statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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has_deriv_within_at_aeval (x : 𝕜) (s : set 𝕜) :
has_deriv_within_at (λx, aeval x q) (aeval x q.derivative) s x | (q.has_deriv_at_aeval x).has_deriv_within_at | theorem | polynomial.has_deriv_within_at_aeval | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at : differentiable_at 𝕜 (λx, p.eval x) x | (p.has_deriv_at x).differentiable_at | lemma | polynomial.differentiable_at | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_aeval : differentiable_at 𝕜 (λx, aeval x q) x | (q.has_deriv_at_aeval x).differentiable_at | lemma | polynomial.differentiable_at_aeval | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at : differentiable_within_at 𝕜 (λx, p.eval x) s x | p.differentiable_at.differentiable_within_at | lemma | polynomial.differentiable_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_aeval : differentiable_within_at 𝕜 (λx, aeval x q) s x | q.differentiable_at_aeval.differentiable_within_at | lemma | polynomial.differentiable_within_at_aeval | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable : differentiable 𝕜 (λx, p.eval x) | λx, p.differentiable_at | lemma | polynomial.differentiable | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_aeval : differentiable 𝕜 (λ x : 𝕜, aeval x q) | λx, q.differentiable_at_aeval | lemma | polynomial.differentiable_aeval | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on : differentiable_on 𝕜 (λx, p.eval x) s | p.differentiable.differentiable_on | lemma | polynomial.differentiable_on | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_aeval : differentiable_on 𝕜 (λx, aeval x q) s | q.differentiable_aeval.differentiable_on | lemma | polynomial.differentiable_on_aeval | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv : deriv (λx, p.eval x) x = p.derivative.eval x | (p.has_deriv_at x).deriv | lemma | polynomial.deriv | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_aeval : deriv (λx, aeval x q) x = aeval x q.derivative | (q.has_deriv_at_aeval x).deriv | lemma | polynomial.deriv_aeval | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"deriv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within (hxs : unique_diff_within_at 𝕜 s x) :
deriv_within (λx, p.eval x) s x = p.derivative.eval x | begin
rw differentiable_at.deriv_within p.differentiable_at hxs,
exact p.deriv
end | lemma | polynomial.deriv_within | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"deriv_within",
"differentiable_at.deriv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_aeval (hxs : unique_diff_within_at 𝕜 s x) :
deriv_within (λx, aeval x q) s x = aeval x q.derivative | by simpa only [aeval_def, eval₂_eq_eval_map, derivative_map]
using (q.map (algebra_map R 𝕜)).deriv_within hxs | lemma | polynomial.deriv_within_aeval | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"algebra_map",
"deriv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at (x : 𝕜) :
has_fderiv_at (λx, p.eval x) (smul_right (1 : 𝕜 →L[𝕜] 𝕜) (p.derivative.eval x)) x | p.has_deriv_at x | lemma | polynomial.has_fderiv_at | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_aeval (x : 𝕜) :
has_fderiv_at (λx, aeval x q) (smul_right (1 : 𝕜 →L[𝕜] 𝕜) (aeval x q.derivative)) x | q.has_deriv_at_aeval x | lemma | polynomial.has_fderiv_at_aeval | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at (x : 𝕜) :
has_fderiv_within_at (λx, p.eval x) (smul_right (1 : 𝕜 →L[𝕜] 𝕜) (p.derivative.eval x)) s x | (p.has_fderiv_at x).has_fderiv_within_at | lemma | polynomial.has_fderiv_within_at | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at_aeval (x : 𝕜) :
has_fderiv_within_at (λx, aeval x q) (smul_right (1 : 𝕜 →L[𝕜] 𝕜) (aeval x q.derivative)) s x | (q.has_fderiv_at_aeval x).has_fderiv_within_at | lemma | polynomial.has_fderiv_within_at_aeval | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv :
fderiv 𝕜 (λx, p.eval x) x = smul_right (1 : 𝕜 →L[𝕜] 𝕜) (p.derivative.eval x) | (p.has_fderiv_at x).fderiv | lemma | polynomial.fderiv | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_aeval :
fderiv 𝕜 (λx, aeval x q) x = smul_right (1 : 𝕜 →L[𝕜] 𝕜) (aeval x q.derivative) | (q.has_fderiv_at_aeval x).fderiv | lemma | polynomial.fderiv_aeval | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (λx, p.eval x) s x = smul_right (1 : 𝕜 →L[𝕜] 𝕜) (p.derivative.eval x) | (p.has_fderiv_within_at x).fderiv_within hxs | lemma | polynomial.fderiv_within | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_aeval (hxs : unique_diff_within_at 𝕜 s x) :
fderiv_within 𝕜 (λx, aeval x q) s x = smul_right (1 : 𝕜 →L[𝕜] 𝕜) (aeval x q.derivative) | (q.has_fderiv_within_at_aeval x).fderiv_within hxs | lemma | polynomial.fderiv_within_aeval | analysis.calculus.deriv | src/analysis/calculus/deriv/polynomial.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.add",
"data.polynomial.algebra_map",
"data.polynomial.derivative"
] | [
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_pow : ∀ (n : ℕ) (x : 𝕜),
has_strict_deriv_at (λx, x^n) ((n : 𝕜) * x^(n-1)) x | | 0 x := by simp [has_strict_deriv_at_const]
| 1 x := by simpa using has_strict_deriv_at_id x
| (n + 1 + 1) x := by simpa [pow_succ', add_mul, mul_assoc]
using (has_strict_deriv_at_pow (n + 1) x).mul (has_strict_deriv_at_id x) | lemma | has_strict_deriv_at_pow | analysis.calculus.deriv | src/analysis/calculus/deriv/pow.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"has_strict_deriv_at",
"has_strict_deriv_at_const",
"has_strict_deriv_at_id",
"mul_assoc",
"pow_succ'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_pow (n : ℕ) (x : 𝕜) : has_deriv_at (λx, x^n) ((n : 𝕜) * x^(n-1)) x | (has_strict_deriv_at_pow n x).has_deriv_at | lemma | has_deriv_at_pow | analysis.calculus.deriv | src/analysis/calculus/deriv/pow.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"has_deriv_at",
"has_strict_deriv_at_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_pow (n : ℕ) (x : 𝕜) (s : set 𝕜) :
has_deriv_within_at (λx, x^n) ((n : 𝕜) * x^(n-1)) s x | (has_deriv_at_pow n x).has_deriv_within_at | theorem | has_deriv_within_at_pow | analysis.calculus.deriv | src/analysis/calculus/deriv/pow.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"has_deriv_at_pow",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_pow : differentiable_at 𝕜 (λx, x^n) x | (has_deriv_at_pow n x).differentiable_at | lemma | differentiable_at_pow | analysis.calculus.deriv | src/analysis/calculus/deriv/pow.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"differentiable_at",
"has_deriv_at_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_pow : differentiable_within_at 𝕜 (λx, x^n) s x | (differentiable_at_pow n).differentiable_within_at | lemma | differentiable_within_at_pow | analysis.calculus.deriv | src/analysis/calculus/deriv/pow.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"differentiable_at_pow",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_pow : differentiable 𝕜 (λx:𝕜, x^n) | λ x, differentiable_at_pow n | lemma | differentiable_pow | analysis.calculus.deriv | src/analysis/calculus/deriv/pow.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"differentiable",
"differentiable_at_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_pow : differentiable_on 𝕜 (λx, x^n) s | (differentiable_pow n).differentiable_on | lemma | differentiable_on_pow | analysis.calculus.deriv | src/analysis/calculus/deriv/pow.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"differentiable_on",
"differentiable_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_pow : deriv (λ x, x^n) x = (n : 𝕜) * x^(n-1) | (has_deriv_at_pow n x).deriv | lemma | deriv_pow | analysis.calculus.deriv | src/analysis/calculus/deriv/pow.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"deriv",
"has_deriv_at_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_pow' : deriv (λ x, x^n) = λ x, (n : 𝕜) * x^(n-1) | funext $ λ x, deriv_pow n | lemma | deriv_pow' | analysis.calculus.deriv | src/analysis/calculus/deriv/pow.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"deriv",
"deriv_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_pow (hxs : unique_diff_within_at 𝕜 s x) :
deriv_within (λx, x^n) s x = (n : 𝕜) * x^(n-1) | (has_deriv_within_at_pow n x s).deriv_within hxs | lemma | deriv_within_pow | analysis.calculus.deriv | src/analysis/calculus/deriv/pow.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"deriv_within",
"has_deriv_within_at_pow",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.pow (hc : has_deriv_within_at c c' s x) :
has_deriv_within_at (λ y, (c y)^n) ((n : 𝕜) * (c x)^(n-1) * c') s x | (has_deriv_at_pow n (c x)).comp_has_deriv_within_at x hc | lemma | has_deriv_within_at.pow | analysis.calculus.deriv | src/analysis/calculus/deriv/pow.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"has_deriv_at_pow",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.pow (hc : has_deriv_at c c' x) :
has_deriv_at (λ y, (c y)^n) ((n : 𝕜) * (c x)^(n-1) * c') x | by { rw ← has_deriv_within_at_univ at *, exact hc.pow n } | lemma | has_deriv_at.pow | analysis.calculus.deriv | src/analysis/calculus/deriv/pow.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"has_deriv_at",
"has_deriv_within_at_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_pow' (hc : differentiable_within_at 𝕜 c s x)
(hxs : unique_diff_within_at 𝕜 s x) :
deriv_within (λx, (c x)^n) s x = (n : 𝕜) * (c x)^(n-1) * (deriv_within c s x) | (hc.has_deriv_within_at.pow n).deriv_within hxs | lemma | deriv_within_pow' | analysis.calculus.deriv | src/analysis/calculus/deriv/pow.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_pow'' (hc : differentiable_at 𝕜 c x) :
deriv (λx, (c x)^n) x = (n : 𝕜) * (c x)^(n-1) * (deriv c x) | (hc.has_deriv_at.pow n).deriv | lemma | deriv_pow'' | analysis.calculus.deriv | src/analysis/calculus/deriv/pow.lean | [
"analysis.calculus.deriv.mul",
"analysis.calculus.deriv.comp"
] | [
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter.prod
(hf₁ : has_deriv_at_filter f₁ f₁' x L) (hf₂ : has_deriv_at_filter f₂ f₂' x L) :
has_deriv_at_filter (λ x, (f₁ x, f₂ x)) (f₁', f₂') x L | hf₁.prod hf₂ | lemma | has_deriv_at_filter.prod | analysis.calculus.deriv | src/analysis/calculus/deriv/prod.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.prod"
] | [
"has_deriv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.prod
(hf₁ : has_deriv_within_at f₁ f₁' s x) (hf₂ : has_deriv_within_at f₂ f₂' s x) :
has_deriv_within_at (λ x, (f₁ x, f₂ x)) (f₁', f₂') s x | hf₁.prod hf₂ | lemma | has_deriv_within_at.prod | analysis.calculus.deriv | src/analysis/calculus/deriv/prod.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.prod"
] | [
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.prod (hf₁ : has_deriv_at f₁ f₁' x) (hf₂ : has_deriv_at f₂ f₂' x) :
has_deriv_at (λ x, (f₁ x, f₂ x)) (f₁', f₂') x | hf₁.prod hf₂ | lemma | has_deriv_at.prod | analysis.calculus.deriv | src/analysis/calculus/deriv/prod.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.prod"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.prod (hf₁ : has_strict_deriv_at f₁ f₁' x)
(hf₂ : has_strict_deriv_at f₂ f₂' x) :
has_strict_deriv_at (λ x, (f₁ x, f₂ x)) (f₁', f₂') x | hf₁.prod hf₂ | lemma | has_strict_deriv_at.prod | analysis.calculus.deriv | src/analysis/calculus/deriv/prod.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.prod"
] | [
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_pi :
has_strict_deriv_at φ φ' x ↔ ∀ i, has_strict_deriv_at (λ x, φ x i) (φ' i) x | has_strict_fderiv_at_pi' | lemma | has_strict_deriv_at_pi | analysis.calculus.deriv | src/analysis/calculus/deriv/prod.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.prod"
] | [
"has_strict_deriv_at",
"has_strict_fderiv_at_pi'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter_pi :
has_deriv_at_filter φ φ' x L ↔
∀ i, has_deriv_at_filter (λ x, φ x i) (φ' i) x L | has_fderiv_at_filter_pi' | lemma | has_deriv_at_filter_pi | analysis.calculus.deriv | src/analysis/calculus/deriv/prod.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.prod"
] | [
"has_deriv_at_filter",
"has_fderiv_at_filter_pi'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_pi :
has_deriv_at φ φ' x ↔ ∀ i, has_deriv_at (λ x, φ x i) (φ' i) x | has_deriv_at_filter_pi | lemma | has_deriv_at_pi | analysis.calculus.deriv | src/analysis/calculus/deriv/prod.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.prod"
] | [
"has_deriv_at",
"has_deriv_at_filter_pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_pi :
has_deriv_within_at φ φ' s x ↔ ∀ i, has_deriv_within_at (λ x, φ x i) (φ' i) s x | has_deriv_at_filter_pi | lemma | has_deriv_within_at_pi | analysis.calculus.deriv | src/analysis/calculus/deriv/prod.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.prod"
] | [
"has_deriv_at_filter_pi",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_pi (h : ∀ i, differentiable_within_at 𝕜 (λ x, φ x i) s x)
(hs : unique_diff_within_at 𝕜 s x) :
deriv_within φ s x = λ i, deriv_within (λ x, φ x i) s x | (has_deriv_within_at_pi.2 (λ i, (h i).has_deriv_within_at)).deriv_within hs | lemma | deriv_within_pi | analysis.calculus.deriv | src/analysis/calculus/deriv/prod.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.prod"
] | [
"deriv_within",
"differentiable_within_at",
"has_deriv_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_pi (h : ∀ i, differentiable_at 𝕜 (λ x, φ x i) x) :
deriv φ x = λ i, deriv (λ x, φ x i) x | (has_deriv_at_pi.2 (λ i, (h i).has_deriv_at)).deriv | lemma | deriv_pi | analysis.calculus.deriv | src/analysis/calculus/deriv/prod.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.prod"
] | [
"deriv",
"differentiable_at",
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_filter_iff_tendsto_slope {x : 𝕜} {L : filter 𝕜} :
has_deriv_at_filter f f' x L ↔ tendsto (slope f x) (L ⊓ 𝓟 {x}ᶜ) (𝓝 f') | begin
conv_lhs { simp only [has_deriv_at_filter_iff_tendsto, (norm_inv _).symm,
(norm_smul _ _).symm, tendsto_zero_iff_norm_tendsto_zero.symm] },
conv_rhs { rw [← nhds_translation_sub f', tendsto_comap_iff] },
refine (tendsto_inf_principal_nhds_iff_of_forall_eq $ by simp).symm.trans (tendsto_congr' _),
refi... | lemma | has_deriv_at_filter_iff_tendsto_slope | analysis.calculus.deriv | src/analysis/calculus/deriv/slope.lean | [
"analysis.calculus.deriv.basic",
"linear_algebra.affine_space.slope"
] | [
"filter",
"has_deriv_at_filter",
"has_deriv_at_filter_iff_tendsto",
"inf_le_right",
"inv_mul_cancel",
"norm_inv",
"norm_smul",
"one_smul",
"slope",
"slope_def_module",
"smul_sub",
"tendsto_inf_principal_nhds_iff_of_forall_eq"
] | If the domain has dimension one, then Fréchet derivative is equivalent to the classical
definition with a limit. In this version we have to take the limit along the subset `-{x}`,
because for `y=x` the slope equals zero due to the convention `0⁻¹=0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_within_at_iff_tendsto_slope :
has_deriv_within_at f f' s x ↔ tendsto (slope f x) (𝓝[s \ {x}] x) (𝓝 f') | begin
simp only [has_deriv_within_at, nhds_within, diff_eq, inf_assoc.symm, inf_principal.symm],
exact has_deriv_at_filter_iff_tendsto_slope
end | lemma | has_deriv_within_at_iff_tendsto_slope | analysis.calculus.deriv | src/analysis/calculus/deriv/slope.lean | [
"analysis.calculus.deriv.basic",
"linear_algebra.affine_space.slope"
] | [
"has_deriv_at_filter_iff_tendsto_slope",
"has_deriv_within_at",
"nhds_within",
"slope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_iff_tendsto_slope' (hs : x ∉ s) :
has_deriv_within_at f f' s x ↔ tendsto (slope f x) (𝓝[s] x) (𝓝 f') | begin
convert ← has_deriv_within_at_iff_tendsto_slope,
exact diff_singleton_eq_self hs
end | lemma | has_deriv_within_at_iff_tendsto_slope' | analysis.calculus.deriv | src/analysis/calculus/deriv/slope.lean | [
"analysis.calculus.deriv.basic",
"linear_algebra.affine_space.slope"
] | [
"has_deriv_within_at",
"has_deriv_within_at_iff_tendsto_slope",
"slope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_iff_tendsto_slope :
has_deriv_at f f' x ↔ tendsto (slope f x) (𝓝[≠] x) (𝓝 f') | has_deriv_at_filter_iff_tendsto_slope | lemma | has_deriv_at_iff_tendsto_slope | analysis.calculus.deriv | src/analysis/calculus/deriv/slope.lean | [
"analysis.calculus.deriv.basic",
"linear_algebra.affine_space.slope"
] | [
"has_deriv_at",
"has_deriv_at_filter_iff_tendsto_slope",
"slope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.limsup_slope_le (hf : has_deriv_within_at f f' s x) (hr : f' < r) :
∀ᶠ z in 𝓝[s \ {x}] x, slope f x z < r | has_deriv_within_at_iff_tendsto_slope.1 hf (is_open.mem_nhds is_open_Iio hr) | lemma | has_deriv_within_at.limsup_slope_le | analysis.calculus.deriv | src/analysis/calculus/deriv/slope.lean | [
"analysis.calculus.deriv.basic",
"linear_algebra.affine_space.slope"
] | [
"has_deriv_within_at",
"is_open.mem_nhds",
"is_open_Iio",
"slope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.limsup_slope_le' (hf : has_deriv_within_at f f' s x)
(hs : x ∉ s) (hr : f' < r) :
∀ᶠ z in 𝓝[s] x, slope f x z < r | (has_deriv_within_at_iff_tendsto_slope' hs).1 hf (is_open.mem_nhds is_open_Iio hr) | lemma | has_deriv_within_at.limsup_slope_le' | analysis.calculus.deriv | src/analysis/calculus/deriv/slope.lean | [
"analysis.calculus.deriv.basic",
"linear_algebra.affine_space.slope"
] | [
"has_deriv_within_at",
"has_deriv_within_at_iff_tendsto_slope'",
"is_open.mem_nhds",
"is_open_Iio",
"slope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.liminf_right_slope_le
(hf : has_deriv_within_at f f' (Ici x) x) (hr : f' < r) :
∃ᶠ z in 𝓝[>] x, slope f x z < r | (hf.Ioi_of_Ici.limsup_slope_le' (lt_irrefl x) hr).frequently | lemma | has_deriv_within_at.liminf_right_slope_le | analysis.calculus.deriv | src/analysis/calculus/deriv/slope.lean | [
"analysis.calculus.deriv.basic",
"linear_algebra.affine_space.slope"
] | [
"has_deriv_within_at",
"slope"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.limsup_norm_slope_le
(hf : has_deriv_within_at f f' s x) (hr : ‖f'‖ < r) :
∀ᶠ z in 𝓝[s] x, ‖z - x‖⁻¹ * ‖f z - f x‖ < r | begin
have hr₀ : 0 < r, from lt_of_le_of_lt (norm_nonneg f') hr,
have A : ∀ᶠ z in 𝓝[s \ {x}] x, ‖(z - x)⁻¹ • (f z - f x)‖ ∈ Iio r,
from (has_deriv_within_at_iff_tendsto_slope.1 hf).norm (is_open.mem_nhds is_open_Iio hr),
have B : ∀ᶠ z in 𝓝[{x}] x, ‖(z - x)⁻¹ • (f z - f x)‖ ∈ Iio r,
from mem_of_superset ... | lemma | has_deriv_within_at.limsup_norm_slope_le | analysis.calculus.deriv | src/analysis/calculus/deriv/slope.lean | [
"analysis.calculus.deriv.basic",
"linear_algebra.affine_space.slope"
] | [
"has_deriv_within_at",
"is_open.mem_nhds",
"is_open_Iio",
"nhds_within_union",
"norm_inv",
"norm_smul",
"self_mem_nhds_within"
] | If `f` has derivative `f'` within `s` at `x`, then for any `r > ‖f'‖` the ratio
`‖f z - f x‖ / ‖z - x‖` is less than `r` in some neighborhood of `x` within `s`.
In other words, the limit superior of this ratio as `z` tends to `x` along `s`
is less than or equal to `‖f'‖`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_within_at.limsup_slope_norm_le
(hf : has_deriv_within_at f f' s x) (hr : ‖f'‖ < r) :
∀ᶠ z in 𝓝[s] x, ‖z - x‖⁻¹ * (‖f z‖ - ‖f x‖) < r | begin
apply (hf.limsup_norm_slope_le hr).mono,
assume z hz,
refine lt_of_le_of_lt (mul_le_mul_of_nonneg_left (norm_sub_norm_le _ _) _) hz,
exact inv_nonneg.2 (norm_nonneg _)
end | lemma | has_deriv_within_at.limsup_slope_norm_le | analysis.calculus.deriv | src/analysis/calculus/deriv/slope.lean | [
"analysis.calculus.deriv.basic",
"linear_algebra.affine_space.slope"
] | [
"has_deriv_within_at",
"mul_le_mul_of_nonneg_left"
] | If `f` has derivative `f'` within `s` at `x`, then for any `r > ‖f'‖` the ratio
`(‖f z‖ - ‖f x‖) / ‖z - x‖` is less than `r` in some neighborhood of `x` within `s`.
In other words, the limit superior of this ratio as `z` tends to `x` along `s`
is less than or equal to `‖f'‖`.
This lemma is a weaker version of `has_der... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_within_at.liminf_right_norm_slope_le
(hf : has_deriv_within_at f f' (Ici x) x) (hr : ‖f'‖ < r) :
∃ᶠ z in 𝓝[>] x, ‖z - x‖⁻¹ * ‖f z - f x‖ < r | (hf.Ioi_of_Ici.limsup_norm_slope_le hr).frequently | lemma | has_deriv_within_at.liminf_right_norm_slope_le | analysis.calculus.deriv | src/analysis/calculus/deriv/slope.lean | [
"analysis.calculus.deriv.basic",
"linear_algebra.affine_space.slope"
] | [
"has_deriv_within_at"
] | If `f` has derivative `f'` within `(x, +∞)` at `x`, then for any `r > ‖f'‖` the ratio
`‖f z - f x‖ / ‖z - x‖` is frequently less than `r` as `z → x+0`.
In other words, the limit inferior of this ratio as `z` tends to `x+0`
is less than or equal to `‖f'‖`. See also `has_deriv_within_at.limsup_norm_slope_le`
for a strong... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_within_at.liminf_right_slope_norm_le
(hf : has_deriv_within_at f f' (Ici x) x) (hr : ‖f'‖ < r) :
∃ᶠ z in 𝓝[>] x, (z - x)⁻¹ * (‖f z‖ - ‖f x‖) < r | begin
have := (hf.Ioi_of_Ici.limsup_slope_norm_le hr).frequently,
refine this.mp (eventually.mono self_mem_nhds_within _),
assume z hxz hz,
rwa [real.norm_eq_abs, abs_of_pos (sub_pos_of_lt hxz)] at hz
end | lemma | has_deriv_within_at.liminf_right_slope_norm_le | analysis.calculus.deriv | src/analysis/calculus/deriv/slope.lean | [
"analysis.calculus.deriv.basic",
"linear_algebra.affine_space.slope"
] | [
"abs_of_pos",
"has_deriv_within_at",
"real.norm_eq_abs",
"self_mem_nhds_within"
] | If `f` has derivative `f'` within `(x, +∞)` at `x`, then for any `r > ‖f'‖` the ratio
`(‖f z‖ - ‖f x‖) / (z - x)` is frequently less than `r` as `z → x+0`.
In other words, the limit inferior of this ratio as `z` tends to `x+0`
is less than or equal to `‖f'‖`.
See also
* `has_deriv_within_at.limsup_norm_slope_le` for ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_filter.star (h : has_deriv_at_filter f f' x L) :
has_deriv_at_filter (λ x, star (f x)) (star f') x L | by simpa using h.star.has_deriv_at_filter | theorem | has_deriv_at_filter.star | analysis.calculus.deriv | src/analysis/calculus/deriv/star.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.star"
] | [
"has_deriv_at_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.star (h : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ x, star (f x)) (star f') s x | h.star | theorem | has_deriv_within_at.star | analysis.calculus.deriv | src/analysis/calculus/deriv/star.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.star"
] | [
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.star (h : has_deriv_at f f' x) :
has_deriv_at (λ x, star (f x)) (star f') x | h.star | theorem | has_deriv_at.star | analysis.calculus.deriv | src/analysis/calculus/deriv/star.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.star"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.star (h : has_strict_deriv_at f f' x) :
has_strict_deriv_at (λ x, star (f x)) (star f') x | by simpa using h.star.has_strict_deriv_at | theorem | has_strict_deriv_at.star | analysis.calculus.deriv | src/analysis/calculus/deriv/star.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.star"
] | [
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within.star (hxs : unique_diff_within_at 𝕜 s x) :
deriv_within (λ y, star (f y)) s x = star (deriv_within f s x) | fun_like.congr_fun (fderiv_within_star hxs) _ | lemma | deriv_within.star | analysis.calculus.deriv | src/analysis/calculus/deriv/star.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.star"
] | [
"deriv_within",
"fderiv_within_star",
"fun_like.congr_fun",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv.star : deriv (λ y, star (f y)) x = star (deriv f x) | fun_like.congr_fun fderiv_star _ | lemma | deriv.star | analysis.calculus.deriv | src/analysis/calculus/deriv/star.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.star"
] | [
"deriv",
"fderiv_star",
"fun_like.congr_fun"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv.star' : deriv (λ y, star (f y)) = (λ x, star (deriv f x)) | funext $ λ x, deriv.star | lemma | deriv.star' | analysis.calculus.deriv | src/analysis/calculus/deriv/star.lean | [
"analysis.calculus.deriv.basic",
"analysis.calculus.fderiv.star"
] | [
"deriv",
"deriv.star"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
support_deriv_subset : support (deriv f) ⊆ tsupport f | begin
intros x,
rw [← not_imp_not],
intro h2x,
rw [not_mem_tsupport_iff_eventually_eq] at h2x,
exact nmem_support.mpr (h2x.deriv_eq.trans (deriv_const x 0))
end | lemma | support_deriv_subset | analysis.calculus.deriv | src/analysis/calculus/deriv/support.lean | [
"analysis.calculus.deriv.basic"
] | [
"deriv",
"deriv_const",
"not_imp_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_compact_support.deriv (hf : has_compact_support f) : has_compact_support (deriv f) | hf.mono' support_deriv_subset | lemma | has_compact_support.deriv | analysis.calculus.deriv | src/analysis/calculus/deriv/support.lean | [
"analysis.calculus.deriv.basic"
] | [
"deriv",
"support_deriv_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_zpow (m : ℤ) (x : 𝕜) (h : x ≠ 0 ∨ 0 ≤ m) :
has_strict_deriv_at (λx, x^m) ((m : 𝕜) * x^(m-1)) x | begin
have : ∀ m : ℤ, 0 < m → has_strict_deriv_at (λx, x^m) ((m:𝕜) * x^(m-1)) x,
{ assume m hm,
lift m to ℕ using (le_of_lt hm),
simp only [zpow_coe_nat, int.cast_coe_nat],
convert has_strict_deriv_at_pow _ _ using 2,
rw [← int.coe_nat_one, ← int.coe_nat_sub, zpow_coe_nat],
norm_cast at hm,
... | lemma | has_strict_deriv_at_zpow | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"has_strict_deriv_at",
"has_strict_deriv_at_const",
"has_strict_deriv_at_inv",
"has_strict_deriv_at_pow",
"int.cast_coe_nat",
"int.cast_neg",
"int.cast_zero",
"inv_inv",
"lift",
"mul_assoc",
"mul_inv",
"neg_mul",
"neg_mul_neg",
"one_div",
"smul_eq_mul",
"zero_mul",
"zpow_add₀",
"zp... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_zpow (m : ℤ) (x : 𝕜) (h : x ≠ 0 ∨ 0 ≤ m) :
has_deriv_at (λx, x^m) ((m : 𝕜) * x^(m-1)) x | (has_strict_deriv_at_zpow m x h).has_deriv_at | lemma | has_deriv_at_zpow | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"has_deriv_at",
"has_strict_deriv_at_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_zpow (m : ℤ) (x : 𝕜) (h : x ≠ 0 ∨ 0 ≤ m) (s : set 𝕜) :
has_deriv_within_at (λx, x^m) ((m : 𝕜) * x^(m-1)) s x | (has_deriv_at_zpow m x h).has_deriv_within_at | theorem | has_deriv_within_at_zpow | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"has_deriv_at_zpow",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_zpow : differentiable_at 𝕜 (λx, x^m) x ↔ x ≠ 0 ∨ 0 ≤ m | ⟨λ H, normed_field.continuous_at_zpow.1 H.continuous_at,
λ H, (has_deriv_at_zpow m x H).differentiable_at⟩ | lemma | differentiable_at_zpow | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"differentiable_at",
"has_deriv_at_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_zpow (m : ℤ) (x : 𝕜) (h : x ≠ 0 ∨ 0 ≤ m) :
differentiable_within_at 𝕜 (λx, x^m) s x | (differentiable_at_zpow.mpr h).differentiable_within_at | lemma | differentiable_within_at_zpow | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_zpow (m : ℤ) (s : set 𝕜) (h : (0 : 𝕜) ∉ s ∨ 0 ≤ m) :
differentiable_on 𝕜 (λx, x^m) s | λ x hxs, differentiable_within_at_zpow m x $ h.imp_left $ ne_of_mem_of_not_mem hxs | lemma | differentiable_on_zpow | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"differentiable_on",
"differentiable_within_at_zpow",
"ne_of_mem_of_not_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_zpow (m : ℤ) (x : 𝕜) : deriv (λ x, x ^ m) x = m * x ^ (m - 1) | begin
by_cases H : x ≠ 0 ∨ 0 ≤ m,
{ exact (has_deriv_at_zpow m x H).deriv },
{ rw deriv_zero_of_not_differentiable_at (mt differentiable_at_zpow.1 H),
push_neg at H, rcases H with ⟨rfl, hm⟩,
rw [zero_zpow _ ((sub_one_lt _).trans hm).ne, mul_zero] }
end | lemma | deriv_zpow | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"deriv",
"deriv_zero_of_not_differentiable_at",
"has_deriv_at_zpow",
"mul_zero",
"sub_one_lt",
"zero_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_zpow' (m : ℤ) : deriv (λ x : 𝕜, x ^ m) = λ x, m * x ^ (m - 1) | funext $ deriv_zpow m | lemma | deriv_zpow' | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"deriv",
"deriv_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_zpow (hxs : unique_diff_within_at 𝕜 s x) (h : x ≠ 0 ∨ 0 ≤ m) :
deriv_within (λx, x^m) s x = (m : 𝕜) * x^(m-1) | (has_deriv_within_at_zpow m x h s).deriv_within hxs | lemma | deriv_within_zpow | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"deriv_within",
"has_deriv_within_at_zpow",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iter_deriv_zpow' (m : ℤ) (k : ℕ) :
deriv^[k] (λ x : 𝕜, x ^ m) = λ x, (∏ i in finset.range k, (m - i)) * x ^ (m - k) | begin
induction k with k ihk,
{ simp only [one_mul, int.coe_nat_zero, id, sub_zero, finset.prod_range_zero,
function.iterate_zero] },
{ simp only [function.iterate_succ_apply', ihk, deriv_const_mul_field', deriv_zpow',
finset.prod_range_succ, int.coe_nat_succ, ← sub_sub, int.cast_sub, int.cast_coe_nat... | lemma | iter_deriv_zpow' | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"deriv",
"deriv_const_mul_field'",
"deriv_zpow'",
"finset.prod_range_succ",
"finset.prod_range_zero",
"finset.range",
"function.iterate_succ_apply'",
"function.iterate_zero",
"int.cast_coe_nat",
"int.cast_sub",
"mul_assoc",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iter_deriv_zpow (m : ℤ) (x : 𝕜) (k : ℕ) :
deriv^[k] (λ y, y ^ m) x = (∏ i in finset.range k, (m - i)) * x ^ (m - k) | congr_fun (iter_deriv_zpow' m k) x | lemma | iter_deriv_zpow | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"deriv",
"finset.range",
"iter_deriv_zpow'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iter_deriv_pow (n : ℕ) (x : 𝕜) (k : ℕ) :
deriv^[k] (λx:𝕜, x^n) x = (∏ i in finset.range k, (n - i)) * x^(n-k) | begin
simp only [← zpow_coe_nat, iter_deriv_zpow, int.cast_coe_nat],
cases le_or_lt k n with hkn hnk,
{ rw int.coe_nat_sub hkn },
{ have : ∏ i in finset.range k, (n - i : 𝕜) = 0,
from finset.prod_eq_zero (finset.mem_range.2 hnk) (sub_self _),
simp only [this, zero_mul] }
end | lemma | iter_deriv_pow | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"deriv",
"finset.prod_eq_zero",
"finset.range",
"int.cast_coe_nat",
"iter_deriv_zpow",
"zero_mul",
"zpow_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iter_deriv_pow' (n k : ℕ) :
deriv^[k] (λ x : 𝕜, x ^ n) = λ x, (∏ i in finset.range k, (n - i)) * x ^ (n - k) | funext $ λ x, iter_deriv_pow n x k | lemma | iter_deriv_pow' | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"deriv",
"finset.range",
"iter_deriv_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iter_deriv_inv (k : ℕ) (x : 𝕜) :
deriv^[k] has_inv.inv x = (∏ i in finset.range k, (-1 - i)) * x ^ (-1 - k : ℤ) | by simpa only [zpow_neg_one, int.cast_neg, int.cast_one] using iter_deriv_zpow (-1) x k | lemma | iter_deriv_inv | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"deriv",
"finset.range",
"int.cast_neg",
"int.cast_one",
"iter_deriv_zpow",
"zpow_neg_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
iter_deriv_inv' (k : ℕ) :
deriv^[k] has_inv.inv = λ x : 𝕜, (∏ i in finset.range k, (-1 - i)) * x ^ (-1 - k : ℤ) | funext (iter_deriv_inv k) | lemma | iter_deriv_inv' | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"deriv",
"finset.range",
"iter_deriv_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.zpow (hf : differentiable_within_at 𝕜 f t a) (h : f a ≠ 0 ∨ 0 ≤ m) :
differentiable_within_at 𝕜 (λ x, f x ^ m) t a | (differentiable_at_zpow.2 h).comp_differentiable_within_at a hf | lemma | differentiable_within_at.zpow | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.zpow (hf : differentiable_at 𝕜 f a) (h : f a ≠ 0 ∨ 0 ≤ m) :
differentiable_at 𝕜 (λ x, f x ^ m) a | (differentiable_at_zpow.2 h).comp a hf | lemma | differentiable_at.zpow | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.zpow (hf : differentiable_on 𝕜 f t) (h : (∀ x ∈ t, f x ≠ 0) ∨ 0 ≤ m) :
differentiable_on 𝕜 (λ x, f x ^ m) t | λ x hx, (hf x hx).zpow $ h.imp_left (λ h, h x hx) | lemma | differentiable_on.zpow | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.zpow (hf : differentiable 𝕜 f) (h : (∀ x, f x ≠ 0) ∨ 0 ≤ m) :
differentiable 𝕜 (λ x, f x ^ m) | λ x, (hf x).zpow $ h.imp_left (λ h, h x) | lemma | differentiable.zpow | analysis.calculus.deriv | src/analysis/calculus/deriv/zpow.lean | [
"analysis.calculus.deriv.pow",
"analysis.calculus.deriv.inv"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.const_smul (h : has_strict_fderiv_at f f' x) (c : R) :
has_strict_fderiv_at (λ x, c • f x) (c • f') x | (c • (1 : F →L[𝕜] F)).has_strict_fderiv_at.comp x h | theorem | has_strict_fderiv_at.const_smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_strict_fderiv_at",
"has_strict_fderiv_at.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.const_smul (h : has_fderiv_at_filter f f' x L) (c : R) :
has_fderiv_at_filter (λ x, c • f x) (c • f') x L | (c • (1 : F →L[𝕜] F)).has_fderiv_at_filter.comp x h tendsto_map | theorem | has_fderiv_at_filter.const_smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at_filter",
"has_fderiv_at_filter.comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.const_smul (h : has_fderiv_within_at f f' s x) (c : R) :
has_fderiv_within_at (λ x, c • f x) (c • f') s x | h.const_smul c | theorem | has_fderiv_within_at.const_smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.const_smul (h : has_fderiv_at f f' x) (c : R) :
has_fderiv_at (λ x, c • f x) (c • f') x | h.const_smul c | theorem | has_fderiv_at.const_smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.const_smul (h : differentiable_within_at 𝕜 f s x) (c : R) :
differentiable_within_at 𝕜 (λy, c • f y) s x | (h.has_fderiv_within_at.const_smul c).differentiable_within_at | lemma | differentiable_within_at.const_smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.const_smul (h : differentiable_at 𝕜 f x) (c : R) :
differentiable_at 𝕜 (λy, c • f y) x | (h.has_fderiv_at.const_smul c).differentiable_at | lemma | differentiable_at.const_smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.const_smul (h : differentiable_on 𝕜 f s) (c : R) :
differentiable_on 𝕜 (λy, c • f y) s | λx hx, (h x hx).const_smul c | lemma | differentiable_on.const_smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.const_smul (h : differentiable 𝕜 f) (c : R) :
differentiable 𝕜 (λy, c • f y) | λx, (h x).const_smul c | lemma | differentiable.const_smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_const_smul (hxs : unique_diff_within_at 𝕜 s x)
(h : differentiable_within_at 𝕜 f s x) (c : R) :
fderiv_within 𝕜 (λy, c • f y) s x = c • fderiv_within 𝕜 f s x | (h.has_fderiv_within_at.const_smul c).fderiv_within hxs | lemma | fderiv_within_const_smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_const_smul (h : differentiable_at 𝕜 f x) (c : R) :
fderiv 𝕜 (λy, c • f y) x = c • fderiv 𝕜 f x | (h.has_fderiv_at.const_smul c).fderiv | lemma | fderiv_const_smul | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.add (hf : has_strict_fderiv_at f f' x)
(hg : has_strict_fderiv_at g g' x) :
has_strict_fderiv_at (λ y, f y + g y) (f' + g') x | (hf.add hg).congr_left $ λ y,
by { simp only [linear_map.sub_apply, linear_map.add_apply, map_sub, map_add, add_apply], abel } | theorem | has_strict_fderiv_at.add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_strict_fderiv_at",
"linear_map.add_apply",
"linear_map.sub_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at_filter.add
(hf : has_fderiv_at_filter f f' x L) (hg : has_fderiv_at_filter g g' x L) :
has_fderiv_at_filter (λ y, f y + g y) (f' + g') x L | (hf.add hg).congr_left $ λ _,
by { simp only [linear_map.sub_apply, linear_map.add_apply, map_sub, map_add, add_apply], abel } | theorem | has_fderiv_at_filter.add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at_filter",
"linear_map.add_apply",
"linear_map.sub_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.add
(hf : has_fderiv_within_at f f' s x) (hg : has_fderiv_within_at g g' s x) :
has_fderiv_within_at (λ y, f y + g y) (f' + g') s x | hf.add hg | theorem | has_fderiv_within_at.add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.add
(hf : has_fderiv_at f f' x) (hg : has_fderiv_at g g' x) :
has_fderiv_at (λ x, f x + g x) (f' + g') x | hf.add hg | theorem | has_fderiv_at.add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.add
(hf : differentiable_within_at 𝕜 f s x) (hg : differentiable_within_at 𝕜 g s x) :
differentiable_within_at 𝕜 (λ y, f y + g y) s x | (hf.has_fderiv_within_at.add hg.has_fderiv_within_at).differentiable_within_at | lemma | differentiable_within_at.add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.add
(hf : differentiable_at 𝕜 f x) (hg : differentiable_at 𝕜 g x) :
differentiable_at 𝕜 (λ y, f y + g y) x | (hf.has_fderiv_at.add hg.has_fderiv_at).differentiable_at | lemma | differentiable_at.add | analysis.calculus.fderiv | src/analysis/calculus/fderiv/add.lean | [
"analysis.calculus.fderiv.linear",
"analysis.calculus.fderiv.comp"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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