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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fderiv_ccos (hc : differentiable_at ℂ f x) :
fderiv ℂ (λx, complex.cos (f x)) x = - complex.sin (f x) • (fderiv ℂ f x) | hc.has_fderiv_at.ccos.fderiv | lemma | fderiv_ccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cos",
"complex.sin",
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.ccos {n} (h : cont_diff ℂ n f) :
cont_diff ℂ n (λ x, complex.cos (f x)) | complex.cont_diff_cos.comp h | lemma | cont_diff.ccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cos",
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.ccos {n} (hf : cont_diff_at ℂ n f x) :
cont_diff_at ℂ n (λ x, complex.cos (f x)) x | complex.cont_diff_cos.cont_diff_at.comp x hf | lemma | cont_diff_at.ccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cos",
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.ccos {n} (hf : cont_diff_on ℂ n f s) :
cont_diff_on ℂ n (λ x, complex.cos (f x)) s | complex.cont_diff_cos.comp_cont_diff_on hf | lemma | cont_diff_on.ccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cos",
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.ccos {n} (hf : cont_diff_within_at ℂ n f s x) :
cont_diff_within_at ℂ n (λ x, complex.cos (f x)) s x | complex.cont_diff_cos.cont_diff_at.comp_cont_diff_within_at x hf | lemma | cont_diff_within_at.ccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cos",
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.csin (hf : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ x, complex.sin (f x)) (complex.cos (f x) • f') x | (complex.has_strict_deriv_at_sin (f x)).comp_has_strict_fderiv_at x hf | lemma | has_strict_fderiv_at.csin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cos",
"complex.has_strict_deriv_at_sin",
"complex.sin",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.csin (hf : has_fderiv_at f f' x) :
has_fderiv_at (λ x, complex.sin (f x)) (complex.cos (f x) • f') x | (complex.has_deriv_at_sin (f x)).comp_has_fderiv_at x hf | lemma | has_fderiv_at.csin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cos",
"complex.has_deriv_at_sin",
"complex.sin",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.csin (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ x, complex.sin (f x)) (complex.cos (f x) • f') s x | (complex.has_deriv_at_sin (f x)).comp_has_fderiv_within_at x hf | lemma | has_fderiv_within_at.csin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cos",
"complex.has_deriv_at_sin",
"complex.sin",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.csin (hf : differentiable_within_at ℂ f s x) :
differentiable_within_at ℂ (λ x, complex.sin (f x)) s x | hf.has_fderiv_within_at.csin.differentiable_within_at | lemma | differentiable_within_at.csin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.sin",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.csin (hc : differentiable_at ℂ f x) :
differentiable_at ℂ (λx, complex.sin (f x)) x | hc.has_fderiv_at.csin.differentiable_at | lemma | differentiable_at.csin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.sin",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.csin (hc : differentiable_on ℂ f s) :
differentiable_on ℂ (λx, complex.sin (f x)) s | λx h, (hc x h).csin | lemma | differentiable_on.csin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.sin",
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.csin (hc : differentiable ℂ f) :
differentiable ℂ (λx, complex.sin (f x)) | λx, (hc x).csin | lemma | differentiable.csin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.sin",
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_csin (hf : differentiable_within_at ℂ f s x)
(hxs : unique_diff_within_at ℂ s x) :
fderiv_within ℂ (λx, complex.sin (f x)) s x = complex.cos (f x) • (fderiv_within ℂ f s x) | hf.has_fderiv_within_at.csin.fderiv_within hxs | lemma | fderiv_within_csin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cos",
"complex.sin",
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_csin (hc : differentiable_at ℂ f x) :
fderiv ℂ (λx, complex.sin (f x)) x = complex.cos (f x) • (fderiv ℂ f x) | hc.has_fderiv_at.csin.fderiv | lemma | fderiv_csin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cos",
"complex.sin",
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.csin {n} (h : cont_diff ℂ n f) :
cont_diff ℂ n (λ x, complex.sin (f x)) | complex.cont_diff_sin.comp h | lemma | cont_diff.csin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.sin",
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.csin {n} (hf : cont_diff_at ℂ n f x) :
cont_diff_at ℂ n (λ x, complex.sin (f x)) x | complex.cont_diff_sin.cont_diff_at.comp x hf | lemma | cont_diff_at.csin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.sin",
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.csin {n} (hf : cont_diff_on ℂ n f s) :
cont_diff_on ℂ n (λ x, complex.sin (f x)) s | complex.cont_diff_sin.comp_cont_diff_on hf | lemma | cont_diff_on.csin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.sin",
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.csin {n} (hf : cont_diff_within_at ℂ n f s x) :
cont_diff_within_at ℂ n (λ x, complex.sin (f x)) s x | complex.cont_diff_sin.cont_diff_at.comp_cont_diff_within_at x hf | lemma | cont_diff_within_at.csin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.sin",
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.ccosh (hf : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ x, complex.cosh (f x)) (complex.sinh (f x) • f') x | (complex.has_strict_deriv_at_cosh (f x)).comp_has_strict_fderiv_at x hf | lemma | has_strict_fderiv_at.ccosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cosh",
"complex.has_strict_deriv_at_cosh",
"complex.sinh",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.ccosh (hf : has_fderiv_at f f' x) :
has_fderiv_at (λ x, complex.cosh (f x)) (complex.sinh (f x) • f') x | (complex.has_deriv_at_cosh (f x)).comp_has_fderiv_at x hf | lemma | has_fderiv_at.ccosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cosh",
"complex.has_deriv_at_cosh",
"complex.sinh",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.ccosh (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ x, complex.cosh (f x)) (complex.sinh (f x) • f') s x | (complex.has_deriv_at_cosh (f x)).comp_has_fderiv_within_at x hf | lemma | has_fderiv_within_at.ccosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cosh",
"complex.has_deriv_at_cosh",
"complex.sinh",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.ccosh (hf : differentiable_within_at ℂ f s x) :
differentiable_within_at ℂ (λ x, complex.cosh (f x)) s x | hf.has_fderiv_within_at.ccosh.differentiable_within_at | lemma | differentiable_within_at.ccosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cosh",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.ccosh (hc : differentiable_at ℂ f x) :
differentiable_at ℂ (λx, complex.cosh (f x)) x | hc.has_fderiv_at.ccosh.differentiable_at | lemma | differentiable_at.ccosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cosh",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.ccosh (hc : differentiable_on ℂ f s) :
differentiable_on ℂ (λx, complex.cosh (f x)) s | λx h, (hc x h).ccosh | lemma | differentiable_on.ccosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cosh",
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.ccosh (hc : differentiable ℂ f) :
differentiable ℂ (λx, complex.cosh (f x)) | λx, (hc x).ccosh | lemma | differentiable.ccosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cosh",
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_ccosh (hf : differentiable_within_at ℂ f s x)
(hxs : unique_diff_within_at ℂ s x) :
fderiv_within ℂ (λx, complex.cosh (f x)) s x = complex.sinh (f x) • (fderiv_within ℂ f s x) | hf.has_fderiv_within_at.ccosh.fderiv_within hxs | lemma | fderiv_within_ccosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cosh",
"complex.sinh",
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_ccosh (hc : differentiable_at ℂ f x) :
fderiv ℂ (λx, complex.cosh (f x)) x = complex.sinh (f x) • (fderiv ℂ f x) | hc.has_fderiv_at.ccosh.fderiv | lemma | fderiv_ccosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cosh",
"complex.sinh",
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.ccosh {n} (h : cont_diff ℂ n f) :
cont_diff ℂ n (λ x, complex.cosh (f x)) | complex.cont_diff_cosh.comp h | lemma | cont_diff.ccosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cosh",
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.ccosh {n} (hf : cont_diff_at ℂ n f x) :
cont_diff_at ℂ n (λ x, complex.cosh (f x)) x | complex.cont_diff_cosh.cont_diff_at.comp x hf | lemma | cont_diff_at.ccosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cosh",
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.ccosh {n} (hf : cont_diff_on ℂ n f s) :
cont_diff_on ℂ n (λ x, complex.cosh (f x)) s | complex.cont_diff_cosh.comp_cont_diff_on hf | lemma | cont_diff_on.ccosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cosh",
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.ccosh {n} (hf : cont_diff_within_at ℂ n f s x) :
cont_diff_within_at ℂ n (λ x, complex.cosh (f x)) s x | complex.cont_diff_cosh.cont_diff_at.comp_cont_diff_within_at x hf | lemma | cont_diff_within_at.ccosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cosh",
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.csinh (hf : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ x, complex.sinh (f x)) (complex.cosh (f x) • f') x | (complex.has_strict_deriv_at_sinh (f x)).comp_has_strict_fderiv_at x hf | lemma | has_strict_fderiv_at.csinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cosh",
"complex.has_strict_deriv_at_sinh",
"complex.sinh",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.csinh (hf : has_fderiv_at f f' x) :
has_fderiv_at (λ x, complex.sinh (f x)) (complex.cosh (f x) • f') x | (complex.has_deriv_at_sinh (f x)).comp_has_fderiv_at x hf | lemma | has_fderiv_at.csinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cosh",
"complex.has_deriv_at_sinh",
"complex.sinh",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.csinh (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ x, complex.sinh (f x)) (complex.cosh (f x) • f') s x | (complex.has_deriv_at_sinh (f x)).comp_has_fderiv_within_at x hf | lemma | has_fderiv_within_at.csinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cosh",
"complex.has_deriv_at_sinh",
"complex.sinh",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.csinh (hf : differentiable_within_at ℂ f s x) :
differentiable_within_at ℂ (λ x, complex.sinh (f x)) s x | hf.has_fderiv_within_at.csinh.differentiable_within_at | lemma | differentiable_within_at.csinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.sinh",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.csinh (hc : differentiable_at ℂ f x) :
differentiable_at ℂ (λx, complex.sinh (f x)) x | hc.has_fderiv_at.csinh.differentiable_at | lemma | differentiable_at.csinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.sinh",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.csinh (hc : differentiable_on ℂ f s) :
differentiable_on ℂ (λx, complex.sinh (f x)) s | λx h, (hc x h).csinh | lemma | differentiable_on.csinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.sinh",
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.csinh (hc : differentiable ℂ f) :
differentiable ℂ (λx, complex.sinh (f x)) | λx, (hc x).csinh | lemma | differentiable.csinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.sinh",
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_csinh (hf : differentiable_within_at ℂ f s x)
(hxs : unique_diff_within_at ℂ s x) :
fderiv_within ℂ (λx, complex.sinh (f x)) s x = complex.cosh (f x) • (fderiv_within ℂ f s x) | hf.has_fderiv_within_at.csinh.fderiv_within hxs | lemma | fderiv_within_csinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cosh",
"complex.sinh",
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_csinh (hc : differentiable_at ℂ f x) :
fderiv ℂ (λx, complex.sinh (f x)) x = complex.cosh (f x) • (fderiv ℂ f x) | hc.has_fderiv_at.csinh.fderiv | lemma | fderiv_csinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.cosh",
"complex.sinh",
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.csinh {n} (h : cont_diff ℂ n f) :
cont_diff ℂ n (λ x, complex.sinh (f x)) | complex.cont_diff_sinh.comp h | lemma | cont_diff.csinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.sinh",
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.csinh {n} (hf : cont_diff_at ℂ n f x) :
cont_diff_at ℂ n (λ x, complex.sinh (f x)) x | complex.cont_diff_sinh.cont_diff_at.comp x hf | lemma | cont_diff_at.csinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.sinh",
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.csinh {n} (hf : cont_diff_on ℂ n f s) :
cont_diff_on ℂ n (λ x, complex.sinh (f x)) s | complex.cont_diff_sinh.comp_cont_diff_on hf | lemma | cont_diff_on.csinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.sinh",
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.csinh {n} (hf : cont_diff_within_at ℂ n f s x) :
cont_diff_within_at ℂ n (λ x, complex.sinh (f x)) s x | complex.cont_diff_sinh.cont_diff_at.comp_cont_diff_within_at x hf | lemma | cont_diff_within_at.csinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.sinh",
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_sin (x : ℝ) : has_strict_deriv_at sin (cos x) x | (complex.has_strict_deriv_at_sin x).real_of_complex | lemma | real.has_strict_deriv_at_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.has_strict_deriv_at_sin",
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_sin (x : ℝ) : has_deriv_at sin (cos x) x | (has_strict_deriv_at_sin x).has_deriv_at | lemma | real.has_deriv_at_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_sin {n} : cont_diff ℝ n sin | complex.cont_diff_sin.real_of_complex | lemma | real.cont_diff_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_sin : differentiable ℝ sin | λx, (has_deriv_at_sin x).differentiable_at | lemma | real.differentiable_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_sin : differentiable_at ℝ sin x | differentiable_sin x | lemma | real.differentiable_at_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_cos (x : ℝ) : has_strict_deriv_at cos (-sin x) x | (complex.has_strict_deriv_at_cos x).real_of_complex | lemma | real.has_strict_deriv_at_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.has_strict_deriv_at_cos",
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_cos (x : ℝ) : has_deriv_at cos (-sin x) x | (complex.has_deriv_at_cos x).real_of_complex | lemma | real.has_deriv_at_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.has_deriv_at_cos",
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_cos {n} : cont_diff ℝ n cos | complex.cont_diff_cos.real_of_complex | lemma | real.cont_diff_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_cos : differentiable ℝ cos | λx, (has_deriv_at_cos x).differentiable_at | lemma | real.differentiable_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_cos : differentiable_at ℝ cos x | differentiable_cos x | lemma | real.differentiable_at_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_cos : deriv cos x = - sin x | (has_deriv_at_cos x).deriv | lemma | real.deriv_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"deriv",
"deriv_cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_cos' : deriv cos = (λ x, - sin x) | funext $ λ _, deriv_cos | lemma | real.deriv_cos' | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"deriv",
"deriv_cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_sinh (x : ℝ) : has_strict_deriv_at sinh (cosh x) x | (complex.has_strict_deriv_at_sinh x).real_of_complex | lemma | real.has_strict_deriv_at_sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.has_strict_deriv_at_sinh",
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_sinh (x : ℝ) : has_deriv_at sinh (cosh x) x | (complex.has_deriv_at_sinh x).real_of_complex | lemma | real.has_deriv_at_sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.has_deriv_at_sinh",
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_sinh {n} : cont_diff ℝ n sinh | complex.cont_diff_sinh.real_of_complex | lemma | real.cont_diff_sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_sinh : differentiable ℝ sinh | λx, (has_deriv_at_sinh x).differentiable_at | lemma | real.differentiable_sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_sinh : differentiable_at ℝ sinh x | differentiable_sinh x | lemma | real.differentiable_at_sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_cosh (x : ℝ) : has_strict_deriv_at cosh (sinh x) x | (complex.has_strict_deriv_at_cosh x).real_of_complex | lemma | real.has_strict_deriv_at_cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.has_strict_deriv_at_cosh",
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_cosh (x : ℝ) : has_deriv_at cosh (sinh x) x | (complex.has_deriv_at_cosh x).real_of_complex | lemma | real.has_deriv_at_cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"complex.has_deriv_at_cosh",
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_cosh {n} : cont_diff ℝ n cosh | complex.cont_diff_cosh.real_of_complex | lemma | real.cont_diff_cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_cosh : differentiable ℝ cosh | λx, (has_deriv_at_cosh x).differentiable_at | lemma | real.differentiable_cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_cosh : differentiable_at ℝ cosh x | differentiable_cosh x | lemma | real.differentiable_at_cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sinh_strict_mono : strict_mono sinh | strict_mono_of_deriv_pos $ by { rw real.deriv_sinh, exact cosh_pos } | lemma | real.sinh_strict_mono | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"real.deriv_sinh",
"strict_mono",
"strict_mono_of_deriv_pos"
] | `sinh` is strictly monotone. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sinh_injective : function.injective sinh | sinh_strict_mono.injective | lemma | real.sinh_injective | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [] | `sinh` is injective, `∀ a b, sinh a = sinh b → a = b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sinh_inj : sinh x = sinh y ↔ x = y | sinh_injective.eq_iff | lemma | real.sinh_inj | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sinh_le_sinh : sinh x ≤ sinh y ↔ x ≤ y | sinh_strict_mono.le_iff_le | lemma | real.sinh_le_sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sinh_lt_sinh : sinh x < sinh y ↔ x < y | sinh_strict_mono.lt_iff_lt | lemma | real.sinh_lt_sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sinh_pos_iff : 0 < sinh x ↔ 0 < x | by simpa only [sinh_zero] using @sinh_lt_sinh 0 x | lemma | real.sinh_pos_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sinh_nonpos_iff : sinh x ≤ 0 ↔ x ≤ 0 | by simpa only [sinh_zero] using @sinh_le_sinh x 0 | lemma | real.sinh_nonpos_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sinh_neg_iff : sinh x < 0 ↔ x < 0 | by simpa only [sinh_zero] using @sinh_lt_sinh x 0 | lemma | real.sinh_neg_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sinh_nonneg_iff : 0 ≤ sinh x ↔ 0 ≤ x | by simpa only [sinh_zero] using @sinh_le_sinh 0 x | lemma | real.sinh_nonneg_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sinh (x : ℝ) : |sinh x| = sinh (|x|) | by cases le_total x 0; simp [abs_of_nonneg, abs_of_nonpos, *] | lemma | real.abs_sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"abs_of_nonneg",
"abs_of_nonpos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cosh_strict_mono_on : strict_mono_on cosh (Ici 0) | (convex_Ici _).strict_mono_on_of_deriv_pos continuous_cosh.continuous_on $ λ x hx,
by { rw [interior_Ici, mem_Ioi] at hx, rwa [deriv_cosh, sinh_pos_iff] } | lemma | real.cosh_strict_mono_on | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"convex_Ici",
"deriv_cosh",
"interior_Ici",
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cosh_le_cosh : cosh x ≤ cosh y ↔ |x| ≤ |y| | cosh_abs x ▸ cosh_abs y ▸ cosh_strict_mono_on.le_iff_le (_root_.abs_nonneg x) (_root_.abs_nonneg y) | lemma | real.cosh_le_cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cosh_lt_cosh : cosh x < cosh y ↔ |x| < |y| | lt_iff_lt_of_le_iff_le cosh_le_cosh | lemma | real.cosh_lt_cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_cosh (x : ℝ) : 1 ≤ cosh x | cosh_zero ▸ cosh_le_cosh.2 (by simp only [_root_.abs_zero, _root_.abs_nonneg]) | lemma | real.one_le_cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_cosh : 1 < cosh x ↔ x ≠ 0 | cosh_zero ▸ cosh_lt_cosh.trans (by simp only [_root_.abs_zero, abs_pos]) | lemma | real.one_lt_cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"abs_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sinh_sub_id_strict_mono : strict_mono (λ x, sinh x - x) | begin
refine strict_mono_of_odd_strict_mono_on_nonneg (λ x, by simp) _,
refine (convex_Ici _).strict_mono_on_of_deriv_pos _ (λ x hx, _),
{ exact (continuous_sinh.sub continuous_id).continuous_on },
{ rw [interior_Ici, mem_Ioi] at hx,
rw [deriv_sub, deriv_sinh, deriv_id'', sub_pos, one_lt_cosh],
exacts [... | lemma | real.sinh_sub_id_strict_mono | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"continuous_id",
"continuous_on",
"convex_Ici",
"deriv_id''",
"deriv_sinh",
"deriv_sub",
"differentiable_at_id",
"interior_Ici",
"strict_mono",
"strict_mono_of_odd_strict_mono_on_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_le_sinh_iff : x ≤ sinh x ↔ 0 ≤ x | calc x ≤ sinh x ↔ sinh 0 - 0 ≤ sinh x - x : by simp
... ↔ 0 ≤ x : sinh_sub_id_strict_mono.le_iff_le | lemma | real.self_le_sinh_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sinh_le_self_iff : sinh x ≤ x ↔ x ≤ 0 | calc sinh x ≤ x ↔ sinh x - x ≤ sinh 0 - 0 : by simp
... ↔ x ≤ 0 : sinh_sub_id_strict_mono.le_iff_le | lemma | real.sinh_le_self_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
self_lt_sinh_iff : x < sinh x ↔ 0 < x | lt_iff_lt_of_le_iff_le sinh_le_self_iff | lemma | real.self_lt_sinh_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sinh_lt_self_iff : sinh x < x ↔ x < 0 | lt_iff_lt_of_le_iff_le self_le_sinh_iff | lemma | real.sinh_lt_self_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.cos (hf : has_strict_deriv_at f f' x) :
has_strict_deriv_at (λ x, real.cos (f x)) (- real.sin (f x) * f') x | (real.has_strict_deriv_at_cos (f x)).comp x hf | lemma | has_strict_deriv_at.cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_strict_deriv_at",
"real.cos",
"real.has_strict_deriv_at_cos",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.cos (hf : has_deriv_at f f' x) :
has_deriv_at (λ x, real.cos (f x)) (- real.sin (f x) * f') x | (real.has_deriv_at_cos (f x)).comp x hf | lemma | has_deriv_at.cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_deriv_at",
"real.cos",
"real.has_deriv_at_cos",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.cos (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ x, real.cos (f x)) (- real.sin (f x) * f') s x | (real.has_deriv_at_cos (f x)).comp_has_deriv_within_at x hf | lemma | has_deriv_within_at.cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_deriv_within_at",
"real.cos",
"real.has_deriv_at_cos",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_cos (hf : differentiable_within_at ℝ f s x)
(hxs : unique_diff_within_at ℝ s x) :
deriv_within (λx, real.cos (f x)) s x = - real.sin (f x) * (deriv_within f s x) | hf.has_deriv_within_at.cos.deriv_within hxs | lemma | deriv_within_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"deriv_within",
"differentiable_within_at",
"real.cos",
"real.sin",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_cos (hc : differentiable_at ℝ f x) :
deriv (λx, real.cos (f x)) x = - real.sin (f x) * (deriv f x) | hc.has_deriv_at.cos.deriv | lemma | deriv_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"deriv",
"differentiable_at",
"real.cos",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.sin (hf : has_strict_deriv_at f f' x) :
has_strict_deriv_at (λ x, real.sin (f x)) (real.cos (f x) * f') x | (real.has_strict_deriv_at_sin (f x)).comp x hf | lemma | has_strict_deriv_at.sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_strict_deriv_at",
"real.cos",
"real.has_strict_deriv_at_sin",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.sin (hf : has_deriv_at f f' x) :
has_deriv_at (λ x, real.sin (f x)) (real.cos (f x) * f') x | (real.has_deriv_at_sin (f x)).comp x hf | lemma | has_deriv_at.sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_deriv_at",
"real.cos",
"real.has_deriv_at_sin",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.sin (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ x, real.sin (f x)) (real.cos (f x) * f') s x | (real.has_deriv_at_sin (f x)).comp_has_deriv_within_at x hf | lemma | has_deriv_within_at.sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_deriv_within_at",
"real.cos",
"real.has_deriv_at_sin",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_sin (hf : differentiable_within_at ℝ f s x)
(hxs : unique_diff_within_at ℝ s x) :
deriv_within (λx, real.sin (f x)) s x = real.cos (f x) * (deriv_within f s x) | hf.has_deriv_within_at.sin.deriv_within hxs | lemma | deriv_within_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"deriv_within",
"differentiable_within_at",
"real.cos",
"real.sin",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_sin (hc : differentiable_at ℝ f x) :
deriv (λx, real.sin (f x)) x = real.cos (f x) * (deriv f x) | hc.has_deriv_at.sin.deriv | lemma | deriv_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"deriv",
"differentiable_at",
"real.cos",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.cosh (hf : has_strict_deriv_at f f' x) :
has_strict_deriv_at (λ x, real.cosh (f x)) (real.sinh (f x) * f') x | (real.has_strict_deriv_at_cosh (f x)).comp x hf | lemma | has_strict_deriv_at.cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_strict_deriv_at",
"real.cosh",
"real.has_strict_deriv_at_cosh",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.cosh (hf : has_deriv_at f f' x) :
has_deriv_at (λ x, real.cosh (f x)) (real.sinh (f x) * f') x | (real.has_deriv_at_cosh (f x)).comp x hf | lemma | has_deriv_at.cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_deriv_at",
"real.cosh",
"real.has_deriv_at_cosh",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.cosh (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ x, real.cosh (f x)) (real.sinh (f x) * f') s x | (real.has_deriv_at_cosh (f x)).comp_has_deriv_within_at x hf | lemma | has_deriv_within_at.cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"has_deriv_within_at",
"real.cosh",
"real.has_deriv_at_cosh",
"real.sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_cosh (hf : differentiable_within_at ℝ f s x)
(hxs : unique_diff_within_at ℝ s x) :
deriv_within (λx, real.cosh (f x)) s x = real.sinh (f x) * (deriv_within f s x) | hf.has_deriv_within_at.cosh.deriv_within hxs | lemma | deriv_within_cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/deriv.lean | [
"order.monotone.odd",
"analysis.special_functions.exp_deriv",
"analysis.special_functions.trigonometric.basic"
] | [
"deriv_within",
"differentiable_within_at",
"real.cosh",
"real.sinh",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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