statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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iterated_fderiv_neg_apply {i : ℕ} {f : E → F} :
iterated_fderiv 𝕜 i (-f) x = -iterated_fderiv 𝕜 i f x | begin
simp_rw [←iterated_fderiv_within_univ],
exact iterated_fderiv_within_neg_apply unique_diff_on_univ (set.mem_univ _),
end | lemma | iterated_fderiv_neg_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"
] | [
"iterated_fderiv",
"iterated_fderiv_within_neg_apply",
"set.mem_univ",
"unique_diff_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.sub {s : set E} {f g : E → F}
(hf : cont_diff_within_at 𝕜 n f s x) (hg : cont_diff_within_at 𝕜 n g s x) :
cont_diff_within_at 𝕜 n (λx, f x - g x) s x | by simpa only [sub_eq_add_neg] using hf.add hg.neg | lemma | cont_diff_within_at.sub | 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_within_at"
] | The difference of two `C^n` functions within a set at a point is `C^n` within this set
at this point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.sub {f g : E → F}
(hf : cont_diff_at 𝕜 n f x) (hg : cont_diff_at 𝕜 n g x) :
cont_diff_at 𝕜 n (λx, f x - g x) x | by simpa only [sub_eq_add_neg] using hf.add hg.neg | lemma | cont_diff_at.sub | 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"
] | The difference of two `C^n` functions at a point is `C^n` at this point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.sub {s : set E} {f g : E → F}
(hf : cont_diff_on 𝕜 n f s) (hg : cont_diff_on 𝕜 n g s) :
cont_diff_on 𝕜 n (λx, f x - g x) s | by simpa only [sub_eq_add_neg] using hf.add hg.neg | lemma | cont_diff_on.sub | 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"
] | The difference of two `C^n` functions on a domain is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.sub {f g : E → F}
(hf : cont_diff 𝕜 n f) (hg : cont_diff 𝕜 n g) : cont_diff 𝕜 n (λx, f x - g x) | by simpa only [sub_eq_add_neg] using hf.add hg.neg | lemma | cont_diff.sub | 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"
] | The difference of two `C^n` functions is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at.sum
{ι : Type*} {f : ι → E → F} {s : finset ι} {t : set E} {x : E}
(h : ∀ i ∈ s, cont_diff_within_at 𝕜 n (λ x, f i x) t x) :
cont_diff_within_at 𝕜 n (λ x, (∑ i in s, f i x)) t x | begin
classical,
induction s using finset.induction_on with i s is IH,
{ simp [cont_diff_within_at_const] },
{ simp only [is, finset.sum_insert, not_false_iff],
exact (h _ (finset.mem_insert_self i s)).add (IH (λ j hj, h _ (finset.mem_insert_of_mem hj))) }
end | lemma | cont_diff_within_at.sum | 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_within_at",
"cont_diff_within_at_const",
"finset",
"finset.induction_on",
"finset.mem_insert_of_mem",
"finset.mem_insert_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.sum
{ι : Type*} {f : ι → E → F} {s : finset ι} {x : E}
(h : ∀ i ∈ s, cont_diff_at 𝕜 n (λ x, f i x) x) :
cont_diff_at 𝕜 n (λ x, (∑ i in s, f i x)) x | by rw [← cont_diff_within_at_univ] at *; exact cont_diff_within_at.sum h | lemma | cont_diff_at.sum | 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",
"cont_diff_within_at.sum",
"cont_diff_within_at_univ",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.sum
{ι : Type*} {f : ι → E → F} {s : finset ι} {t : set E}
(h : ∀ i ∈ s, cont_diff_on 𝕜 n (λ x, f i x) t) :
cont_diff_on 𝕜 n (λ x, (∑ i in s, f i x)) t | λ x hx, cont_diff_within_at.sum (λ i hi, h i hi x hx) | lemma | cont_diff_on.sum | 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",
"cont_diff_within_at.sum",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.sum
{ι : Type*} {f : ι → E → F} {s : finset ι}
(h : ∀ i ∈ s, cont_diff 𝕜 n (λ x, f i x)) :
cont_diff 𝕜 n (λ x, (∑ i in s, f i x)) | by simp only [← cont_diff_on_univ] at *; exact cont_diff_on.sum h | lemma | cont_diff.sum | 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",
"cont_diff_on.sum",
"cont_diff_on_univ",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_mul : cont_diff 𝕜 n (λ p : 𝔸 × 𝔸, p.1 * p.2) | (continuous_linear_map.mul 𝕜 𝔸).is_bounded_bilinear_map.cont_diff | lemma | cont_diff_mul | 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.mul",
"is_bounded_bilinear_map.cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.mul {s : set E} {f g : E → 𝔸}
(hf : cont_diff_within_at 𝕜 n f s x) (hg : cont_diff_within_at 𝕜 n g s x) :
cont_diff_within_at 𝕜 n (λ x, f x * g x) s x | cont_diff_mul.comp_cont_diff_within_at (hf.prod hg) | lemma | cont_diff_within_at.mul | 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_within_at"
] | The product of two `C^n` functions within a set at a point is `C^n` within this set
at this point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.mul {f g : E → 𝔸} (hf : cont_diff_at 𝕜 n f x) (hg : cont_diff_at 𝕜 n g x) :
cont_diff_at 𝕜 n (λ x, f x * g x) x | hf.mul hg | lemma | cont_diff_at.mul | 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"
] | The product of two `C^n` functions at a point is `C^n` at this point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.mul {f g : E → 𝔸} (hf : cont_diff_on 𝕜 n f s) (hg : cont_diff_on 𝕜 n g s) :
cont_diff_on 𝕜 n (λ x, f x * g x) s | λ x hx, (hf x hx).mul (hg x hx) | lemma | cont_diff_on.mul | 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"
] | The product of two `C^n` functions on a domain is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.mul {f g : E → 𝔸} (hf : cont_diff 𝕜 n f) (hg : cont_diff 𝕜 n g) :
cont_diff 𝕜 n (λ x, f x * g x) | cont_diff_mul.comp (hf.prod hg) | lemma | cont_diff.mul | 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"
] | The product of two `C^n`functions is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at_prod' {t : finset ι} {f : ι → E → 𝔸'}
(h : ∀ i ∈ t, cont_diff_within_at 𝕜 n (f i) s x) :
cont_diff_within_at 𝕜 n (∏ i in t, f i) s x | finset.prod_induction f (λ f, cont_diff_within_at 𝕜 n f s x) (λ _ _, cont_diff_within_at.mul)
(@cont_diff_within_at_const _ _ _ _ _ _ _ _ _ _ _ 1) h | lemma | cont_diff_within_at_prod' | 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_within_at",
"cont_diff_within_at.mul",
"cont_diff_within_at_const",
"finset",
"finset.prod_induction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at_prod {t : finset ι} {f : ι → E → 𝔸'}
(h : ∀ i ∈ t, cont_diff_within_at 𝕜 n (f i) s x) :
cont_diff_within_at 𝕜 n (λ y, ∏ i in t, f i y) s x | by simpa only [← finset.prod_apply] using cont_diff_within_at_prod' h | lemma | cont_diff_within_at_prod | 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_within_at",
"cont_diff_within_at_prod'",
"finset",
"finset.prod_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_prod' {t : finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, cont_diff_at 𝕜 n (f i) x) :
cont_diff_at 𝕜 n (∏ i in t, f i) x | cont_diff_within_at_prod' h | lemma | cont_diff_at_prod' | 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",
"cont_diff_within_at_prod'",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_prod {t : finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, cont_diff_at 𝕜 n (f i) x) :
cont_diff_at 𝕜 n (λ y, ∏ i in t, f i y) x | cont_diff_within_at_prod h | lemma | cont_diff_at_prod | 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",
"cont_diff_within_at_prod",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_prod' {t : finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, cont_diff_on 𝕜 n (f i) s) :
cont_diff_on 𝕜 n (∏ i in t, f i) s | λ x hx, cont_diff_within_at_prod' (λ i hi, h i hi x hx) | lemma | cont_diff_on_prod' | 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",
"cont_diff_within_at_prod'",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_prod {t : finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, cont_diff_on 𝕜 n (f i) s) :
cont_diff_on 𝕜 n (λ y, ∏ i in t, f i y) s | λ x hx, cont_diff_within_at_prod (λ i hi, h i hi x hx) | lemma | cont_diff_on_prod | 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",
"cont_diff_within_at_prod",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_prod' {t : finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, cont_diff 𝕜 n (f i)) :
cont_diff 𝕜 n (∏ i in t, f i) | cont_diff_iff_cont_diff_at.mpr $ λ x, cont_diff_at_prod' $ λ i hi, (h i hi).cont_diff_at | lemma | cont_diff_prod' | 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",
"cont_diff_at",
"cont_diff_at_prod'",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_prod {t : finset ι} {f : ι → E → 𝔸'} (h : ∀ i ∈ t, cont_diff 𝕜 n (f i)) :
cont_diff 𝕜 n (λ y, ∏ i in t, f i y) | cont_diff_iff_cont_diff_at.mpr $ λ x, cont_diff_at_prod $ λ i hi, (h i hi).cont_diff_at | lemma | cont_diff_prod | 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",
"cont_diff_at",
"cont_diff_at_prod",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.pow {f : E → 𝔸} (hf : cont_diff 𝕜 n f) :
∀ m : ℕ, cont_diff 𝕜 n (λ x, (f x) ^ m) | | 0 := by simpa using cont_diff_const
| (m + 1) := by simpa [pow_succ] using hf.mul (cont_diff.pow m) | lemma | cont_diff.pow | 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",
"cont_diff_const",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.pow {f : E → 𝔸} (hf : cont_diff_within_at 𝕜 n f s x) (m : ℕ) :
cont_diff_within_at 𝕜 n (λ y, f y ^ m) s x | (cont_diff_id.pow m).comp_cont_diff_within_at hf | lemma | cont_diff_within_at.pow | 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_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.pow {f : E → 𝔸} (hf : cont_diff_at 𝕜 n f x) (m : ℕ) :
cont_diff_at 𝕜 n (λ y, f y ^ m) x | hf.pow m | lemma | cont_diff_at.pow | 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_on.pow {f : E → 𝔸} (hf : cont_diff_on 𝕜 n f s) (m : ℕ) :
cont_diff_on 𝕜 n (λ y, f y ^ m) s | λ y hy, (hf y hy).pow m | lemma | cont_diff_on.pow | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.div_const {f : E → 𝕜'} {n}
(hf : cont_diff_within_at 𝕜 n f s x) (c : 𝕜') :
cont_diff_within_at 𝕜 n (λ x, f x / c) s x | by simpa only [div_eq_mul_inv] using hf.mul cont_diff_within_at_const | lemma | cont_diff_within_at.div_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_within_at",
"cont_diff_within_at_const",
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.div_const {f : E → 𝕜'} {n} (hf : cont_diff_at 𝕜 n f x) (c : 𝕜') :
cont_diff_at 𝕜 n (λ x, f x / c) x | hf.div_const c | lemma | cont_diff_at.div_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_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.div_const {f : E → 𝕜'} {n} (hf : cont_diff_on 𝕜 n f s) (c : 𝕜') :
cont_diff_on 𝕜 n (λ x, f x / c) s | λ x hx, (hf x hx).div_const c | lemma | cont_diff_on.div_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.div_const {f : E → 𝕜'} {n} (hf : cont_diff 𝕜 n f) (c : 𝕜') :
cont_diff 𝕜 n (λ x, f x / c) | by simpa only [div_eq_mul_inv] using hf.mul cont_diff_const | lemma | cont_diff.div_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",
"cont_diff_const",
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_smul : cont_diff 𝕜 n (λ p : 𝕜 × F, p.1 • p.2) | is_bounded_bilinear_map_smul.cont_diff | lemma | cont_diff_smul | 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 | |
cont_diff_within_at.smul {s : set E} {f : E → 𝕜} {g : E → F}
(hf : cont_diff_within_at 𝕜 n f s x) (hg : cont_diff_within_at 𝕜 n g s x) :
cont_diff_within_at 𝕜 n (λ x, f x • g x) s x | cont_diff_smul.cont_diff_within_at.comp x (hf.prod hg) subset_preimage_univ | lemma | cont_diff_within_at.smul | 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_within_at"
] | The scalar multiplication of two `C^n` functions within a set at a point is `C^n` within this
set at this point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.smul {f : E → 𝕜} {g : E → F}
(hf : cont_diff_at 𝕜 n f x) (hg : cont_diff_at 𝕜 n g x) :
cont_diff_at 𝕜 n (λ x, f x • g x) x | by rw [← cont_diff_within_at_univ] at *; exact hf.smul hg | lemma | cont_diff_at.smul | 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",
"cont_diff_within_at_univ"
] | The scalar multiplication of two `C^n` functions at a point is `C^n` at this point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.smul {f : E → 𝕜} {g : E → F} (hf : cont_diff 𝕜 n f) (hg : cont_diff 𝕜 n g) :
cont_diff 𝕜 n (λ x, f x • g x) | cont_diff_smul.comp (hf.prod hg) | lemma | cont_diff.smul | 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"
] | The scalar multiplication of two `C^n` functions is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.smul {s : set E} {f : E → 𝕜} {g : E → F}
(hf : cont_diff_on 𝕜 n f s) (hg : cont_diff_on 𝕜 n g s) :
cont_diff_on 𝕜 n (λ x, f x • g x) s | λ x hx, (hf x hx).smul (hg x hx) | lemma | cont_diff_on.smul | 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"
] | The scalar multiplication of two `C^n` functions on a domain is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_const_smul (c : R) : cont_diff 𝕜 n (λ p : F, c • p) | (c • continuous_linear_map.id 𝕜 F).cont_diff | lemma | cont_diff_const_smul | 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.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.const_smul {s : set E} {f : E → F} {x : E} (c : R)
(hf : cont_diff_within_at 𝕜 n f s x) : cont_diff_within_at 𝕜 n (λ y, c • f y) s x | (cont_diff_const_smul c).cont_diff_at.comp_cont_diff_within_at x hf | lemma | cont_diff_within_at.const_smul | 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.comp_cont_diff_within_at",
"cont_diff_const_smul",
"cont_diff_within_at"
] | The scalar multiplication of a constant and a `C^n` function within a set at a point is `C^n`
within this set at this point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.const_smul {f : E → F} {x : E} (c : R)
(hf : cont_diff_at 𝕜 n f x) : cont_diff_at 𝕜 n (λ y, c • f y) x | by rw [←cont_diff_within_at_univ] at *; exact hf.const_smul c | lemma | cont_diff_at.const_smul | 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"
] | The scalar multiplication of a constant and a `C^n` function at a point is `C^n` at this
point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.const_smul {f : E → F} (c : R)
(hf : cont_diff 𝕜 n f) : cont_diff 𝕜 n (λ y, c • f y) | (cont_diff_const_smul c).comp hf | lemma | cont_diff.const_smul | 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",
"cont_diff_const_smul"
] | The scalar multiplication of a constant and a `C^n` function is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.const_smul {s : set E} {f : E → F} (c : R)
(hf : cont_diff_on 𝕜 n f s) : cont_diff_on 𝕜 n (λ y, c • f y) s | λ x hx, (hf x hx).const_smul c | lemma | cont_diff_on.const_smul | 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"
] | The scalar multiplication of a constant and a `C^n` on a domain is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iterated_fderiv_within_const_smul_apply (hf : cont_diff_on 𝕜 i f s)
(hu : unique_diff_on 𝕜 s) (hx : x ∈ s) :
iterated_fderiv_within 𝕜 i (a • f) s x = a • (iterated_fderiv_within 𝕜 i f s x) | begin
induction i with i hi generalizing x,
{ ext, simp },
{ ext h,
have hi' : (i : ℕ∞) < i+1 :=
with_top.coe_lt_coe.mpr (nat.lt_succ_self _),
have hdf : differentiable_on 𝕜 (iterated_fderiv_within 𝕜 i f s) s :=
hf.differentiable_on_iterated_fderiv_within hi' hu,
have hcdf : cont_diff_on... | lemma | iterated_fderiv_within_const_smul_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",
"differentiable_on",
"fderiv_within",
"fderiv_within_congr'",
"fderiv_within_const_smul",
"fin.tail",
"iterated_fderiv_within",
"pi.smul_def",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iterated_fderiv_const_smul_apply {x : E} (hf : cont_diff 𝕜 i f) :
iterated_fderiv 𝕜 i (a • f) x = a • iterated_fderiv 𝕜 i f x | begin
simp_rw [←cont_diff_on_univ, ←iterated_fderiv_within_univ] at *,
refine iterated_fderiv_within_const_smul_apply hf unique_diff_on_univ (set.mem_univ _),
end | lemma | iterated_fderiv_const_smul_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",
"iterated_fderiv",
"iterated_fderiv_within_const_smul_apply",
"set.mem_univ",
"unique_diff_on_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.prod_map'
{s : set E} {t : set E'} {f : E → F} {g : E' → F'} {p : E × E'}
(hf : cont_diff_within_at 𝕜 n f s p.1) (hg : cont_diff_within_at 𝕜 n g t p.2) :
cont_diff_within_at 𝕜 n (prod.map f g) (s ×ˢ t) p | (hf.comp p cont_diff_within_at_fst (prod_subset_preimage_fst _ _)).prod
(hg.comp p cont_diff_within_at_snd (prod_subset_preimage_snd _ _)) | lemma | cont_diff_within_at.prod_map' | 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_within_at",
"cont_diff_within_at_fst",
"cont_diff_within_at_snd"
] | The product map of two `C^n` functions within a set at a point is `C^n`
within the product set at the product point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at.prod_map
{s : set E} {t : set E'} {f : E → F} {g : E' → F'} {x : E} {y : E'}
(hf : cont_diff_within_at 𝕜 n f s x) (hg : cont_diff_within_at 𝕜 n g t y) :
cont_diff_within_at 𝕜 n (prod.map f g) (s ×ˢ t) (x, y) | cont_diff_within_at.prod_map' hf hg | lemma | cont_diff_within_at.prod_map | 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_within_at",
"cont_diff_within_at.prod_map'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.prod_map {E' : Type*} [normed_add_comm_group E'] [normed_space 𝕜 E']
{F' : Type*} [normed_add_comm_group F'] [normed_space 𝕜 F']
{s : set E} {t : set E'} {f : E → F} {g : E' → F'}
(hf : cont_diff_on 𝕜 n f s) (hg : cont_diff_on 𝕜 n g t) :
cont_diff_on 𝕜 n (prod.map f g) (s ×ˢ t) | (hf.comp cont_diff_on_fst (prod_subset_preimage_fst _ _)).prod
(hg.comp (cont_diff_on_snd) (prod_subset_preimage_snd _ _)) | lemma | cont_diff_on.prod_map | 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",
"cont_diff_on_fst",
"cont_diff_on_snd",
"normed_add_comm_group",
"normed_space"
] | The product map of two `C^n` functions on a set is `C^n` on the product set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.prod_map {f : E → F} {g : E' → F'} {x : E} {y : E'}
(hf : cont_diff_at 𝕜 n f x) (hg : cont_diff_at 𝕜 n g y) :
cont_diff_at 𝕜 n (prod.map f g) (x, y) | begin
rw cont_diff_at at *,
convert hf.prod_map hg,
simp only [univ_prod_univ]
end | lemma | cont_diff_at.prod_map | 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"
] | The product map of two `C^n` functions within a set at a point is `C^n`
within the product set at the product point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.prod_map' {f : E → F} {g : E' → F'} {p : E × E'}
(hf : cont_diff_at 𝕜 n f p.1) (hg : cont_diff_at 𝕜 n g p.2) :
cont_diff_at 𝕜 n (prod.map f g) p | begin
rcases p,
exact cont_diff_at.prod_map hf hg
end | lemma | cont_diff_at.prod_map' | 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",
"cont_diff_at.prod_map"
] | The product map of two `C^n` functions within a set at a point is `C^n`
within the product set at the product point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.prod_map {f : E → F} {g : E' → F'}
(hf : cont_diff 𝕜 n f) (hg : cont_diff 𝕜 n g) :
cont_diff 𝕜 n (prod.map f g) | begin
rw cont_diff_iff_cont_diff_at at *,
exact λ ⟨x, y⟩, (hf x).prod_map (hg y)
end | lemma | cont_diff.prod_map | 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",
"cont_diff_iff_cont_diff_at",
"prod_map"
] | The product map of two `C^n` functions is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_prod_mk_left (f₀ : F) : cont_diff 𝕜 n (λ e : E, (e, f₀)) | cont_diff_id.prod cont_diff_const | lemma | cont_diff_prod_mk_left | 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",
"cont_diff_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_prod_mk_right (e₀ : E) : cont_diff 𝕜 n (λ f : F, (e₀, f)) | cont_diff_const.prod cont_diff_id | lemma | cont_diff_prod_mk_right | 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",
"cont_diff_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_ring_inverse [complete_space R] (x : Rˣ) :
cont_diff_at 𝕜 n ring.inverse (x : R) | begin
induction n using enat.nat_induction with n IH Itop,
{ intros m hm,
refine ⟨{y : R | is_unit y}, _, _⟩,
{ simp [nhds_within_univ],
exact x.nhds },
{ use (ftaylor_series_within 𝕜 inverse univ),
rw [le_antisymm hm bot_le, has_ftaylor_series_up_to_on_zero_iff],
split,
{ rintr... | lemma | cont_diff_at_ring_inverse | 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"
] | [
"bot_le",
"complete_space",
"cont_diff_at",
"continuous_within_at",
"enat.nat_induction",
"ftaylor_series_within",
"has_fderiv_at_ring_inverse",
"has_ftaylor_series_up_to_on_zero_iff",
"is_unit",
"nhds_within_univ",
"ring.inverse"
] | In a complete normed algebra, the operation of inversion is `C^n`, for all `n`, at each
invertible element. The proof is by induction, bootstrapping using an identity expressing the
derivative of inversion as a bilinear map of inversion itself. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at_inv {x : 𝕜'} (hx : x ≠ 0) {n} :
cont_diff_at 𝕜 n has_inv.inv x | by simpa only [ring.inverse_eq_inv'] using cont_diff_at_ring_inverse 𝕜 (units.mk0 x hx) | lemma | cont_diff_at_inv | 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",
"cont_diff_at_ring_inverse",
"ring.inverse_eq_inv'",
"units.mk0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_inv {n} : cont_diff_on 𝕜 n (has_inv.inv : 𝕜' → 𝕜') {0}ᶜ | λ x hx, (cont_diff_at_inv 𝕜 hx).cont_diff_within_at | lemma | cont_diff_on_inv | 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_inv",
"cont_diff_on",
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.inv {f : E → 𝕜'} {n} (hf : cont_diff_within_at 𝕜 n f s x)
(hx : f x ≠ 0) :
cont_diff_within_at 𝕜 n (λ x, (f x)⁻¹) s x | (cont_diff_at_inv 𝕜 hx).comp_cont_diff_within_at x hf | lemma | cont_diff_within_at.inv | 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_inv",
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.inv {f : E → 𝕜'} {n} (hf : cont_diff_on 𝕜 n f s)
(h : ∀ x ∈ s, f x ≠ 0) :
cont_diff_on 𝕜 n (λ x, (f x)⁻¹) s | λ x hx, (hf.cont_diff_within_at hx).inv (h x hx) | lemma | cont_diff_on.inv | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.inv {f : E → 𝕜'} {n} (hf : cont_diff_at 𝕜 n f x) (hx : f x ≠ 0) :
cont_diff_at 𝕜 n (λ x, (f x)⁻¹) x | hf.inv hx | lemma | cont_diff_at.inv | 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.inv {f : E → 𝕜'} {n} (hf : cont_diff 𝕜 n f) (h : ∀ x, f x ≠ 0) :
cont_diff 𝕜 n (λ x, (f x)⁻¹) | by { rw cont_diff_iff_cont_diff_at, exact λ x, hf.cont_diff_at.inv (h x) } | lemma | cont_diff.inv | 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",
"cont_diff_iff_cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.div [complete_space 𝕜] {f g : E → 𝕜} {n}
(hf : cont_diff_within_at 𝕜 n f s x) (hg : cont_diff_within_at 𝕜 n g s x)
(hx : g x ≠ 0) :
cont_diff_within_at 𝕜 n (λ x, f x / g x) s x | by simpa only [div_eq_mul_inv] using hf.mul (hg.inv hx) | lemma | cont_diff_within_at.div | 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"
] | [
"complete_space",
"cont_diff_within_at",
"div_eq_mul_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.div [complete_space 𝕜] {f g : E → 𝕜} {n}
(hf : cont_diff_on 𝕜 n f s) (hg : cont_diff_on 𝕜 n g s) (h₀ : ∀ x ∈ s, g x ≠ 0) :
cont_diff_on 𝕜 n (f / g) s | λ x hx, (hf x hx).div (hg x hx) (h₀ x hx) | lemma | cont_diff_on.div | 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"
] | [
"complete_space",
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.div [complete_space 𝕜] {f g : E → 𝕜} {n}
(hf : cont_diff_at 𝕜 n f x) (hg : cont_diff_at 𝕜 n g x)
(hx : g x ≠ 0) :
cont_diff_at 𝕜 n (λ x, f x / g x) x | hf.div hg hx | lemma | cont_diff_at.div | 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"
] | [
"complete_space",
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.div [complete_space 𝕜] {f g : E → 𝕜} {n}
(hf : cont_diff 𝕜 n f) (hg : cont_diff 𝕜 n g)
(h0 : ∀ x, g x ≠ 0) :
cont_diff 𝕜 n (λ x, f x / g x) | begin
simp only [cont_diff_iff_cont_diff_at] at *,
exact λ x, (hf x).div (hg x) (h0 x)
end | lemma | cont_diff.div | 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"
] | [
"complete_space",
"cont_diff",
"cont_diff_iff_cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_map_inverse [complete_space E] (e : E ≃L[𝕜] F) :
cont_diff_at 𝕜 n inverse (e : E →L[𝕜] F) | begin
nontriviality E,
-- first, we use the lemma `to_ring_inverse` to rewrite in terms of `ring.inverse` in the ring
-- `E →L[𝕜] E`
let O₁ : (E →L[𝕜] E) → (F →L[𝕜] E) := λ f, f.comp (e.symm : (F →L[𝕜] E)),
let O₂ : (E →L[𝕜] F) → (E →L[𝕜] E) := λ f, (e.symm : (F →L[𝕜] E)).comp f,
have : continuous_li... | lemma | cont_diff_at_map_inverse | 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"
] | [
"complete_space",
"cont_diff",
"cont_diff_at",
"cont_diff_at.comp",
"cont_diff_at_ring_inverse",
"cont_diff_const",
"cont_diff_id",
"continuous_linear_map.inverse",
"ring.inverse"
] | At a continuous linear equivalence `e : E ≃L[𝕜] F` between Banach spaces, the operation of
inversion is `C^n`, for all `n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_homeomorph.cont_diff_at_symm [complete_space E]
(f : local_homeomorph E F) {f₀' : E ≃L[𝕜] F} {a : F} (ha : a ∈ f.target)
(hf₀' : has_fderiv_at f (f₀' : E →L[𝕜] F) (f.symm a)) (hf : cont_diff_at 𝕜 n f (f.symm a)) :
cont_diff_at 𝕜 n f.symm a | begin
-- We prove this by induction on `n`
induction n using enat.nat_induction with n IH Itop,
{ rw cont_diff_at_zero,
exact ⟨f.target, is_open.mem_nhds f.open_target ha, f.continuous_inv_fun⟩ },
{ obtain ⟨f', ⟨u, hu, hff'⟩, hf'⟩ := cont_diff_at_succ_iff_has_fderiv_at.mp hf,
apply cont_diff_at_succ_iff... | theorem | local_homeomorph.cont_diff_at_symm | 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"
] | [
"complete_space",
"cont_diff_at",
"cont_diff_at_map_inverse",
"cont_diff_at_zero",
"enat.nat_induction",
"filter.inter_mem",
"has_fderiv_at",
"is_open.mem_nhds",
"local_homeomorph",
"mem_of_mem_nhds",
"unique"
] | If `f` is a local homeomorphism and the point `a` is in its target,
and if `f` is `n` times continuously differentiable at `f.symm a`,
and if the derivative at `f.symm a` is a continuous linear equivalence,
then `f.symm` is `n` times continuously differentiable at the point `a`.
This is one of the easy parts of the in... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homeomorph.cont_diff_symm [complete_space E] (f : E ≃ₜ F) {f₀' : E → E ≃L[𝕜] F}
(hf₀' : ∀ a, has_fderiv_at f (f₀' a : E →L[𝕜] F) a) (hf : cont_diff 𝕜 n (f : E → F)) :
cont_diff 𝕜 n (f.symm : F → E) | cont_diff_iff_cont_diff_at.2 $ λ x,
f.to_local_homeomorph.cont_diff_at_symm (mem_univ x) (hf₀' _) hf.cont_diff_at | theorem | homeomorph.cont_diff_symm | 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"
] | [
"complete_space",
"cont_diff",
"has_fderiv_at"
] | If `f` is an `n` times continuously differentiable homeomorphism,
and if the derivative of `f` at each point is a continuous linear equivalence,
then `f.symm` is `n` times continuously differentiable.
This is one of the easy parts of the inverse function theorem: it assumes that we already have
an inverse function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
local_homeomorph.cont_diff_at_symm_deriv [complete_space 𝕜]
(f : local_homeomorph 𝕜 𝕜) {f₀' a : 𝕜} (h₀ : f₀' ≠ 0) (ha : a ∈ f.target)
(hf₀' : has_deriv_at f f₀' (f.symm a)) (hf : cont_diff_at 𝕜 n f (f.symm a)) :
cont_diff_at 𝕜 n f.symm a | f.cont_diff_at_symm ha (hf₀'.has_fderiv_at_equiv h₀) hf | theorem | local_homeomorph.cont_diff_at_symm_deriv | 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"
] | [
"complete_space",
"cont_diff_at",
"has_deriv_at",
"local_homeomorph"
] | Let `f` be a local homeomorphism of a nontrivially normed field, let `a` be a point in its
target. if `f` is `n` times continuously differentiable at `f.symm a`, and if the derivative at
`f.symm a` is nonzero, then `f.symm` is `n` times continuously differentiable at the point `a`.
This is one of the easy parts of the... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
homeomorph.cont_diff_symm_deriv [complete_space 𝕜] (f : 𝕜 ≃ₜ 𝕜) {f' : 𝕜 → 𝕜}
(h₀ : ∀ x, f' x ≠ 0) (hf' : ∀ x, has_deriv_at f (f' x) x) (hf : cont_diff 𝕜 n (f : 𝕜 → 𝕜)) :
cont_diff 𝕜 n (f.symm : 𝕜 → 𝕜) | cont_diff_iff_cont_diff_at.2 $ λ x,
f.to_local_homeomorph.cont_diff_at_symm_deriv (h₀ _) (mem_univ x) (hf' _) hf.cont_diff_at | theorem | homeomorph.cont_diff_symm_deriv | 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"
] | [
"complete_space",
"cont_diff",
"has_deriv_at"
] | Let `f` be an `n` times continuously differentiable homeomorphism of a nontrivially normed
field. Suppose that the derivative of `f` is never equal to zero. Then `f.symm` is `n` times
continuously differentiable.
This is one of the easy parts of the inverse function theorem: it assumes that we already have
an inverse... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_clm_apply {n : ℕ∞} {f : E → F →L[𝕜] G}
{s : set E} [finite_dimensional 𝕜 F] :
cont_diff_on 𝕜 n f s ↔ ∀ y, cont_diff_on 𝕜 n (λ x, f x y) s | begin
refine ⟨λ h y, h.clm_apply cont_diff_on_const, λ h, _⟩,
let d := finrank 𝕜 F,
have hd : d = finrank 𝕜 (fin d → 𝕜) := (finrank_fin_fun 𝕜).symm,
let e₁ := continuous_linear_equiv.of_finrank_eq hd,
let e₂ := (e₁.arrow_congr (1 : G ≃L[𝕜] G)).trans (continuous_linear_equiv.pi_ring (fin d)),
rw [← comp... | lemma | cont_diff_on_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",
"cont_diff_on_const",
"continuous_linear_equiv.of_finrank_eq",
"continuous_linear_equiv.pi_ring",
"finite_dimensional"
] | A family of continuous linear maps is `C^n` on `s` if all its applications are. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_clm_apply_iff {n : ℕ∞} {f : E → F →L[𝕜] G} [finite_dimensional 𝕜 F] :
cont_diff 𝕜 n f ↔ ∀ y, cont_diff 𝕜 n (λ x, f x y) | by simp_rw [← cont_diff_on_univ, cont_diff_on_clm_apply] | lemma | cont_diff_clm_apply_iff | 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",
"cont_diff_on_clm_apply",
"cont_diff_on_univ",
"finite_dimensional"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_succ_iff_fderiv_apply [finite_dimensional 𝕜 E] {n : ℕ} {f : E → F} :
cont_diff 𝕜 ((n + 1) : ℕ) f ↔
differentiable 𝕜 f ∧ ∀ y, cont_diff 𝕜 n (λ x, fderiv 𝕜 f x y) | by rw [cont_diff_succ_iff_fderiv, cont_diff_clm_apply_iff] | lemma | cont_diff_succ_iff_fderiv_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",
"cont_diff_clm_apply_iff",
"cont_diff_succ_iff_fderiv",
"differentiable",
"fderiv",
"finite_dimensional"
] | This is a useful lemma to prove that a certain operation preserves functions being `C^n`.
When you do induction on `n`, this gives a useful characterization of a function being `C^(n+1)`,
assuming you have already computed the derivative. The advantage of this version over
`cont_diff_succ_iff_fderiv` is that both occur... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_succ_of_fderiv_apply [finite_dimensional 𝕜 E] {n : ℕ} {f : E → F}
{s : set E} (hf : differentiable_on 𝕜 f s)
(h : ∀ y, cont_diff_on 𝕜 n (λ x, fderiv_within 𝕜 f s x y) s) :
cont_diff_on 𝕜 ((n + 1) : ℕ) f s | cont_diff_on_succ_of_fderiv_within hf $ cont_diff_on_clm_apply.mpr h | lemma | cont_diff_on_succ_of_fderiv_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",
"cont_diff_on_succ_of_fderiv_within",
"differentiable_on",
"fderiv_within",
"finite_dimensional"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_succ_iff_fderiv_apply [finite_dimensional 𝕜 E] {n : ℕ} {f : E → F}
{s : set E} (hs : unique_diff_on 𝕜 s) : cont_diff_on 𝕜 ((n + 1) : ℕ) f s ↔
differentiable_on 𝕜 f s ∧ ∀ y, cont_diff_on 𝕜 n (λ x, fderiv_within 𝕜 f s x y) s | by rw [cont_diff_on_succ_iff_fderiv_within hs, cont_diff_on_clm_apply] | lemma | cont_diff_on_succ_iff_fderiv_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",
"cont_diff_on_clm_apply",
"cont_diff_on_succ_iff_fderiv_within",
"differentiable_on",
"fderiv_within",
"finite_dimensional",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to_on.has_strict_fderiv_at
{s : set E'} {f : E' → F'} {x : E'} {p : E' → formal_multilinear_series 𝕂 E' F'}
(hf : has_ftaylor_series_up_to_on n f p s) (hn : 1 ≤ n) (hs : s ∈ 𝓝 x) :
has_strict_fderiv_at f ((continuous_multilinear_curry_fin1 𝕂 E' F') (p x 1)) x | has_strict_fderiv_at_of_has_fderiv_at_of_continuous_at (hf.eventually_has_fderiv_at hn hs) $
(continuous_multilinear_curry_fin1 𝕂 E' F').continuous_at.comp $
(hf.cont 1 hn).continuous_at hs | lemma | has_ftaylor_series_up_to_on.has_strict_fderiv_at | 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"
] | [
"continuous_at",
"continuous_at.comp",
"continuous_multilinear_curry_fin1",
"formal_multilinear_series",
"has_ftaylor_series_up_to_on",
"has_strict_fderiv_at",
"has_strict_fderiv_at_of_has_fderiv_at_of_continuous_at"
] | If a function has a Taylor series at order at least 1, then at points in the interior of the
domain of definition, the term of order 1 of this series is a strict derivative of `f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.has_strict_fderiv_at'
{f : E' → F'} {f' : E' →L[𝕂] F'} {x : E'}
(hf : cont_diff_at 𝕂 n f x) (hf' : has_fderiv_at f f' x) (hn : 1 ≤ n) :
has_strict_fderiv_at f f' x | begin
rcases hf 1 hn with ⟨u, H, p, hp⟩,
simp only [nhds_within_univ, mem_univ, insert_eq_of_mem] at H,
have := hp.has_strict_fderiv_at le_rfl H,
rwa hf'.unique this.has_fderiv_at
end | lemma | cont_diff_at.has_strict_fderiv_at' | 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",
"has_fderiv_at",
"has_strict_fderiv_at",
"le_rfl",
"nhds_within_univ"
] | If a function is `C^n` with `1 ≤ n` around a point, and its derivative at that point is given to
us as `f'`, then `f'` is also a strict derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.has_strict_deriv_at' {f : 𝕂 → F'} {f' : F'} {x : 𝕂}
(hf : cont_diff_at 𝕂 n f x) (hf' : has_deriv_at f f' x) (hn : 1 ≤ n) :
has_strict_deriv_at f f' x | hf.has_strict_fderiv_at' hf' hn | lemma | cont_diff_at.has_strict_deriv_at' | 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",
"has_deriv_at",
"has_strict_deriv_at"
] | If a function is `C^n` with `1 ≤ n` around a point, and its derivative at that point is given to
us as `f'`, then `f'` is also a strict derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.has_strict_fderiv_at {f : E' → F'} {x : E'}
(hf : cont_diff_at 𝕂 n f x) (hn : 1 ≤ n) :
has_strict_fderiv_at f (fderiv 𝕂 f x) x | hf.has_strict_fderiv_at' (hf.differentiable_at hn).has_fderiv_at hn | lemma | cont_diff_at.has_strict_fderiv_at | 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",
"fderiv",
"has_fderiv_at",
"has_strict_fderiv_at"
] | If a function is `C^n` with `1 ≤ n` around a point, then the derivative of `f` at this point
is also a strict derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.has_strict_deriv_at {f : 𝕂 → F'} {x : 𝕂}
(hf : cont_diff_at 𝕂 n f x) (hn : 1 ≤ n) :
has_strict_deriv_at f (deriv f x) x | (hf.has_strict_fderiv_at hn).has_strict_deriv_at | lemma | cont_diff_at.has_strict_deriv_at | 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",
"deriv",
"has_strict_deriv_at"
] | If a function is `C^n` with `1 ≤ n` around a point, then the derivative of `f` at this point
is also a strict derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.has_strict_fderiv_at
{f : E' → F'} {x : E'} (hf : cont_diff 𝕂 n f) (hn : 1 ≤ n) :
has_strict_fderiv_at f (fderiv 𝕂 f x) x | hf.cont_diff_at.has_strict_fderiv_at hn | lemma | cont_diff.has_strict_fderiv_at | 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",
"fderiv",
"has_strict_fderiv_at"
] | If a function is `C^n` with `1 ≤ n`, then the derivative of `f` is also a strict derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.has_strict_deriv_at
{f : 𝕂 → F'} {x : 𝕂} (hf : cont_diff 𝕂 n f) (hn : 1 ≤ n) :
has_strict_deriv_at f (deriv f x) x | hf.cont_diff_at.has_strict_deriv_at hn | lemma | cont_diff.has_strict_deriv_at | 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",
"deriv",
"has_strict_deriv_at"
] | If a function is `C^n` with `1 ≤ n`, then the derivative of `f` is also a strict derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to_on.exists_lipschitz_on_with_of_nnnorm_lt {E F : Type*}
[normed_add_comm_group E] [normed_space ℝ E] [normed_add_comm_group F] [normed_space ℝ F]
{f : E → F} {p : E → formal_multilinear_series ℝ E F} {s : set E} {x : E}
(hf : has_ftaylor_series_up_to_on 1 f p (insert x s)) (hs : convex ℝ s... | begin
set f' := λ y, continuous_multilinear_curry_fin1 ℝ E F (p y 1),
have hder : ∀ y ∈ s, has_fderiv_within_at f (f' y) s y,
from λ y hy, (hf.has_fderiv_within_at le_rfl (subset_insert x s hy)).mono (subset_insert x s),
have hcont : continuous_within_at f' s x,
from (continuous_multilinear_curry_fin1 ℝ E... | lemma | has_ftaylor_series_up_to_on.exists_lipschitz_on_with_of_nnnorm_lt | 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"
] | [
"continuous_at.comp_continuous_within_at",
"continuous_multilinear_curry_fin1",
"continuous_within_at",
"convex",
"formal_multilinear_series",
"has_fderiv_within_at",
"has_ftaylor_series_up_to_on",
"le_rfl",
"linear_isometry_equiv.nnnorm_map",
"lipschitz_on_with",
"normed_add_comm_group",
"nor... | If `f` has a formal Taylor series `p` up to order `1` on `{x} ∪ s`, where `s` is a convex set,
and `‖p x 1‖₊ < K`, then `f` is `K`-Lipschitz in a neighborhood of `x` within `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_ftaylor_series_up_to_on.exists_lipschitz_on_with {E F : Type*} [normed_add_comm_group E]
[normed_space ℝ E] [normed_add_comm_group F] [normed_space ℝ F] {f : E → F}
{p : E → formal_multilinear_series ℝ E F} {s : set E} {x : E}
(hf : has_ftaylor_series_up_to_on 1 f p (insert x s)) (hs : convex ℝ s) :
∃ K (t ... | (exists_gt _).imp $ hf.exists_lipschitz_on_with_of_nnnorm_lt hs | lemma | has_ftaylor_series_up_to_on.exists_lipschitz_on_with | 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"
] | [
"convex",
"formal_multilinear_series",
"has_ftaylor_series_up_to_on",
"lipschitz_on_with",
"normed_add_comm_group",
"normed_space"
] | If `f` has a formal Taylor series `p` up to order `1` on `{x} ∪ s`, where `s` is a convex set,
then `f` is Lipschitz in a neighborhood of `x` within `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_within_at.exists_lipschitz_on_with {E F : Type*} [normed_add_comm_group E]
[normed_space ℝ E] [normed_add_comm_group F] [normed_space ℝ F] {f : E → F} {s : set E}
{x : E} (hf : cont_diff_within_at ℝ 1 f s x) (hs : convex ℝ s) :
∃ (K : ℝ≥0) (t ∈ 𝓝[s] x), lipschitz_on_with K f t | begin
rcases hf 1 le_rfl with ⟨t, hst, p, hp⟩,
rcases metric.mem_nhds_within_iff.mp hst with ⟨ε, ε0, hε⟩,
replace hp : has_ftaylor_series_up_to_on 1 f p (metric.ball x ε ∩ insert x s) := hp.mono hε,
clear hst hε t,
rw [← insert_eq_of_mem (metric.mem_ball_self ε0), ← insert_inter_distrib] at hp,
rcases hp.ex... | lemma | cont_diff_within_at.exists_lipschitz_on_with | 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_within_at",
"convex",
"convex_ball",
"has_ftaylor_series_up_to_on",
"le_rfl",
"lipschitz_on_with",
"metric.ball",
"metric.ball_mem_nhds",
"metric.mem_ball_self",
"nhds_within_restrict'",
"normed_add_comm_group",
"normed_space"
] | If `f` is `C^1` within a conves set `s` at `x`, then it is Lipschitz on a neighborhood of `x`
within `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.exists_lipschitz_on_with_of_nnnorm_lt {f : E' → F'} {x : E'}
(hf : cont_diff_at 𝕂 1 f x) (K : ℝ≥0) (hK : ‖fderiv 𝕂 f x‖₊ < K) :
∃ t ∈ 𝓝 x, lipschitz_on_with K f t | (hf.has_strict_fderiv_at le_rfl).exists_lipschitz_on_with_of_nnnorm_lt K hK | lemma | cont_diff_at.exists_lipschitz_on_with_of_nnnorm_lt | 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",
"le_rfl",
"lipschitz_on_with"
] | If `f` is `C^1` at `x` and `K > ‖fderiv 𝕂 f x‖`, then `f` is `K`-Lipschitz in a neighborhood of
`x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_at.exists_lipschitz_on_with {f : E' → F'} {x : E'}
(hf : cont_diff_at 𝕂 1 f x) :
∃ K (t ∈ 𝓝 x), lipschitz_on_with K f t | (hf.has_strict_fderiv_at le_rfl).exists_lipschitz_on_with | lemma | cont_diff_at.exists_lipschitz_on_with | 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",
"le_rfl",
"lipschitz_on_with"
] | If `f` is `C^1` at `x`, then `f` is Lipschitz in a neighborhood of `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_succ_iff_deriv_within {n : ℕ} (hs : unique_diff_on 𝕜 s₂) :
cont_diff_on 𝕜 ((n + 1) : ℕ) f₂ s₂ ↔
differentiable_on 𝕜 f₂ s₂ ∧ cont_diff_on 𝕜 n (deriv_within f₂ s₂) s₂ | begin
rw cont_diff_on_succ_iff_fderiv_within hs,
congr' 2,
apply le_antisymm,
{ assume h,
have : deriv_within f₂ s₂ = (λ u : 𝕜 →L[𝕜] F, u 1) ∘ (fderiv_within 𝕜 f₂ s₂),
by { ext x, refl },
simp only [this],
apply cont_diff.comp_cont_diff_on _ h,
exact (is_bounded_bilinear_map_apply.is_bo... | theorem | cont_diff_on_succ_iff_deriv_within | 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",
"cont_diff.comp_cont_diff_on",
"cont_diff_on",
"cont_diff_on_succ_iff_fderiv_within",
"deriv_within",
"differentiable_on",
"fderiv_within",
"is_bounded_bilinear_map",
"is_bounded_bilinear_map_smul_right",
"unique_diff_on"
] | A function is `C^(n + 1)` on a domain with unique derivatives if and only if it is
differentiable there, and its derivative (formulated with `deriv_within`) is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_succ_iff_deriv_of_open {n : ℕ} (hs : is_open s₂) :
cont_diff_on 𝕜 ((n + 1) : ℕ) f₂ s₂ ↔
differentiable_on 𝕜 f₂ s₂ ∧ cont_diff_on 𝕜 n (deriv f₂) s₂ | begin
rw cont_diff_on_succ_iff_deriv_within hs.unique_diff_on,
congrm _ ∧ _,
exact cont_diff_on_congr (λ _, deriv_within_of_open hs)
end | theorem | cont_diff_on_succ_iff_deriv_of_open | 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",
"cont_diff_on_congr",
"cont_diff_on_succ_iff_deriv_within",
"deriv",
"deriv_within_of_open",
"differentiable_on",
"is_open"
] | A function is `C^(n + 1)` on an open domain if and only if it is
differentiable there, and its derivative (formulated with `deriv`) is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_top_iff_deriv_within (hs : unique_diff_on 𝕜 s₂) :
cont_diff_on 𝕜 ∞ f₂ s₂ ↔
differentiable_on 𝕜 f₂ s₂ ∧ cont_diff_on 𝕜 ∞ (deriv_within f₂ s₂) s₂ | begin
split,
{ assume h,
refine ⟨h.differentiable_on le_top, _⟩,
apply cont_diff_on_top.2 (λ n, ((cont_diff_on_succ_iff_deriv_within hs).1 _).2),
exact h.of_le le_top },
{ assume h,
refine cont_diff_on_top.2 (λ n, _),
have A : (n : ℕ∞) ≤ ∞ := le_top,
apply ((cont_diff_on_succ_iff_deriv_wit... | theorem | cont_diff_on_top_iff_deriv_within | 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",
"cont_diff_on_succ_iff_deriv_within",
"deriv_within",
"differentiable_on",
"le_top",
"unique_diff_on"
] | A function is `C^∞` on a domain with unique derivatives if and only if it is differentiable
there, and its derivative (formulated with `deriv_within`) is `C^∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on_top_iff_deriv_of_open (hs : is_open s₂) :
cont_diff_on 𝕜 ∞ f₂ s₂ ↔
differentiable_on 𝕜 f₂ s₂ ∧ cont_diff_on 𝕜 ∞ (deriv f₂) s₂ | begin
rw cont_diff_on_top_iff_deriv_within hs.unique_diff_on,
congrm _ ∧ _,
exact cont_diff_on_congr (λ _, deriv_within_of_open hs)
end | theorem | cont_diff_on_top_iff_deriv_of_open | 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",
"cont_diff_on_congr",
"cont_diff_on_top_iff_deriv_within",
"deriv",
"deriv_within_of_open",
"differentiable_on",
"is_open"
] | A function is `C^∞` on an open domain if and only if it is differentiable
there, and its derivative (formulated with `deriv`) is `C^∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_on.deriv_within
(hf : cont_diff_on 𝕜 n f₂ s₂) (hs : unique_diff_on 𝕜 s₂) (hmn : m + 1 ≤ n) :
cont_diff_on 𝕜 m (deriv_within f₂ s₂) s₂ | begin
cases m,
{ change ∞ + 1 ≤ n at hmn,
have : n = ∞, by simpa using hmn,
rw this at hf,
exact ((cont_diff_on_top_iff_deriv_within hs).1 hf).2 },
{ change (m.succ : ℕ∞) ≤ n at hmn,
exact ((cont_diff_on_succ_iff_deriv_within hs).1 (hf.of_le hmn)).2 }
end | lemma | cont_diff_on.deriv_within | 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",
"cont_diff_on_succ_iff_deriv_within",
"cont_diff_on_top_iff_deriv_within",
"deriv_within",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.deriv_of_open
(hf : cont_diff_on 𝕜 n f₂ s₂) (hs : is_open s₂) (hmn : m + 1 ≤ n) :
cont_diff_on 𝕜 m (deriv f₂) s₂ | (hf.deriv_within hs.unique_diff_on hmn).congr (λ x hx, (deriv_within_of_open hs hx).symm) | lemma | cont_diff_on.deriv_of_open | 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",
"deriv",
"deriv_within_of_open",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.continuous_on_deriv_within
(h : cont_diff_on 𝕜 n f₂ s₂) (hs : unique_diff_on 𝕜 s₂) (hn : 1 ≤ n) :
continuous_on (deriv_within f₂ s₂) s₂ | ((cont_diff_on_succ_iff_deriv_within hs).1 (h.of_le hn)).2.continuous_on | lemma | cont_diff_on.continuous_on_deriv_within | 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",
"cont_diff_on_succ_iff_deriv_within",
"continuous_on",
"deriv_within",
"unique_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.continuous_on_deriv_of_open
(h : cont_diff_on 𝕜 n f₂ s₂) (hs : is_open s₂) (hn : 1 ≤ n) :
continuous_on (deriv f₂) s₂ | ((cont_diff_on_succ_iff_deriv_of_open hs).1 (h.of_le hn)).2.continuous_on | lemma | cont_diff_on.continuous_on_deriv_of_open | 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",
"cont_diff_on_succ_iff_deriv_of_open",
"continuous_on",
"deriv",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_succ_iff_deriv {n : ℕ} :
cont_diff 𝕜 ((n + 1) : ℕ) f₂ ↔
differentiable 𝕜 f₂ ∧ cont_diff 𝕜 n (deriv f₂) | by simp only [← cont_diff_on_univ, cont_diff_on_succ_iff_deriv_of_open, is_open_univ,
differentiable_on_univ] | theorem | cont_diff_succ_iff_deriv | 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",
"cont_diff_on_succ_iff_deriv_of_open",
"cont_diff_on_univ",
"deriv",
"differentiable",
"differentiable_on_univ",
"is_open_univ"
] | A function is `C^(n + 1)` if and only if it is differentiable,
and its derivative (formulated in terms of `deriv`) is `C^n`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_one_iff_deriv :
cont_diff 𝕜 1 f₂ ↔ differentiable 𝕜 f₂ ∧ continuous (deriv f₂) | cont_diff_succ_iff_deriv.trans $ iff.rfl.and cont_diff_zero | theorem | cont_diff_one_iff_deriv | 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",
"cont_diff_zero",
"continuous",
"deriv",
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_top_iff_deriv :
cont_diff 𝕜 ∞ f₂ ↔
differentiable 𝕜 f₂ ∧ cont_diff 𝕜 ∞ (deriv f₂) | begin
simp only [← cont_diff_on_univ, ← differentiable_on_univ, ← deriv_within_univ],
rw cont_diff_on_top_iff_deriv_within unique_diff_on_univ,
end | theorem | cont_diff_top_iff_deriv | 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",
"cont_diff_on_top_iff_deriv_within",
"cont_diff_on_univ",
"deriv",
"deriv_within_univ",
"differentiable",
"differentiable_on_univ",
"unique_diff_on_univ"
] | A function is `C^∞` if and only if it is differentiable,
and its derivative (formulated in terms of `deriv`) is `C^∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff.continuous_deriv (h : cont_diff 𝕜 n f₂) (hn : 1 ≤ n) :
continuous (deriv f₂) | (cont_diff_succ_iff_deriv.mp (h.of_le hn)).2.continuous | lemma | cont_diff.continuous_deriv | 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",
"deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.iterate_deriv :
∀ (n : ℕ) {f₂ : 𝕜 → F} (hf : cont_diff 𝕜 ∞ f₂), cont_diff 𝕜 ∞ (deriv^[n] f₂) | | 0 f₂ hf := hf
| (n + 1) f₂ hf := cont_diff.iterate_deriv n (cont_diff_top_iff_deriv.mp hf).2 | lemma | cont_diff.iterate_deriv | 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",
"deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.iterate_deriv' (n : ℕ) :
∀ (k : ℕ) {f₂ : 𝕜 → F} (hf : cont_diff 𝕜 (n + k : ℕ) f₂), cont_diff 𝕜 n (deriv^[k] f₂) | | 0 f₂ hf := hf
| (n + 1) f₂ hf := cont_diff.iterate_deriv' n (cont_diff_succ_iff_deriv.mp hf).2 | lemma | cont_diff.iterate_deriv' | 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",
"deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_ftaylor_series_up_to_on.restrict_scalars
(h : has_ftaylor_series_up_to_on n f p' s) :
has_ftaylor_series_up_to_on n f (λ x, (p' x).restrict_scalars 𝕜) s | { zero_eq := λ x hx, h.zero_eq x hx,
fderiv_within :=
begin
intros m hm x hx,
convert ((continuous_multilinear_map.restrict_scalars_linear 𝕜).has_fderiv_at)
.comp_has_fderiv_within_at _ ((h.fderiv_within m hm x hx).restrict_scalars 𝕜),
end,
cont := λ m hm, continuous_multilinear_map.co... | lemma | has_ftaylor_series_up_to_on.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",
"continuous_multilinear_map.restrict_scalars_linear",
"fderiv_within",
"has_fderiv_at",
"has_ftaylor_series_up_to_on",
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.restrict_scalars (h : cont_diff_within_at 𝕜' n f s x) :
cont_diff_within_at 𝕜 n f s x | begin
intros m hm,
rcases h m hm with ⟨u, u_mem, p', hp'⟩,
exact ⟨u, u_mem, _, hp'.restrict_scalars _⟩
end | lemma | cont_diff_within_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_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.restrict_scalars (h : cont_diff_on 𝕜' n f s) :
cont_diff_on 𝕜 n f s | λ x hx, (h x hx).restrict_scalars _ | lemma | cont_diff_on.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_on",
"restrict_scalars"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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