i f x‖ *
‖iterated_... | begin
simp only [← iterated_fderiv_within_univ],
exact norm_iterated_fderiv_within_clm_apply hf.cont_diff_on hg.cont_diff_on unique_diff_on_univ
(set.mem_univ x) hn,
end | lemma | norm_iterated_fderiv_clm_apply | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"finset.range",
"iterated_fderiv_within_univ",
"norm_iterated_fderiv_within_clm_apply",
"set.mem_univ",
"unique_diff_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_iterated_fderiv_within_clm_apply_const {f : E → (F →L[𝕜] G)} {c : F} {s : set E} {x : E}
{N : ℕ∞} {n : ℕ} (hf : cont_diff_on 𝕜 N f s) (hs : unique_diff_on 𝕜 s) (hx : x ∈ s)
(hn : ↑n ≤ N) : ‖iterated_fderiv_within 𝕜 n (λ (y : E), (f y) c) s x‖ ≤
‖c‖ * ‖iterated_fderiv_within 𝕜 n f s x‖ | begin
let g : (F →L[𝕜] G) →L[𝕜] G := continuous_linear_map.apply 𝕜 G c,
have h := g.norm_comp_continuous_multilinear_map_le (iterated_fderiv_within 𝕜 n f s x),
rw ← g.iterated_fderiv_within_comp_left hf hs hx hn at h,
refine h.trans (mul_le_mul_of_nonneg_right _ (norm_nonneg _)),
refine g.op_norm_le_bound... | lemma | norm_iterated_fderiv_within_clm_apply_const | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff_on",
"continuous_linear_map.apply",
"continuous_linear_map.apply_apply",
"iterated_fderiv_within",
"mul_comm",
"mul_le_mul_of_nonneg_right",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_iterated_fderiv_clm_apply_const {f : E → (F →L[𝕜] G)} {c : F} {x : E} {N : ℕ∞} {n : ℕ}
(hf : cont_diff 𝕜 N f) (hn : ↑n ≤ N) :
‖iterated_fderiv 𝕜 n (λ (y : E), (f y) c) x‖ ≤ ‖c‖ * ‖iterated_fderiv 𝕜 n f x‖ | begin
simp only [← iterated_fderiv_within_univ],
refine norm_iterated_fderiv_within_clm_apply_const hf.cont_diff_on unique_diff_on_univ
(set.mem_univ x) hn,
end | lemma | norm_iterated_fderiv_clm_apply_const | analysis.calculus | src/analysis/calculus/cont_diff.lean | [
"analysis.calculus.cont_diff_def",
"analysis.calculus.deriv.inverse",
"analysis.calculus.mean_value",
"analysis.normed_space.finite_dimension",
"data.nat.choose.cast"
] | [
"cont_diff",
"iterated_fderiv_within_univ",
"norm_iterated_fderiv_within_clm_apply_const",
"set.mem_univ",
"unique_diff_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to_on (n : ℕ∞)
(f : E → F) (p : E → formal_multilinear_series 𝕜 E F) (s : set E) : Prop | (zero_eq : ∀ x ∈ s, (p x 0).uncurry0 = f x)
(fderiv_within : ∀ (m : ℕ) (hm : (m : ℕ∞) < n), ∀ x ∈ s,
has_fderiv_within_at (λ y, p y m) (p x m.succ).curry_left s x)
(cont : ∀ (m : ℕ) (hm : (m : ℕ∞) ≤ n), continuous_on (λ x, p x m) s) | structure | has_ftaylor_series_up_to_on | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont",
"continuous_on",
"fderiv_within",
"formal_multilinear_series",
"has_fderiv_within_at"
] | `has_ftaylor_series_up_to_on n f p s` registers the fact that `p 0 = f` and `p (m+1)` is a
derivative of `p m` for `m < n`, and is continuous for `m ≤ n`. This is a predicate analogous to
`has_fderiv_within_at` but for higher order derivatives. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to_on.zero_eq'
(h : has_ftaylor_series_up_to_on n f p s) {x : E} (hx : x ∈ s) :
p x 0 = (continuous_multilinear_curry_fin0 𝕜 E F).symm (f x) | by { rw ← h.zero_eq x hx, symmetry, exact continuous_multilinear_map.uncurry0_curry0 _ } | lemma | has_ftaylor_series_up_to_on.zero_eq' | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_multilinear_curry_fin0",
"continuous_multilinear_map.uncurry0_curry0",
"has_ftaylor_series_up_to_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to_on.congr
(h : has_ftaylor_series_up_to_on n f p s) (h₁ : ∀ x ∈ s, f₁ x = f x) :
has_ftaylor_series_up_to_on n f₁ p s | begin
refine ⟨λ x hx, _, h.fderiv_within, h.cont⟩,
rw h₁ x hx,
exact h.zero_eq x hx
end | lemma | has_ftaylor_series_up_to_on.congr | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"has_ftaylor_series_up_to_on"
] | If two functions coincide on a set `s`, then a Taylor series for the first one is as well a
Taylor series for the second one. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to_on.mono
(h : has_ftaylor_series_up_to_on n f p s) {t : set E} (hst : t ⊆ s) :
has_ftaylor_series_up_to_on n f p t | ⟨λ x hx, h.zero_eq x (hst hx),
λ m hm x hx, (h.fderiv_within m hm x (hst hx)).mono hst,
λ m hm, (h.cont m hm).mono hst⟩ | lemma | has_ftaylor_series_up_to_on.mono | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"has_ftaylor_series_up_to_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to_on.of_le
(h : has_ftaylor_series_up_to_on n f p s) (hmn : m ≤ n) :
has_ftaylor_series_up_to_on m f p s | ⟨h.zero_eq,
λ k hk x hx, h.fderiv_within k (lt_of_lt_of_le hk hmn) x hx,
λ k hk, h.cont k (le_trans hk hmn)⟩ | lemma | has_ftaylor_series_up_to_on.of_le | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"has_ftaylor_series_up_to_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to_on.continuous_on
(h : has_ftaylor_series_up_to_on n f p s) : continuous_on f s | begin
have := (h.cont 0 bot_le).congr (λ x hx, (h.zero_eq' hx).symm),
rwa linear_isometry_equiv.comp_continuous_on_iff at this
end | lemma | has_ftaylor_series_up_to_on.continuous_on | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"bot_le",
"continuous_on",
"has_ftaylor_series_up_to_on",
"linear_isometry_equiv.comp_continuous_on_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to_on_zero_iff :
has_ftaylor_series_up_to_on 0 f p s ↔ continuous_on f s ∧ (∀ x ∈ s, (p x 0).uncurry0 = f x) | begin
refine ⟨λ H, ⟨H.continuous_on, H.zero_eq⟩,
λ H, ⟨H.2, λ m hm, false.elim (not_le.2 hm bot_le), _⟩⟩,
assume m hm,
obtain rfl : m = 0, by exact_mod_cast (hm.antisymm (zero_le _)),
have : ∀ x ∈ s, p x 0 = (continuous_multilinear_curry_fin0 𝕜 E F).symm (f x),
by { assume x hx, rw ← H.2 x hx, sy... | lemma | has_ftaylor_series_up_to_on_zero_iff | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"bot_le",
"continuous_multilinear_curry_fin0",
"continuous_multilinear_map.uncurry0_curry0",
"continuous_on",
"continuous_on_congr",
"has_ftaylor_series_up_to_on",
"linear_isometry_equiv.comp_continuous_on_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to_on_top_iff :
(has_ftaylor_series_up_to_on ∞ f p s) ↔ (∀ (n : ℕ), has_ftaylor_series_up_to_on n f p s) | begin
split,
{ assume H n, exact H.of_le le_top },
{ assume H,
split,
{ exact (H 0).zero_eq },
{ assume m hm,
apply (H m.succ).fderiv_within m (with_top.coe_lt_coe.2 (lt_add_one m)) },
{ assume m hm,
apply (H m).cont m le_rfl } }
end | lemma | has_ftaylor_series_up_to_on_top_iff | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont",
"fderiv_within",
"has_ftaylor_series_up_to_on",
"le_rfl",
"le_top",
"lt_add_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to_on_top_iff' : has_ftaylor_series_up_to_on ∞ f p s ↔
(∀ x ∈ s, (p x 0).uncurry0 = f x) ∧
(∀ (m : ℕ), ∀ x ∈ s, has_fderiv_within_at (λ y, p y m) (p x m.succ).curry_left s x) | -- Everything except for the continuity is trivial:
⟨λ h, ⟨h.1, λ m, h.2 m (with_top.coe_lt_top m)⟩, λ h, ⟨h.1, λ m _, h.2 m, λ m _ x hx,
-- The continuity follows from the existence of a derivative:
(h.2 m x hx).continuous_within_at⟩⟩ | lemma | has_ftaylor_series_up_to_on_top_iff' | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"has_fderiv_within_at",
"has_ftaylor_series_up_to_on",
"with_top.coe_lt_top"
] | In the case that `n = ∞` we don't need the continuity assumption in
`has_ftaylor_series_up_to_on`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to_on.has_fderiv_within_at
(h : has_ftaylor_series_up_to_on n f p s) (hn : 1 ≤ n) (hx : x ∈ s) :
has_fderiv_within_at f (continuous_multilinear_curry_fin1 𝕜 E F (p x 1)) s x | begin
have A : ∀ y ∈ s, f y = (continuous_multilinear_curry_fin0 𝕜 E F) (p y 0),
{ assume y hy, rw ← h.zero_eq y hy, refl },
suffices H : has_fderiv_within_at
(λ y, continuous_multilinear_curry_fin0 𝕜 E F (p y 0))
(continuous_multilinear_curry_fin1 𝕜 E F (p x 1)) s x,
by exact H.congr A (A x hx... | lemma | has_ftaylor_series_up_to_on.has_fderiv_within_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_multilinear_curry_fin0",
"continuous_multilinear_curry_fin1",
"has_fderiv_within_at",
"has_ftaylor_series_up_to_on",
"linear_isometry_equiv.comp_has_fderiv_within_at_iff'",
"unique.eq_default"
] | If a function has a Taylor series at order at least `1`, then the term of order `1` of this
series is a derivative of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to_on.differentiable_on
(h : has_ftaylor_series_up_to_on n f p s) (hn : 1 ≤ n) : differentiable_on 𝕜 f s | λ x hx, (h.has_fderiv_within_at hn hx).differentiable_within_at | lemma | has_ftaylor_series_up_to_on.differentiable_on | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"differentiable_on",
"differentiable_within_at",
"has_ftaylor_series_up_to_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to_on.has_fderiv_at
(h : has_ftaylor_series_up_to_on n f p s) (hn : 1 ≤ n) (hx : s ∈ 𝓝 x) :
has_fderiv_at f (continuous_multilinear_curry_fin1 𝕜 E F (p x 1)) x | (h.has_fderiv_within_at hn (mem_of_mem_nhds hx)).has_fderiv_at hx | lemma | has_ftaylor_series_up_to_on.has_fderiv_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_multilinear_curry_fin1",
"has_fderiv_at",
"has_ftaylor_series_up_to_on",
"mem_of_mem_nhds"
] | If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then the term
of order `1` of this series is a derivative of `f` at `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to_on.eventually_has_fderiv_at
(h : has_ftaylor_series_up_to_on n f p s) (hn : 1 ≤ n) (hx : s ∈ 𝓝 x) :
∀ᶠ y in 𝓝 x, has_fderiv_at f (continuous_multilinear_curry_fin1 𝕜 E F (p y 1)) y | (eventually_eventually_nhds.2 hx).mono $ λ y hy, h.has_fderiv_at hn hy | lemma | has_ftaylor_series_up_to_on.eventually_has_fderiv_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_multilinear_curry_fin1",
"has_fderiv_at",
"has_ftaylor_series_up_to_on"
] | If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then
in a neighborhood of `x`, the term of order `1` of this series is a derivative of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to_on.differentiable_at
(h : has_ftaylor_series_up_to_on n f p s) (hn : 1 ≤ n) (hx : s ∈ 𝓝 x) :
differentiable_at 𝕜 f x | (h.has_fderiv_at hn hx).differentiable_at | lemma | has_ftaylor_series_up_to_on.differentiable_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"differentiable_at",
"has_ftaylor_series_up_to_on"
] | If a function has a Taylor series at order at least `1` on a neighborhood of `x`, then
it is differentiable at `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to_on_succ_iff_left {n : ℕ} :
has_ftaylor_series_up_to_on (n + 1) f p s ↔
has_ftaylor_series_up_to_on n f p s
∧ (∀ x ∈ s, has_fderiv_within_at (λ y, p y n) (p x n.succ).curry_left s x)
∧ continuous_on (λ x, p x (n + 1)) s | begin
split,
{ assume h,
exact ⟨h.of_le (with_top.coe_le_coe.2 (nat.le_succ n)),
h.fderiv_within _ (with_top.coe_lt_coe.2 (lt_add_one n)),
h.cont (n + 1) le_rfl⟩ },
{ assume h,
split,
{ exact h.1.zero_eq },
{ assume m hm,
by_cases h' : m < n,
{ exact h.1.fderiv_wi... | theorem | has_ftaylor_series_up_to_on_succ_iff_left | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_on",
"has_fderiv_within_at",
"has_ftaylor_series_up_to_on",
"lt_add_one",
"nat.eq_of_lt_succ_of_not_lt"
] | `p` is a Taylor series of `f` up to `n+1` if and only if `p` is a Taylor series up to `n`, and
`p (n + 1)` is a derivative of `p n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to_on_succ_iff_right {n : ℕ} :
has_ftaylor_series_up_to_on ((n + 1) : ℕ) f p s ↔
(∀ x ∈ s, (p x 0).uncurry0 = f x)
∧ (∀ x ∈ s, has_fderiv_within_at (λ y, p y 0) (p x 1).curry_left s x)
∧ has_ftaylor_series_up_to_on n
(λ x, continuous_multilinear_curry_fin1 𝕜 E F (p x 1)) (λ x, (p x).s... | begin
split,
{ assume H,
refine ⟨H.zero_eq, H.fderiv_within 0 (with_top.coe_lt_coe.2 (nat.succ_pos n)), _⟩,
split,
{ assume x hx, refl },
{ assume m (hm : (m : ℕ∞) < n) x (hx : x ∈ s),
have A : (m.succ : ℕ∞) < n.succ,
by { rw with_top.coe_lt_coe at ⊢ hm, exact nat.lt_succ_iff.mpr hm },... | theorem | has_ftaylor_series_up_to_on_succ_iff_right | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_multilinear_curry_fin1",
"continuous_multilinear_curry_right_equiv'",
"continuous_on",
"differentiable_on",
"differentiable_within_at",
"has_fderiv_within_at",
"has_ftaylor_series_up_to_on",
"linear_isometry_equiv.comp_continuous_on_iff",
"linear_isometry_equiv.comp_has_fderiv_within_at_... | `p` is a Taylor series of `f` up to `n+1` if and only if `p.shift` is a Taylor series up to `n`
for `p 1`, which is a derivative of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at (n : ℕ∞) (f : E → F) (s : set E) (x : E) : Prop | ∀ (m : ℕ), (m : ℕ∞) ≤ n →
∃ u ∈ 𝓝[insert x s] x, ∃ p : E → formal_multilinear_series 𝕜 E F,
has_ftaylor_series_up_to_on m f p u | def | cont_diff_within_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"formal_multilinear_series",
"has_ftaylor_series_up_to_on"
] | A function is continuously differentiable up to order `n` within a set `s` at a point `x` if
it admits continuous derivatives up to order `n` in a neighborhood of `x` in `s ∪ {x}`.
For `n = ∞`, we only require that this holds up to any finite order (where the neighborhood may
depend on the finite order we consider).
F... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at_nat {n : ℕ} :
cont_diff_within_at 𝕜 n f s x ↔
∃ u ∈ 𝓝[insert x s] x, ∃ p : E → formal_multilinear_series 𝕜 E F,
has_ftaylor_series_up_to_on n f p u | ⟨λ H, H n le_rfl, λ ⟨u, hu, p, hp⟩ m hm, ⟨u, hu, p, hp.of_le hm⟩⟩ | lemma | cont_diff_within_at_nat | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"formal_multilinear_series",
"has_ftaylor_series_up_to_on",
"le_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.of_le
(h : cont_diff_within_at 𝕜 n f s x) (hmn : m ≤ n) :
cont_diff_within_at 𝕜 m f s x | λ k hk, h k (le_trans hk hmn) | lemma | cont_diff_within_at.of_le | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at_iff_forall_nat_le :
cont_diff_within_at 𝕜 n f s x ↔ ∀ m : ℕ, ↑m ≤ n → cont_diff_within_at 𝕜 m f s x | ⟨λ H m hm, H.of_le hm, λ H m hm, H m hm _ le_rfl⟩ | lemma | cont_diff_within_at_iff_forall_nat_le | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at_top :
cont_diff_within_at 𝕜 ∞ f s x ↔ ∀ (n : ℕ), cont_diff_within_at 𝕜 n f s x | cont_diff_within_at_iff_forall_nat_le.trans $ by simp only [forall_prop_of_true, le_top] | lemma | cont_diff_within_at_top | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"forall_prop_of_true",
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.continuous_within_at
(h : cont_diff_within_at 𝕜 n f s x) : continuous_within_at f s x | begin
rcases h 0 bot_le with ⟨u, hu, p, H⟩,
rw [mem_nhds_within_insert] at hu,
exact (H.continuous_on.continuous_within_at hu.1).mono_of_mem hu.2
end | lemma | cont_diff_within_at.continuous_within_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"bot_le",
"cont_diff_within_at",
"continuous_within_at",
"mem_nhds_within_insert"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.congr_of_eventually_eq
(h : cont_diff_within_at 𝕜 n f s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
cont_diff_within_at 𝕜 n f₁ s x | λ m hm, let ⟨u, hu, p, H⟩ := h m hm in
⟨{x ∈ u | f₁ x = f x}, filter.inter_mem hu (mem_nhds_within_insert.2 ⟨hx, h₁⟩), p,
(H.mono (sep_subset _ _)).congr (λ _, and.right)⟩ | lemma | cont_diff_within_at.congr_of_eventually_eq | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"filter.inter_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.congr_of_eventually_eq_insert
(h : cont_diff_within_at 𝕜 n f s x) (h₁ : f₁ =ᶠ[𝓝[insert x s] x] f) :
cont_diff_within_at 𝕜 n f₁ s x | h.congr_of_eventually_eq (nhds_within_mono x (subset_insert x s) h₁)
(mem_of_mem_nhds_within (mem_insert x s) h₁ : _) | lemma | cont_diff_within_at.congr_of_eventually_eq_insert | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"mem_of_mem_nhds_within",
"nhds_within_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.congr_of_eventually_eq'
(h : cont_diff_within_at 𝕜 n f s x) (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : x ∈ s) :
cont_diff_within_at 𝕜 n f₁ s x | h.congr_of_eventually_eq h₁ $ h₁.self_of_nhds_within hx | lemma | cont_diff_within_at.congr_of_eventually_eq' | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.cont_diff_within_at_iff
(h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) :
cont_diff_within_at 𝕜 n f₁ s x ↔ cont_diff_within_at 𝕜 n f s x | ⟨λ H, cont_diff_within_at.congr_of_eventually_eq H h₁.symm hx.symm,
λ H, H.congr_of_eventually_eq h₁ hx⟩ | lemma | filter.eventually_eq.cont_diff_within_at_iff | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"cont_diff_within_at.congr_of_eventually_eq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.congr
(h : cont_diff_within_at 𝕜 n f s x) (h₁ : ∀ y ∈ s, f₁ y = f y) (hx : f₁ x = f x) :
cont_diff_within_at 𝕜 n f₁ s x | h.congr_of_eventually_eq (filter.eventually_eq_of_mem self_mem_nhds_within h₁) hx | lemma | cont_diff_within_at.congr | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"filter.eventually_eq_of_mem",
"self_mem_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.congr'
(h : cont_diff_within_at 𝕜 n f s x) (h₁ : ∀ y ∈ s, f₁ y = f y) (hx : x ∈ s) :
cont_diff_within_at 𝕜 n f₁ s x | h.congr h₁ (h₁ _ hx) | lemma | cont_diff_within_at.congr' | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.mono_of_mem
(h : cont_diff_within_at 𝕜 n f s x) {t : set E} (hst : s ∈ 𝓝[t] x) :
cont_diff_within_at 𝕜 n f t x | begin
assume m hm,
rcases h m hm with ⟨u, hu, p, H⟩,
exact ⟨u, nhds_within_le_of_mem (insert_mem_nhds_within_insert hst) hu, p, H⟩
end | lemma | cont_diff_within_at.mono_of_mem | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"insert_mem_nhds_within_insert",
"nhds_within_le_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.mono
(h : cont_diff_within_at 𝕜 n f s x) {t : set E} (hst : t ⊆ s) :
cont_diff_within_at 𝕜 n f t x | h.mono_of_mem $ filter.mem_of_superset self_mem_nhds_within hst | lemma | cont_diff_within_at.mono | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"filter.mem_of_superset",
"self_mem_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.congr_nhds
(h : cont_diff_within_at 𝕜 n f s x) {t : set E} (hst : 𝓝[s] x = 𝓝[t] x) :
cont_diff_within_at 𝕜 n f t x | h.mono_of_mem $ hst ▸ self_mem_nhds_within | lemma | cont_diff_within_at.congr_nhds | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"self_mem_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at_congr_nhds {t : set E} (hst : 𝓝[s] x = 𝓝[t] x) :
cont_diff_within_at 𝕜 n f s x ↔ cont_diff_within_at 𝕜 n f t x | ⟨λ h, h.congr_nhds hst, λ h, h.congr_nhds hst.symm⟩ | lemma | cont_diff_within_at_congr_nhds | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at_inter' (h : t ∈ 𝓝[s] x) :
cont_diff_within_at 𝕜 n f (s ∩ t) x ↔ cont_diff_within_at 𝕜 n f s x | cont_diff_within_at_congr_nhds $ eq.symm $ nhds_within_restrict'' _ h | lemma | cont_diff_within_at_inter' | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"cont_diff_within_at_congr_nhds",
"nhds_within_restrict''"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at_inter (h : t ∈ 𝓝 x) :
cont_diff_within_at 𝕜 n f (s ∩ t) x ↔ cont_diff_within_at 𝕜 n f s x | cont_diff_within_at_inter' (mem_nhds_within_of_mem_nhds h) | lemma | cont_diff_within_at_inter | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"cont_diff_within_at_inter'",
"mem_nhds_within_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at_insert {y : E} :
cont_diff_within_at 𝕜 n f (insert y s) x ↔ cont_diff_within_at 𝕜 n f s x | begin
simp_rw [cont_diff_within_at],
rcases eq_or_ne x y with rfl|h,
{ simp_rw [insert_eq_of_mem (mem_insert _ _)] },
simp_rw [insert_comm x y, nhds_within_insert_of_ne h]
end | lemma | cont_diff_within_at_insert | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"eq_or_ne",
"nhds_within_insert_of_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.insert (h : cont_diff_within_at 𝕜 n f s x) :
cont_diff_within_at 𝕜 n f (insert x s) x | h.insert' | lemma | cont_diff_within_at.insert | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.differentiable_within_at'
(h : cont_diff_within_at 𝕜 n f s x) (hn : 1 ≤ n) :
differentiable_within_at 𝕜 f (insert x s) x | begin
rcases h 1 hn with ⟨u, hu, p, H⟩,
rcases mem_nhds_within.1 hu with ⟨t, t_open, xt, tu⟩,
rw inter_comm at tu,
have := ((H.mono tu).differentiable_on le_rfl) x ⟨mem_insert x s, xt⟩,
exact (differentiable_within_at_inter (is_open.mem_nhds t_open xt)).1 this,
end | lemma | cont_diff_within_at.differentiable_within_at' | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"differentiable_on",
"differentiable_within_at",
"differentiable_within_at_inter",
"is_open.mem_nhds",
"le_rfl"
] | If a function is `C^n` within a set at a point, with `n ≥ 1`, then it is differentiable
within this set at this point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at.differentiable_within_at
(h : cont_diff_within_at 𝕜 n f s x) (hn : 1 ≤ n) :
differentiable_within_at 𝕜 f s x | (h.differentiable_within_at' hn).mono (subset_insert x s) | lemma | cont_diff_within_at.differentiable_within_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at_succ_iff_has_fderiv_within_at {n : ℕ} :
cont_diff_within_at 𝕜 ((n + 1) : ℕ) f s x
↔ ∃ u ∈ 𝓝[insert x s] x, ∃ f' : E → (E →L[𝕜] F),
(∀ x ∈ u, has_fderiv_within_at f (f' x) u x) ∧ (cont_diff_within_at 𝕜 n f' u x) | begin
split,
{ assume h,
rcases h n.succ le_rfl with ⟨u, hu, p, Hp⟩,
refine ⟨u, hu, λ y, (continuous_multilinear_curry_fin1 𝕜 E F) (p y 1),
λ y hy, Hp.has_fderiv_within_at (with_top.coe_le_coe.2 (nat.le_add_left 1 n)) hy, _⟩,
assume m hm,
refine ⟨u, _, λ (y : E), (p y).shift, _⟩,
{ conver... | theorem | cont_diff_within_at_succ_iff_has_fderiv_within_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"cont_diff_within_at_nat",
"continuous_multilinear_curry_fin0",
"continuous_multilinear_curry_fin1",
"filter.inter_mem",
"formal_multilinear_series.unshift",
"has_fderiv_within_at",
"has_ftaylor_series_up_to_on_succ_iff_right",
"le_rfl",
"linear_isometry_equiv.comp_has_fderi... | A function is `C^(n + 1)` on a domain iff locally, it has a derivative which is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at_succ_iff_has_fderiv_within_at' {n : ℕ} :
cont_diff_within_at 𝕜 (n + 1 : ℕ) f s x
↔ ∃ u ∈ 𝓝[insert x s] x, u ⊆ insert x s ∧ ∃ f' : E → E →L[𝕜] F,
(∀ x ∈ u, has_fderiv_within_at f (f' x) s x) ∧ cont_diff_within_at 𝕜 n f' s x | begin
refine ⟨λ hf, _, _⟩,
{ obtain ⟨u, hu, f', huf', hf'⟩ := cont_diff_within_at_succ_iff_has_fderiv_within_at.mp hf,
obtain ⟨w, hw, hxw, hwu⟩ := mem_nhds_within.mp hu,
rw [inter_comm] at hwu,
refine ⟨insert x s ∩ w, inter_mem_nhds_within _ (hw.mem_nhds hxw), inter_subset_left _ _,
f', λ y hy, _,... | lemma | cont_diff_within_at_succ_iff_has_fderiv_within_at' | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"cont_diff_within_at_insert",
"cont_diff_within_at_succ_iff_has_fderiv_within_at",
"has_fderiv_within_at",
"inter_mem_nhds_within",
"nhds_within_mono"
] | A version of `cont_diff_within_at_succ_iff_has_fderiv_within_at` where all derivatives
are taken within the same set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on (n : ℕ∞) (f : E → F) (s : set E) : Prop | ∀ x ∈ s, cont_diff_within_at 𝕜 n f s x | def | cont_diff_on | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at"
] | A function is continuously differentiable up to `n` on `s` if, for any point `x` in `s`, it
admits continuous derivatives up to order `n` on a neighborhood of `x` in `s`.
For `n = ∞`, we only require that this holds up to any finite order (where the neighborhood may
depend on the finite order we consider). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to_on.cont_diff_on {f' : E → formal_multilinear_series 𝕜 E F}
(hf : has_ftaylor_series_up_to_on n f f' s) : cont_diff_on 𝕜 n f s | begin
intros x hx m hm,
use s,
simp only [set.insert_eq_of_mem hx, self_mem_nhds_within, true_and],
exact ⟨f', hf.of_le hm⟩,
end | lemma | has_ftaylor_series_up_to_on.cont_diff_on | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"formal_multilinear_series",
"has_ftaylor_series_up_to_on",
"self_mem_nhds_within",
"set.insert_eq_of_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.cont_diff_within_at (h : cont_diff_on 𝕜 n f s) (hx : x ∈ s) :
cont_diff_within_at 𝕜 n f s x | h x hx | lemma | cont_diff_on.cont_diff_within_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.cont_diff_on' {m : ℕ}
(hm : (m : ℕ∞) ≤ n) (h : cont_diff_within_at 𝕜 n f s x) :
∃ u, is_open u ∧ x ∈ u ∧ cont_diff_on 𝕜 m f (insert x s ∩ u) | begin
rcases h m hm with ⟨t, ht, p, hp⟩,
rcases mem_nhds_within.1 ht with ⟨u, huo, hxu, hut⟩,
rw [inter_comm] at hut,
exact ⟨u, huo, hxu, (hp.mono hut).cont_diff_on⟩
end | lemma | cont_diff_within_at.cont_diff_on' | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"cont_diff_within_at",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.cont_diff_on {m : ℕ}
(hm : (m : ℕ∞) ≤ n) (h : cont_diff_within_at 𝕜 n f s x) :
∃ u ∈ 𝓝[insert x s] x, u ⊆ insert x s ∧ cont_diff_on 𝕜 m f u | let ⟨u, uo, xu, h⟩ := h.cont_diff_on' hm
in ⟨_, inter_mem_nhds_within _ (uo.mem_nhds xu), inter_subset_left _ _, h⟩ | lemma | cont_diff_within_at.cont_diff_on | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"cont_diff_within_at",
"inter_mem_nhds_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.eventually {n : ℕ}
(h : cont_diff_within_at 𝕜 n f s x) :
∀ᶠ y in 𝓝[insert x s] x, cont_diff_within_at 𝕜 n f s y | begin
rcases h.cont_diff_on le_rfl with ⟨u, hu, hu_sub, hd⟩,
have : ∀ᶠ (y : E) in 𝓝[insert x s] x, u ∈ 𝓝[insert x s] y ∧ y ∈ u,
from (eventually_nhds_within_nhds_within.2 hu).and hu,
refine this.mono (λ y hy, (hd y hy.2).mono_of_mem _),
exact nhds_within_mono y (subset_insert _ _) hy.1
end | lemma | cont_diff_within_at.eventually | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_within_at",
"le_rfl",
"nhds_within_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.of_le (h : cont_diff_on 𝕜 n f s) (hmn : m ≤ n) :
cont_diff_on 𝕜 m f s | λ x hx, (h x hx).of_le hmn | lemma | cont_diff_on.of_le | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.of_succ {n : ℕ} (h : cont_diff_on 𝕜 (n + 1) f s) : cont_diff_on 𝕜 n f s | h.of_le $ with_top.coe_le_coe.mpr le_self_add | lemma | cont_diff_on.of_succ | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.one_of_succ {n : ℕ} (h : cont_diff_on 𝕜 (n + 1) f s) : cont_diff_on 𝕜 1 f s | h.of_le $ with_top.coe_le_coe.mpr le_add_self | lemma | cont_diff_on.one_of_succ | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_iff_forall_nat_le :
cont_diff_on 𝕜 n f s ↔ ∀ m : ℕ, ↑m ≤ n → cont_diff_on 𝕜 m f s | ⟨λ H m hm, H.of_le hm, λ H x hx m hm, H m hm x hx m le_rfl⟩ | lemma | cont_diff_on_iff_forall_nat_le | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_top :
cont_diff_on 𝕜 ∞ f s ↔ ∀ (n : ℕ), cont_diff_on 𝕜 n f s | cont_diff_on_iff_forall_nat_le.trans $ by simp only [le_top, forall_prop_of_true] | lemma | cont_diff_on_top | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"forall_prop_of_true",
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_all_iff_nat :
(∀ n, cont_diff_on 𝕜 n f s) ↔ (∀ n : ℕ, cont_diff_on 𝕜 n f s) | begin
refine ⟨λ H n, H n, _⟩,
rintro H (_|n),
exacts [cont_diff_on_top.2 H, H n]
end | lemma | cont_diff_on_all_iff_nat | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.continuous_on
(h : cont_diff_on 𝕜 n f s) : continuous_on f s | λ x hx, (h x hx).continuous_within_at | lemma | cont_diff_on.continuous_on | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"continuous_on",
"continuous_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.congr
(h : cont_diff_on 𝕜 n f s) (h₁ : ∀ x ∈ s, f₁ x = f x) :
cont_diff_on 𝕜 n f₁ s | λ x hx, (h x hx).congr h₁ (h₁ x hx) | lemma | cont_diff_on.congr | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_congr (h₁ : ∀ x ∈ s, f₁ x = f x) :
cont_diff_on 𝕜 n f₁ s ↔ cont_diff_on 𝕜 n f s | ⟨λ H, H.congr (λ x hx, (h₁ x hx).symm), λ H, H.congr h₁⟩ | lemma | cont_diff_on_congr | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.mono
(h : cont_diff_on 𝕜 n f s) {t : set E} (hst : t ⊆ s) :
cont_diff_on 𝕜 n f t | λ x hx, (h x (hst hx)).mono hst | lemma | cont_diff_on.mono | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.congr_mono
(hf : cont_diff_on 𝕜 n f s) (h₁ : ∀ x ∈ s₁, f₁ x = f x) (hs : s₁ ⊆ s) :
cont_diff_on 𝕜 n f₁ s₁ | (hf.mono hs).congr h₁ | lemma | cont_diff_on.congr_mono | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.differentiable_on
(h : cont_diff_on 𝕜 n f s) (hn : 1 ≤ n) : differentiable_on 𝕜 f s | λ x hx, (h x hx).differentiable_within_at hn | lemma | cont_diff_on.differentiable_on | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"differentiable_on",
"differentiable_within_at"
] | If a function is `C^n` on a set with `n ≥ 1`, then it is differentiable there. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_of_locally_cont_diff_on
(h : ∀ x ∈ s, ∃u, is_open u ∧ x ∈ u ∧ cont_diff_on 𝕜 n f (s ∩ u)) :
cont_diff_on 𝕜 n f s | begin
assume x xs,
rcases h x xs with ⟨u, u_open, xu, hu⟩,
apply (cont_diff_within_at_inter _).1 (hu x ⟨xs, xu⟩),
exact is_open.mem_nhds u_open xu
end | lemma | cont_diff_on_of_locally_cont_diff_on | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"cont_diff_within_at_inter",
"is_open",
"is_open.mem_nhds"
] | If a function is `C^n` around each point in a set, then it is `C^n` on the set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_succ_iff_has_fderiv_within_at {n : ℕ} :
cont_diff_on 𝕜 ((n + 1) : ℕ) f s
↔ ∀ x ∈ s, ∃ u ∈ 𝓝[insert x s] x, ∃ f' : E → (E →L[𝕜] F),
(∀ x ∈ u, has_fderiv_within_at f (f' x) u x) ∧ (cont_diff_on 𝕜 n f' u) | begin
split,
{ assume h x hx,
rcases (h x hx) n.succ le_rfl with ⟨u, hu, p, Hp⟩,
refine ⟨u, hu, λ y, (continuous_multilinear_curry_fin1 𝕜 E F) (p y 1),
λ y hy, Hp.has_fderiv_within_at (with_top.coe_le_coe.2 (nat.le_add_left 1 n)) hy, _⟩,
rw has_ftaylor_series_up_to_on_succ_iff_right at Hp,
as... | theorem | cont_diff_on_succ_iff_has_fderiv_within_at | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"cont_diff_on",
"cont_diff_within_at_succ_iff_has_fderiv_within_at",
"continuous_multilinear_curry_fin1",
"has_fderiv_within_at",
"has_ftaylor_series_up_to_on_succ_iff_right",
"le_rfl",
"mem_of_mem_nhds_within",
"self_mem_nhds_within"
] | A function is `C^(n + 1)` on a domain iff locally, it has a derivative which is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_fderiv_within (n : ℕ) (f : E → F) (s : set E) :
E → (E [×n]→L[𝕜] F) | nat.rec_on n
(λ x, continuous_multilinear_map.curry0 𝕜 E (f x))
(λ n rec x, continuous_linear_map.uncurry_left (fderiv_within 𝕜 rec s x)) | def | iterated_fderiv_within | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_linear_map.uncurry_left",
"continuous_multilinear_map.curry0",
"fderiv_within"
] | The `n`-th derivative of a function along a set, defined inductively by saying that the `n+1`-th
derivative of `f` is the derivative of the `n`-th derivative of `f` along this set, together with
an uncurrying step to see it as a multilinear map in `n+1` variables.. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ftaylor_series_within (f : E → F) (s : set E) (x : E) : formal_multilinear_series 𝕜 E F | λ n, iterated_fderiv_within 𝕜 n f s x | def | ftaylor_series_within | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"formal_multilinear_series",
"iterated_fderiv_within"
] | Formal Taylor series associated to a function within a set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_fderiv_within_zero_apply (m : (fin 0) → E) :
(iterated_fderiv_within 𝕜 0 f s x : ((fin 0) → E) → F) m = f x | rfl | lemma | iterated_fderiv_within_zero_apply | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"iterated_fderiv_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_within_zero_eq_comp :
iterated_fderiv_within 𝕜 0 f s = (continuous_multilinear_curry_fin0 𝕜 E F).symm ∘ f | rfl | lemma | iterated_fderiv_within_zero_eq_comp | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_multilinear_curry_fin0",
"iterated_fderiv_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_iterated_fderiv_within_zero :
‖iterated_fderiv_within 𝕜 0 f s x‖ = ‖f x‖ | by rw [iterated_fderiv_within_zero_eq_comp, linear_isometry_equiv.norm_map] | lemma | norm_iterated_fderiv_within_zero | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"iterated_fderiv_within_zero_eq_comp",
"linear_isometry_equiv.norm_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_within_succ_apply_left {n : ℕ} (m : fin (n + 1) → E):
(iterated_fderiv_within 𝕜 (n + 1) f s x : (fin (n + 1) → E) → F) m
= (fderiv_within 𝕜 (iterated_fderiv_within 𝕜 n f s) s x : E → (E [×n]→L[𝕜] F))
(m 0) (tail m) | rfl | lemma | iterated_fderiv_within_succ_apply_left | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"fderiv_within",
"iterated_fderiv_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_within_succ_eq_comp_left {n : ℕ} :
iterated_fderiv_within 𝕜 (n + 1) f s =
(continuous_multilinear_curry_left_equiv 𝕜 (λ(i : fin (n + 1)), E) F)
∘ (fderiv_within 𝕜 (iterated_fderiv_within 𝕜 n f s) s) | rfl | lemma | iterated_fderiv_within_succ_eq_comp_left | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_multilinear_curry_left_equiv",
"fderiv_within",
"iterated_fderiv_within"
] | Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
and the derivative of the `n`-th derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_fderiv_within_iterated_fderiv_within {n : ℕ} :
‖fderiv_within 𝕜 (iterated_fderiv_within 𝕜 n f s) s x‖ =
‖iterated_fderiv_within 𝕜 (n + 1) f s x‖ | by rw [iterated_fderiv_within_succ_eq_comp_left, linear_isometry_equiv.norm_map] | lemma | norm_fderiv_within_iterated_fderiv_within | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"iterated_fderiv_within",
"iterated_fderiv_within_succ_eq_comp_left",
"linear_isometry_equiv.norm_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_within_succ_apply_right {n : ℕ}
(hs : unique_diff_on 𝕜 s) (hx : x ∈ s) (m : fin (n + 1) → E) :
(iterated_fderiv_within 𝕜 (n + 1) f s x : (fin (n + 1) → E) → F) m
= iterated_fderiv_within 𝕜 n (λy, fderiv_within 𝕜 f s y) s x (init m) (m (last n)) | begin
induction n with n IH generalizing x,
{ rw [iterated_fderiv_within_succ_eq_comp_left, iterated_fderiv_within_zero_eq_comp,
iterated_fderiv_within_zero_apply,
function.comp_apply, linear_isometry_equiv.comp_fderiv_within _ (hs x hx)],
refl },
{ let I := continuous_multilinear_curry_right_... | theorem | iterated_fderiv_within_succ_apply_right | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_multilinear_curry_right_equiv'",
"fderiv_within",
"fderiv_within_congr",
"function.comp_apply",
"iterated_fderiv_within",
"iterated_fderiv_within_succ_apply_left",
"iterated_fderiv_within_succ_eq_comp_left",
"iterated_fderiv_within_zero_apply",
"iterated_fderiv_within_zero_eq_comp",
"l... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_within_succ_eq_comp_right {n : ℕ} (hs : unique_diff_on 𝕜 s) (hx : x ∈ s) :
iterated_fderiv_within 𝕜 (n + 1) f s x =
((continuous_multilinear_curry_right_equiv' 𝕜 n E F)
∘ (iterated_fderiv_within 𝕜 n (λy, fderiv_within 𝕜 f s y) s)) x | by { ext m, rw iterated_fderiv_within_succ_apply_right hs hx, refl } | lemma | iterated_fderiv_within_succ_eq_comp_right | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_multilinear_curry_right_equiv'",
"fderiv_within",
"iterated_fderiv_within",
"iterated_fderiv_within_succ_apply_right",
"unique_diff_on"
] | Writing explicitly the `n+1`-th derivative as the composition of a currying linear equiv,
and the `n`-th derivative of the derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
norm_iterated_fderiv_within_fderiv_within {n : ℕ} (hs : unique_diff_on 𝕜 s) (hx : x ∈ s) :
‖iterated_fderiv_within 𝕜 n (fderiv_within 𝕜 f s) s x‖ =
‖iterated_fderiv_within 𝕜 (n + 1) f s x‖ | by rw [iterated_fderiv_within_succ_eq_comp_right hs hx, linear_isometry_equiv.norm_map] | lemma | norm_iterated_fderiv_within_fderiv_within | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"fderiv_within",
"iterated_fderiv_within_succ_eq_comp_right",
"linear_isometry_equiv.norm_map",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_within_one_apply
(h : unique_diff_within_at 𝕜 s x) (m : fin 1 → E) :
(iterated_fderiv_within 𝕜 1 f s x : ((fin 1) → E) → F) m
= (fderiv_within 𝕜 f s x : E → F) (m 0) | begin
simp only [iterated_fderiv_within_succ_apply_left, iterated_fderiv_within_zero_eq_comp,
(continuous_multilinear_curry_fin0 𝕜 E F).symm.comp_fderiv_within h],
refl
end | lemma | iterated_fderiv_within_one_apply | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"continuous_multilinear_curry_fin0",
"fderiv_within",
"iterated_fderiv_within",
"iterated_fderiv_within_succ_apply_left",
"iterated_fderiv_within_zero_eq_comp",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.iterated_fderiv_within' (h : f₁ =ᶠ[𝓝[s] x] f) (ht : t ⊆ s) (n : ℕ) :
iterated_fderiv_within 𝕜 n f₁ t =ᶠ[𝓝[s] x] iterated_fderiv_within 𝕜 n f t | begin
induction n with n ihn,
{ exact h.mono (λ y hy, fun_like.ext _ _ $ λ _, hy) },
{ have : fderiv_within 𝕜 _ t =ᶠ[𝓝[s] x] fderiv_within 𝕜 _ t := ihn.fderiv_within' ht,
apply this.mono,
intros y hy,
simp only [iterated_fderiv_within_succ_eq_comp_left, hy, (∘)] }
end | lemma | filter.eventually_eq.iterated_fderiv_within' | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"fderiv_within",
"fun_like.ext",
"iterated_fderiv_within",
"iterated_fderiv_within_succ_eq_comp_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.iterated_fderiv_within (h : f₁ =ᶠ[𝓝[s] x] f) (n : ℕ) :
iterated_fderiv_within 𝕜 n f₁ s =ᶠ[𝓝[s] x] iterated_fderiv_within 𝕜 n f s | h.iterated_fderiv_within' subset.rfl n | lemma | filter.eventually_eq.iterated_fderiv_within | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"iterated_fderiv_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.eventually_eq.iterated_fderiv_within_eq (h : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x)
(n : ℕ) : iterated_fderiv_within 𝕜 n f₁ s x = iterated_fderiv_within 𝕜 n f s x | have f₁ =ᶠ[𝓝[insert x s] x] f, by simpa [eventually_eq, hx],
(this.iterated_fderiv_within' (subset_insert _ _) n).self_of_nhds_within (mem_insert _ _) | lemma | filter.eventually_eq.iterated_fderiv_within_eq | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"iterated_fderiv_within"
] | If two functions coincide in a neighborhood of `x` within a set `s` and at `x`, then their
iterated differentials within this set at `x` coincide. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_fderiv_within_congr (hs : eq_on f₁ f s) (hx : x ∈ s) (n : ℕ) :
iterated_fderiv_within 𝕜 n f₁ s x = iterated_fderiv_within 𝕜 n f s x | (hs.eventually_eq.filter_mono inf_le_right).iterated_fderiv_within_eq (hs hx) _ | lemma | iterated_fderiv_within_congr | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"inf_le_right",
"iterated_fderiv_within"
] | If two functions coincide on a set `s`, then their iterated differentials within this set
coincide. See also `filter.eventually_eq.iterated_fderiv_within_eq` and
`filter.eventually_eq.iterated_fderiv_within`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.eq_on.iterated_fderiv_within (hs : eq_on f₁ f s) (n : ℕ) :
eq_on (iterated_fderiv_within 𝕜 n f₁ s) (iterated_fderiv_within 𝕜 n f s) s | λ x hx, iterated_fderiv_within_congr hs hx n | lemma | set.eq_on.iterated_fderiv_within | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"iterated_fderiv_within",
"iterated_fderiv_within_congr"
] | If two functions coincide on a set `s`, then their iterated differentials within this set
coincide. See also `filter.eventually_eq.iterated_fderiv_within_eq` and
`filter.eventually_eq.iterated_fderiv_within`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_fderiv_within_eventually_congr_set' (y : E) (h : s =ᶠ[𝓝[{y}ᶜ] x] t) (n : ℕ) :
iterated_fderiv_within 𝕜 n f s =ᶠ[𝓝 x] iterated_fderiv_within 𝕜 n f t | begin
induction n with n ihn generalizing x,
{ refl },
{ refine (eventually_nhds_nhds_within.2 h).mono (λ y hy, _),
simp only [iterated_fderiv_within_succ_eq_comp_left, (∘)],
rw [(ihn hy).fderiv_within_eq_nhds, fderiv_within_congr_set' _ hy] }
end | lemma | iterated_fderiv_within_eventually_congr_set' | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"fderiv_within_congr_set'",
"iterated_fderiv_within",
"iterated_fderiv_within_succ_eq_comp_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_within_eventually_congr_set (h : s =ᶠ[𝓝 x] t) (n : ℕ) :
iterated_fderiv_within 𝕜 n f s =ᶠ[𝓝 x] iterated_fderiv_within 𝕜 n f t | iterated_fderiv_within_eventually_congr_set' x (h.filter_mono inf_le_left) n | lemma | iterated_fderiv_within_eventually_congr_set | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"inf_le_left",
"iterated_fderiv_within",
"iterated_fderiv_within_eventually_congr_set'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_within_congr_set (h : s =ᶠ[𝓝 x] t) (n : ℕ) :
iterated_fderiv_within 𝕜 n f s x = iterated_fderiv_within 𝕜 n f t x | (iterated_fderiv_within_eventually_congr_set h n).self_of_nhds | lemma | iterated_fderiv_within_congr_set | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"iterated_fderiv_within",
"iterated_fderiv_within_eventually_congr_set"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_within_inter' {n : ℕ} (hu : u ∈ 𝓝[s] x) :
iterated_fderiv_within 𝕜 n f (s ∩ u) x = iterated_fderiv_within 𝕜 n f s x | iterated_fderiv_within_congr_set (nhds_within_eq_iff_eventually_eq.1 $ nhds_within_inter_of_mem' hu)
_ | lemma | iterated_fderiv_within_inter' | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"iterated_fderiv_within",
"iterated_fderiv_within_congr_set",
"nhds_within_inter_of_mem'"
] | The iterated differential within a set `s` at a point `x` is not modified if one intersects
`s` with a neighborhood of `x` within `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_fderiv_within_inter {n : ℕ} (hu : u ∈ 𝓝 x) :
iterated_fderiv_within 𝕜 n f (s ∩ u) x = iterated_fderiv_within 𝕜 n f s x | iterated_fderiv_within_inter' (mem_nhds_within_of_mem_nhds hu) | lemma | iterated_fderiv_within_inter | analysis.calculus | src/analysis/calculus/cont_diff_def.lean | [
"analysis.calculus.fderiv.add",
"analysis.calculus.fderiv.mul",
"analysis.calculus.fderiv.equiv",
"analysis.calculus.fderiv.restrict_scalars",
"analysis.calculus.formal_multilinear_series"
] | [
"iterated_fderiv_within",
"iterated_fderiv_within_inter'",
"mem_nhds_within_of_mem_nhds"
] | The iterated differential within a set `s` at a point `x` is not modified if one intersects
`s` with a neighborhood of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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