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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le_tan {x : ℝ} (h1 : 0 ≤ x) (h2 : x < π / 2) : x ≤ tan x | begin
rcases eq_or_lt_of_le h1 with rfl | h1',
{ rw tan_zero },
{ exact le_of_lt (lt_tan h1' h2) }
end | lemma | real.le_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/bounds.lean | [
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"eq_or_lt_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_lt_one_div_sqrt_sq_add_one {x : ℝ}
(hx1 : -(3 * π / 2) ≤ x) (hx2 : x ≤ 3 * π / 2) (hx3 : x ≠ 0) :
cos x < 1 / sqrt (x ^ 2 + 1) | begin
suffices : ∀ {y : ℝ} (hy1 : 0 < y) (hy2 : y ≤ 3 * π / 2), cos y < 1 / sqrt (y ^ 2 + 1),
{ rcases lt_or_lt_iff_ne.mpr hx3.symm,
{ exact this h hx2 },
{ convert this (by linarith : 0 < -x) (by linarith) using 1,
{ rw cos_neg }, { rw neg_sq } } },
intros y hy1 hy2,
have hy3 : 0 < y ^ 2 + 1, by ... | lemma | real.cos_lt_one_div_sqrt_sq_add_one | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/bounds.lean | [
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"abs_of_nonneg",
"abs_of_pos",
"div_pow",
"inv_inv",
"lt_one_div",
"neg_sq",
"one_div",
"one_pow",
"pow_pos",
"real.lt_tan",
"sq_lt_sq",
"sq_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_le_one_div_sqrt_sq_add_one {x : ℝ} (hx1 : -(3 * π / 2) ≤ x) (hx2 : x ≤ 3 * π / 2) :
cos x ≤ 1 / sqrt (x ^ 2 + 1) | begin
rcases eq_or_ne x 0 with rfl | hx3,
{ simp },
{ exact (cos_lt_one_div_sqrt_sq_add_one hx1 hx2 hx3).le }
end | lemma | real.cos_le_one_div_sqrt_sq_add_one | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/bounds.lean | [
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"eq_or_ne"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_T (x : A) (n : ℕ) : aeval x (T R n) = (T A n).eval x | by rw [aeval_def, eval₂_eq_eval_map, map_T] | lemma | polynomial.chebyshev.aeval_T | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/chebyshev.lean | [
"data.complex.exponential",
"data.complex.module",
"data.polynomial.algebra_map",
"ring_theory.polynomial.chebyshev"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
aeval_U (x : A) (n : ℕ) : aeval x (U R n) = (U A n).eval x | by rw [aeval_def, eval₂_eq_eval_map, map_U] | lemma | polynomial.chebyshev.aeval_U | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/chebyshev.lean | [
"data.complex.exponential",
"data.complex.module",
"data.polynomial.algebra_map",
"ring_theory.polynomial.chebyshev"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_eval_T (x : R) (n : ℕ) :
algebra_map R A ((T R n).eval x) = (T A n).eval (algebra_map R A x) | by rw [←aeval_algebra_map_apply_eq_algebra_map_eval, aeval_T] | lemma | polynomial.chebyshev.algebra_map_eval_T | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/chebyshev.lean | [
"data.complex.exponential",
"data.complex.module",
"data.polynomial.algebra_map",
"ring_theory.polynomial.chebyshev"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
algebra_map_eval_U (x : R) (n : ℕ) :
algebra_map R A ((U R n).eval x) = (U A n).eval (algebra_map R A x) | by rw [←aeval_algebra_map_apply_eq_algebra_map_eval, aeval_U] | lemma | polynomial.chebyshev.algebra_map_eval_U | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/chebyshev.lean | [
"data.complex.exponential",
"data.complex.module",
"data.polynomial.algebra_map",
"ring_theory.polynomial.chebyshev"
] | [
"algebra_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complex_of_real_eval_T : ∀ x n, ((T ℝ n).eval x : ℂ) = (T ℂ n).eval x | @algebra_map_eval_T ℝ ℂ _ _ _ | lemma | polynomial.chebyshev.complex_of_real_eval_T | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/chebyshev.lean | [
"data.complex.exponential",
"data.complex.module",
"data.polynomial.algebra_map",
"ring_theory.polynomial.chebyshev"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complex_of_real_eval_U : ∀ x n, ((U ℝ n).eval x : ℂ) = (U ℂ n).eval x | @algebra_map_eval_U ℝ ℂ _ _ _ | lemma | polynomial.chebyshev.complex_of_real_eval_U | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/chebyshev.lean | [
"data.complex.exponential",
"data.complex.module",
"data.polynomial.algebra_map",
"ring_theory.polynomial.chebyshev"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
T_complex_cos : ∀ n, (T ℂ n).eval (cos θ) = cos (n * θ) | | 0 := by simp only [T_zero, eval_one, nat.cast_zero, zero_mul, cos_zero]
| 1 := by simp only [eval_X, one_mul, T_one, nat.cast_one]
| (n + 2) :=
begin
simp only [eval_X, eval_one, T_add_two, eval_sub, eval_bit0, nat.cast_succ, eval_mul],
rw [T_complex_cos (n + 1), T_complex_cos n],
have aux : sin θ *... | lemma | polynomial.chebyshev.T_complex_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/chebyshev.lean | [
"data.complex.exponential",
"data.complex.module",
"data.polynomial.algebra_map",
"ring_theory.polynomial.chebyshev"
] | [
"aux",
"mul_assoc",
"nat.cast_add",
"nat.cast_one",
"nat.cast_succ",
"nat.cast_zero",
"one_mul",
"ring",
"zero_mul"
] | The `n`-th Chebyshev polynomial of the first kind evaluates on `cos θ` to the
value `cos (n * θ)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
U_complex_cos (n : ℕ) : (U ℂ n).eval (cos θ) * sin θ = sin ((n + 1) * θ) | begin
induction n with d hd,
{ simp only [U_zero, nat.cast_zero, eval_one, mul_one, zero_add, one_mul] },
{ rw U_eq_X_mul_U_add_T,
simp only [eval_add, eval_mul, eval_X, T_complex_cos, add_mul, mul_assoc, hd, one_mul],
conv_rhs { rw [sin_add, mul_comm] },
push_cast,
simp only [add_mul, one_mul] }
... | lemma | polynomial.chebyshev.U_complex_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/chebyshev.lean | [
"data.complex.exponential",
"data.complex.module",
"data.polynomial.algebra_map",
"ring_theory.polynomial.chebyshev"
] | [
"mul_assoc",
"mul_comm",
"mul_one",
"nat.cast_zero",
"one_mul"
] | The `n`-th Chebyshev polynomial of the second kind evaluates on `cos θ` to the
value `sin ((n + 1) * θ) / sin θ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
T_real_cos : (T ℝ n).eval (cos θ) = cos (n * θ) | by exact_mod_cast T_complex_cos θ n | lemma | polynomial.chebyshev.T_real_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/chebyshev.lean | [
"data.complex.exponential",
"data.complex.module",
"data.polynomial.algebra_map",
"ring_theory.polynomial.chebyshev"
] | [] | The `n`-th Chebyshev polynomial of the first kind evaluates on `cos θ` to the
value `cos (n * θ)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
U_real_cos : (U ℝ n).eval (cos θ) * sin θ = sin ((n + 1) * θ) | by exact_mod_cast U_complex_cos θ n | lemma | polynomial.chebyshev.U_real_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/chebyshev.lean | [
"data.complex.exponential",
"data.complex.module",
"data.polynomial.algebra_map",
"ring_theory.polynomial.chebyshev"
] | [] | The `n`-th Chebyshev polynomial of the second kind evaluates on `cos θ` to the
value `sin ((n + 1) * θ) / sin θ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cos_eq_zero_iff {θ : ℂ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 | begin
have h : (exp (θ * I) + exp (-θ * I)) / 2 = 0 ↔ exp (2 * θ * I) = -1,
{ rw [@div_eq_iff _ _ (exp (θ * I) + exp (-θ * I)) 2 0 two_ne_zero, zero_mul,
add_eq_zero_iff_eq_neg, neg_eq_neg_one_mul, ← div_eq_iff (exp_ne_zero _), ← exp_sub],
field_simp only, congr' 3, ring },
rw [cos, h, ← exp_pi_mul_I, e... | theorem | complex.cos_eq_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"div_eq_iff",
"exp",
"iff_of_eq",
"mul_ne_zero",
"mul_right_comm",
"mul_right_inj'",
"neg_eq_neg_one_mul",
"ring",
"two_ne_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_ne_zero_iff {θ : ℂ} : cos θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ (2 * k + 1) * π / 2 | by rw [← not_exists, not_iff_not, cos_eq_zero_iff] | theorem | complex.cos_ne_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"not_exists",
"not_iff_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_eq_zero_iff {θ : ℂ} : sin θ = 0 ↔ ∃ k : ℤ, θ = k * π | begin
rw [← complex.cos_sub_pi_div_two, cos_eq_zero_iff],
split,
{ rintros ⟨k, hk⟩,
use k + 1,
field_simp [eq_add_of_sub_eq hk],
ring },
{ rintros ⟨k, rfl⟩,
use k - 1,
field_simp,
ring }
end | theorem | complex.sin_eq_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"complex.cos_sub_pi_div_two",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_ne_zero_iff {θ : ℂ} : sin θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ k * π | by rw [← not_exists, not_iff_not, sin_eq_zero_iff] | theorem | complex.sin_ne_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"not_exists",
"not_iff_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_eq_zero_iff {θ : ℂ} : tan θ = 0 ↔ ∃ k : ℤ, θ = k * π / 2 | begin
have h := (sin_two_mul θ).symm,
rw mul_assoc at h,
rw [tan, div_eq_zero_iff, ← mul_eq_zero, ← zero_mul ((1/2):ℂ), mul_one_div,
cancel_factors.cancel_factors_eq_div h two_ne_zero, mul_comm],
simpa only [zero_div, zero_mul, ne.def, not_false_iff] with field_simps using sin_eq_zero_iff,
end | lemma | complex.tan_eq_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"cancel_factors.cancel_factors_eq_div",
"div_eq_zero_iff",
"mul_assoc",
"mul_comm",
"mul_eq_zero",
"mul_one_div",
"two_ne_zero",
"zero_div",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_ne_zero_iff {θ : ℂ} : tan θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ k * π / 2 | by rw [← not_exists, not_iff_not, tan_eq_zero_iff] | lemma | complex.tan_ne_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"not_exists",
"not_iff_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_eq_cos_iff {x y : ℂ} :
cos x = cos y ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = 2 * k * π - x | calc cos x = cos y ↔ cos x - cos y = 0 : sub_eq_zero.symm
... ↔ -2 * sin((x + y)/2) * sin((x - y)/2) = 0 : by rw cos_sub_cos
... ↔ sin((x + y)/2) = 0 ∨ sin((x - y)/2) = 0 : by simp [(by norm_num : (2:ℂ) ≠ 0)]
... ↔ sin((x - y)/2) = 0 ∨ sin((x + y)/2) = 0 : or.comm
... ↔ (∃ k : ℤ, y = 2 * k * π + x) ∨ (∃ k :ℤ, y = 2 * k... | lemma | complex.cos_eq_cos_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"mul_comm",
"mul_right_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_eq_sin_iff {x y : ℂ} :
sin x = sin y ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = (2 * k + 1) * π - x | begin
simp only [← complex.cos_sub_pi_div_two, cos_eq_cos_iff, sub_eq_iff_eq_add],
refine exists_congr (λ k, or_congr _ _); refine eq.congr rfl _; field_simp; ring
end | lemma | complex.sin_eq_sin_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"complex.cos_sub_pi_div_two",
"eq.congr",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_add {x y : ℂ}
(h : ((∀ k : ℤ, x ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, y ≠ (2 * l + 1) * π / 2)
∨ ((∃ k : ℤ, x = (2 * k + 1) * π / 2) ∧ ∃ l : ℤ, y = (2 * l + 1) * π / 2)) :
tan (x + y) = (tan x + tan y) / (1 - tan x * tan y) | begin
rcases h with ⟨h1, h2⟩ | ⟨⟨k, rfl⟩, ⟨l, rfl⟩⟩,
{ rw [tan, sin_add, cos_add,
← div_div_div_cancel_right (sin x * cos y + cos x * sin y)
(mul_ne_zero (cos_ne_zero_iff.mpr h1) (cos_ne_zero_iff.mpr h2)),
add_div, sub_div],
simp only [←div_mul_div_comm, ←tan, mul_one, one_mul,
... | lemma | complex.tan_add | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"add_div",
"div_div_div_cancel_right",
"div_self",
"int.cast_add",
"int.cast_bit0",
"int.cast_mul",
"int.cast_one",
"mul_div_assoc",
"mul_ne_zero",
"mul_one",
"one_mul",
"sub_div",
"zero_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_add' {x y : ℂ}
(h : ((∀ k : ℤ, x ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, y ≠ (2 * l + 1) * π / 2)) :
tan (x + y) = (tan x + tan y) / (1 - tan x * tan y) | tan_add (or.inl h) | lemma | complex.tan_add' | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_two_mul {z : ℂ} : tan (2 * z) = 2 * tan z / (1 - tan z ^ 2) | begin
by_cases h : ∀ k : ℤ, z ≠ (2 * k + 1) * π / 2,
{ rw [two_mul, two_mul, sq, tan_add (or.inl ⟨h, h⟩)] },
{ rw not_forall_not at h,
rw [two_mul, two_mul, sq, tan_add (or.inr ⟨h, h⟩)] },
end | lemma | complex.tan_two_mul | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"not_forall_not",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_add_mul_I {x y : ℂ}
(h : ((∀ k : ℤ, x ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, y * I ≠ (2 * l + 1) * π / 2)
∨ ((∃ k : ℤ, x = (2 * k + 1) * π / 2) ∧ ∃ l : ℤ, y * I = (2 * l + 1) * π / 2)) :
tan (x + y*I) = (tan x + tanh y * I) / (1 - tan x * tanh y * I) | by rw [tan_add h, tan_mul_I, mul_assoc] | lemma | complex.tan_add_mul_I | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_eq {z : ℂ}
(h : ((∀ k : ℤ, (z.re:ℂ) ≠ (2 * k + 1) * π / 2) ∧ ∀ l : ℤ, (z.im:ℂ) * I ≠ (2 * l + 1) * π / 2)
∨ ((∃ k : ℤ, (z.re:ℂ) = (2 * k + 1) * π / 2) ∧ ∃ l : ℤ, (z.im:ℂ) * I = (2 * l + 1) * π / 2)) :
tan z = (tan z.re + tanh z.im * I) / (1 - tan z.re * tanh z.im * I) | by convert tan_add_mul_I h; exact (re_add_im z).symm | lemma | complex.tan_eq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_tan : continuous_on tan {x | cos x ≠ 0} | continuous_on_sin.div continuous_on_cos $ λ x, id | lemma | complex.continuous_on_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_eq_iff_quadratic {z w : ℂ} :
cos z = w ↔ (exp (z * I)) ^ 2 - 2 * w * exp (z * I) + 1 = 0 | begin
rw ← sub_eq_zero,
field_simp [cos, exp_neg, exp_ne_zero],
refine eq.congr _ rfl,
ring
end | lemma | complex.cos_eq_iff_quadratic | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"eq.congr",
"exp",
"exp_neg",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_surjective : function.surjective cos | begin
intro x,
obtain ⟨w, w₀, hw⟩ : ∃ w ≠ 0, 1 * w * w + (-2 * x) * w + 1 = 0,
{ rcases exists_quadratic_eq_zero one_ne_zero
⟨_, ((cpow_nat_inv_pow _ two_ne_zero).symm.trans $ pow_two _)⟩ with ⟨w, hw⟩,
refine ⟨w, _, hw⟩,
rintro rfl,
simpa only [zero_add, one_ne_zero, mul_zero] using hw },
refi... | lemma | complex.cos_surjective | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"div_mul_cancel",
"exists_quadratic_eq_zero",
"mul_zero",
"one_ne_zero",
"pow_two",
"ring",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_cos : range cos = set.univ | cos_surjective.range_eq | lemma | complex.range_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_surjective : function.surjective sin | begin
intro x,
rcases cos_surjective x with ⟨z, rfl⟩,
exact ⟨z + π / 2, sin_add_pi_div_two z⟩
end | lemma | complex.sin_surjective | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_sin : range sin = set.univ | sin_surjective.range_eq | lemma | complex.range_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_eq_zero_iff {θ : ℝ} : cos θ = 0 ↔ ∃ k : ℤ, θ = (2 * k + 1) * π / 2 | by exact_mod_cast @complex.cos_eq_zero_iff θ | theorem | real.cos_eq_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"complex.cos_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_ne_zero_iff {θ : ℝ} : cos θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ (2 * k + 1) * π / 2 | by rw [← not_exists, not_iff_not, cos_eq_zero_iff] | theorem | real.cos_ne_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"not_exists",
"not_iff_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_eq_cos_iff {x y : ℝ} :
cos x = cos y ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = 2 * k * π - x | by exact_mod_cast @complex.cos_eq_cos_iff x y | lemma | real.cos_eq_cos_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"complex.cos_eq_cos_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_eq_sin_iff {x y : ℝ} :
sin x = sin y ↔ ∃ k : ℤ, y = 2 * k * π + x ∨ y = (2 * k + 1) * π - x | by exact_mod_cast @complex.sin_eq_sin_iff x y | lemma | real.sin_eq_sin_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"complex.sin_eq_sin_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_sin_mul {x : ℝ} (hx : 0 < x) (hx' : x < 1) : x < sin ((π / 2) * x) | by simpa [mul_comm x] using strict_concave_on_sin_Icc.2 ⟨le_rfl, pi_pos.le⟩
⟨pi_div_two_pos.le, half_le_self pi_pos.le⟩ pi_div_two_pos.ne (sub_pos.2 hx') hx | lemma | real.lt_sin_mul | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_sin_mul {x : ℝ} (hx : 0 ≤ x) (hx' : x ≤ 1) : x ≤ sin ((π / 2) * x) | by simpa [mul_comm x] using strict_concave_on_sin_Icc.concave_on.2 ⟨le_rfl, pi_pos.le⟩
⟨pi_div_two_pos.le, half_le_self pi_pos.le⟩ (sub_nonneg.2 hx') hx | lemma | real.le_sin_mul | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_lt_sin {x : ℝ} (hx : 0 < x) (hx' : x < π / 2) : (2 / π) * x < sin x | begin
rw [←inv_div],
simpa [-inv_div, pi_div_two_pos.ne'] using @lt_sin_mul ((π / 2)⁻¹ * x) _ _,
{ exact mul_pos (inv_pos.2 pi_div_two_pos) hx },
{ rwa [←div_eq_inv_mul, div_lt_one pi_div_two_pos] },
end | lemma | real.mul_lt_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"div_lt_one",
"inv_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_le_sin {x : ℝ} (hx : 0 ≤ x) (hx' : x ≤ π / 2) : (2 / π) * x ≤ sin x | begin
rw [←inv_div],
simpa [-inv_div, pi_div_two_pos.ne'] using @le_sin_mul ((π / 2)⁻¹ * x) _ _,
{ exact mul_nonneg (inv_nonneg.2 pi_div_two_pos.le) hx },
{ rwa [←div_eq_inv_mul, div_le_one pi_div_two_pos] },
end | lemma | real.mul_le_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex.lean | [
"algebra.quadratic_discriminant",
"analysis.convex.specific_functions.deriv"
] | [
"div_le_one",
"inv_div"
] | In the range `[0, π / 2]`, we have a linear lower bound on `sin`. This inequality forms one half
of Jordan's inequality, the other half is `real.sin_lt` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_strict_deriv_at_tan {x : ℂ} (h : cos x ≠ 0) :
has_strict_deriv_at tan (1 / (cos x)^2) x | begin
convert (has_strict_deriv_at_sin x).div (has_strict_deriv_at_cos x) h,
rw ← sin_sq_add_cos_sq x,
ring,
end | lemma | complex.has_strict_deriv_at_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex_deriv.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"has_strict_deriv_at",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_tan {x : ℂ} (h : cos x ≠ 0) :
has_deriv_at tan (1 / (cos x)^2) x | (has_strict_deriv_at_tan h).has_deriv_at | lemma | complex.has_deriv_at_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex_deriv.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_abs_tan_of_cos_eq_zero {x : ℂ} (hx : cos x = 0) :
tendsto (λ x, abs (tan x)) (𝓝[≠] x) at_top | begin
simp only [tan_eq_sin_div_cos, ← norm_eq_abs, norm_div],
have A : sin x ≠ 0 := λ h, by simpa [*, sq] using sin_sq_add_cos_sq x,
have B : tendsto cos (𝓝[≠] (x)) (𝓝[≠] 0),
from hx ▸ (has_deriv_at_cos x).tendsto_punctured_nhds (neg_ne_zero.2 A),
exact continuous_sin.continuous_within_at.norm.mul_at_top... | lemma | complex.tendsto_abs_tan_of_cos_eq_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex_deriv.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"norm_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_tan {x : ℂ} : continuous_at tan x ↔ cos x ≠ 0 | begin
refine ⟨λ hc h₀, _, λ h, (has_deriv_at_tan h).continuous_at⟩,
exact not_tendsto_nhds_of_tendsto_at_top (tendsto_abs_tan_of_cos_eq_zero h₀) _
(hc.norm.tendsto.mono_left inf_le_left)
end | lemma | complex.continuous_at_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex_deriv.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"continuous_at",
"inf_le_left",
"not_tendsto_nhds_of_tendsto_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_tan {x : ℂ} : differentiable_at ℂ tan x ↔ cos x ≠ 0 | ⟨λ h, continuous_at_tan.1 h.continuous_at, λ h, (has_deriv_at_tan h).differentiable_at⟩ | lemma | complex.differentiable_at_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex_deriv.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_tan (x : ℂ) : deriv tan x = 1 / (cos x)^2 | if h : cos x = 0 then
have ¬differentiable_at ℂ tan x := mt differentiable_at_tan.1 (not_not.2 h),
by simp [deriv_zero_of_not_differentiable_at this, h, sq]
else (has_deriv_at_tan h).deriv | lemma | complex.deriv_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex_deriv.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"deriv",
"deriv_zero_of_not_differentiable_at",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_tan {x : ℂ} {n : ℕ∞} :
cont_diff_at ℂ n tan x ↔ cos x ≠ 0 | ⟨λ h, continuous_at_tan.1 h.continuous_at,
cont_diff_sin.cont_diff_at.div cont_diff_cos.cont_diff_at⟩ | lemma | complex.cont_diff_at_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/complex_deriv.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_sin (x : ℂ) : has_strict_deriv_at sin (cos x) x | begin
simp only [cos, div_eq_mul_inv],
convert ((((has_strict_deriv_at_id x).neg.mul_const I).cexp.sub
((has_strict_deriv_at_id x).mul_const I).cexp).mul_const I).mul_const (2:ℂ)⁻¹,
simp only [function.comp, id],
rw [sub_mul, mul_assoc, mul_assoc, I_mul_I, neg_one_mul, neg_neg, mul_one, one_mul, mul_assoc,
... | lemma | complex.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"
] | [
"div_eq_mul_inv",
"has_strict_deriv_at",
"has_strict_deriv_at_id",
"mul_assoc",
"mul_neg_one",
"mul_one",
"neg_one_mul",
"one_mul"
] | The complex sine function is everywhere strictly differentiable, with the derivative `cos x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_sin (x : ℂ) : has_deriv_at sin (cos x) x | (has_strict_deriv_at_sin x).has_deriv_at | lemma | complex.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"
] | The complex sine function is everywhere differentiable, with the derivative `cos x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_sin {n} : cont_diff ℂ n sin | (((cont_diff_neg.mul cont_diff_const).cexp.sub
(cont_diff_id.mul cont_diff_const).cexp).mul cont_diff_const).div_const _ | lemma | complex.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",
"cont_diff_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_sin : differentiable ℂ sin | λx, (has_deriv_at_sin x).differentiable_at | lemma | complex.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 {x : ℂ} : differentiable_at ℂ sin x | differentiable_sin x | lemma | complex.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 | |
deriv_sin : deriv sin = cos | funext $ λ x, (has_deriv_at_sin x).deriv | lemma | complex.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",
"deriv_sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_cos (x : ℂ) : has_strict_deriv_at cos (-sin x) x | begin
simp only [sin, div_eq_mul_inv, neg_mul_eq_neg_mul],
convert (((has_strict_deriv_at_id x).mul_const I).cexp.add
((has_strict_deriv_at_id x).neg.mul_const I).cexp).mul_const (2:ℂ)⁻¹,
simp only [function.comp, id],
ring
end | lemma | complex.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"
] | [
"div_eq_mul_inv",
"has_strict_deriv_at",
"has_strict_deriv_at_id",
"neg_mul_eq_neg_mul",
"ring"
] | The complex cosine function is everywhere strictly differentiable, with the derivative
`-sin x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_cos (x : ℂ) : has_deriv_at cos (-sin x) x | (has_strict_deriv_at_cos x).has_deriv_at | lemma | complex.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"
] | The complex cosine function is everywhere differentiable, with the derivative `-sin x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_cos {n} : cont_diff ℂ n cos | ((cont_diff_id.mul cont_diff_const).cexp.add
(cont_diff_neg.mul cont_diff_const).cexp).div_const _ | lemma | complex.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",
"cont_diff_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_cos : differentiable ℂ cos | λx, (has_deriv_at_cos x).differentiable_at | lemma | complex.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 {x : ℂ} : differentiable_at ℂ cos x | differentiable_cos x | lemma | complex.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 {x : ℂ} : deriv cos x = -sin x | (has_deriv_at_cos x).deriv | lemma | complex.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 x, deriv_cos | lemma | complex.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 : ℂ) : has_strict_deriv_at sinh (cosh x) x | begin
simp only [cosh, div_eq_mul_inv],
convert ((has_strict_deriv_at_exp x).sub (has_strict_deriv_at_id x).neg.cexp).mul_const (2:ℂ)⁻¹,
rw [id, mul_neg_one, sub_eq_add_neg, neg_neg]
end | lemma | complex.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"
] | [
"div_eq_mul_inv",
"has_strict_deriv_at",
"has_strict_deriv_at_exp",
"has_strict_deriv_at_id",
"mul_neg_one"
] | The complex hyperbolic sine function is everywhere strictly differentiable, with the derivative
`cosh x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_sinh (x : ℂ) : has_deriv_at sinh (cosh x) x | (has_strict_deriv_at_sinh x).has_deriv_at | lemma | complex.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"
] | [
"has_deriv_at"
] | The complex hyperbolic sine function is everywhere differentiable, with the derivative
`cosh x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_sinh {n} : cont_diff ℂ n sinh | (cont_diff_exp.sub cont_diff_neg.cexp).div_const _ | lemma | complex.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 ℂ sinh | λx, (has_deriv_at_sinh x).differentiable_at | lemma | complex.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 {x : ℂ} : differentiable_at ℂ sinh x | differentiable_sinh x | lemma | complex.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 | |
deriv_sinh : deriv sinh = cosh | funext $ λ x, (has_deriv_at_sinh x).deriv | lemma | complex.deriv_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"
] | [
"deriv",
"deriv_sinh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_cosh (x : ℂ) : has_strict_deriv_at cosh (sinh x) x | begin
simp only [sinh, div_eq_mul_inv],
convert ((has_strict_deriv_at_exp x).add (has_strict_deriv_at_id x).neg.cexp).mul_const (2:ℂ)⁻¹,
rw [id, mul_neg_one, sub_eq_add_neg]
end | lemma | complex.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"
] | [
"div_eq_mul_inv",
"has_strict_deriv_at",
"has_strict_deriv_at_exp",
"has_strict_deriv_at_id",
"mul_neg_one"
] | The complex hyperbolic cosine function is everywhere strictly differentiable, with the
derivative `sinh x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_deriv_at_cosh (x : ℂ) : has_deriv_at cosh (sinh x) x | (has_strict_deriv_at_cosh x).has_deriv_at | lemma | complex.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"
] | The complex hyperbolic cosine function is everywhere differentiable, with the derivative
`sinh x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cont_diff_cosh {n} : cont_diff ℂ n cosh | (cont_diff_exp.add cont_diff_neg.cexp).div_const _ | lemma | complex.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 ℂ cosh | λx, (has_deriv_at_cosh x).differentiable_at | lemma | complex.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 {x : ℂ} : differentiable_at ℂ cosh x | differentiable_cosh x | lemma | complex.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 | |
deriv_cosh : deriv cosh = sinh | funext $ λ x, (has_deriv_at_cosh x).deriv | lemma | complex.deriv_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",
"deriv_cosh"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.ccos (hf : has_strict_deriv_at f f' x) :
has_strict_deriv_at (λ x, complex.cos (f x)) (- complex.sin (f x) * f') x | (complex.has_strict_deriv_at_cos (f x)).comp x hf | lemma | has_strict_deriv_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",
"complex.has_strict_deriv_at_cos",
"complex.sin",
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.ccos (hf : has_deriv_at f f' x) :
has_deriv_at (λ x, complex.cos (f x)) (- complex.sin (f x) * f') x | (complex.has_deriv_at_cos (f x)).comp x hf | lemma | has_deriv_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",
"complex.has_deriv_at_cos",
"complex.sin",
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.ccos (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ x, complex.cos (f x)) (- complex.sin (f x) * f') s x | (complex.has_deriv_at_cos (f x)).comp_has_deriv_within_at x hf | lemma | has_deriv_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",
"complex.has_deriv_at_cos",
"complex.sin",
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_ccos (hf : differentiable_within_at ℂ f s x)
(hxs : unique_diff_within_at ℂ s x) :
deriv_within (λx, complex.cos (f x)) s x = - complex.sin (f x) * (deriv_within f s x) | hf.has_deriv_within_at.ccos.deriv_within hxs | lemma | deriv_within_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",
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_ccos (hc : differentiable_at ℂ f x) :
deriv (λx, complex.cos (f x)) x = - complex.sin (f x) * (deriv f x) | hc.has_deriv_at.ccos.deriv | lemma | deriv_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",
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.csin (hf : has_strict_deriv_at f f' x) :
has_strict_deriv_at (λ x, complex.sin (f x)) (complex.cos (f x) * f') x | (complex.has_strict_deriv_at_sin (f x)).comp x hf | lemma | has_strict_deriv_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_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.csin (hf : has_deriv_at f f' x) :
has_deriv_at (λ x, complex.sin (f x)) (complex.cos (f x) * f') x | (complex.has_deriv_at_sin (f x)).comp x hf | lemma | has_deriv_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_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.csin (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ x, complex.sin (f x)) (complex.cos (f x) * f') s x | (complex.has_deriv_at_sin (f x)).comp_has_deriv_within_at x hf | lemma | has_deriv_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_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_csin (hf : differentiable_within_at ℂ f s x)
(hxs : unique_diff_within_at ℂ s x) :
deriv_within (λx, complex.sin (f x)) s x = complex.cos (f x) * (deriv_within f s x) | hf.has_deriv_within_at.csin.deriv_within hxs | lemma | deriv_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",
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_csin (hc : differentiable_at ℂ f x) :
deriv (λx, complex.sin (f x)) x = complex.cos (f x) * (deriv f x) | hc.has_deriv_at.csin.deriv | lemma | deriv_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",
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.ccosh (hf : has_strict_deriv_at f f' x) :
has_strict_deriv_at (λ x, complex.cosh (f x)) (complex.sinh (f x) * f') x | (complex.has_strict_deriv_at_cosh (f x)).comp x hf | lemma | has_strict_deriv_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_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.ccosh (hf : has_deriv_at f f' x) :
has_deriv_at (λ x, complex.cosh (f x)) (complex.sinh (f x) * f') x | (complex.has_deriv_at_cosh (f x)).comp x hf | lemma | has_deriv_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_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.ccosh (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ x, complex.cosh (f x)) (complex.sinh (f x) * f') s x | (complex.has_deriv_at_cosh (f x)).comp_has_deriv_within_at x hf | lemma | has_deriv_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_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_ccosh (hf : differentiable_within_at ℂ f s x)
(hxs : unique_diff_within_at ℂ s x) :
deriv_within (λx, complex.cosh (f x)) s x = complex.sinh (f x) * (deriv_within f s x) | hf.has_deriv_within_at.ccosh.deriv_within hxs | lemma | deriv_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",
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_ccosh (hc : differentiable_at ℂ f x) :
deriv (λx, complex.cosh (f x)) x = complex.sinh (f x) * (deriv f x) | hc.has_deriv_at.ccosh.deriv | lemma | deriv_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",
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.csinh (hf : has_strict_deriv_at f f' x) :
has_strict_deriv_at (λ x, complex.sinh (f x)) (complex.cosh (f x) * f') x | (complex.has_strict_deriv_at_sinh (f x)).comp x hf | lemma | has_strict_deriv_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_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.csinh (hf : has_deriv_at f f' x) :
has_deriv_at (λ x, complex.sinh (f x)) (complex.cosh (f x) * f') x | (complex.has_deriv_at_sinh (f x)).comp x hf | lemma | has_deriv_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_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.csinh (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ x, complex.sinh (f x)) (complex.cosh (f x) * f') s x | (complex.has_deriv_at_sinh (f x)).comp_has_deriv_within_at x hf | lemma | has_deriv_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_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_csinh (hf : differentiable_within_at ℂ f s x)
(hxs : unique_diff_within_at ℂ s x) :
deriv_within (λx, complex.sinh (f x)) s x = complex.cosh (f x) * (deriv_within f s x) | hf.has_deriv_within_at.csinh.deriv_within hxs | lemma | deriv_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",
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_csinh (hc : differentiable_at ℂ f x) :
deriv (λx, complex.sinh (f x)) x = complex.cosh (f x) * (deriv f x) | hc.has_deriv_at.csinh.deriv | lemma | deriv_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",
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.ccos (hf : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ x, complex.cos (f x)) (- complex.sin (f x) • f') x | (complex.has_strict_deriv_at_cos (f x)).comp_has_strict_fderiv_at x hf | lemma | has_strict_fderiv_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",
"complex.has_strict_deriv_at_cos",
"complex.sin",
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.ccos (hf : has_fderiv_at f f' x) :
has_fderiv_at (λ x, complex.cos (f x)) (- complex.sin (f x) • f') x | (complex.has_deriv_at_cos (f x)).comp_has_fderiv_at x hf | lemma | has_fderiv_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",
"complex.has_deriv_at_cos",
"complex.sin",
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.ccos (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ x, complex.cos (f x)) (- complex.sin (f x) • f') s x | (complex.has_deriv_at_cos (f x)).comp_has_fderiv_within_at x hf | lemma | has_fderiv_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",
"complex.has_deriv_at_cos",
"complex.sin",
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.ccos (hf : differentiable_within_at ℂ f s x) :
differentiable_within_at ℂ (λ x, complex.cos (f x)) s x | hf.has_fderiv_within_at.ccos.differentiable_within_at | lemma | differentiable_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",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.ccos (hc : differentiable_at ℂ f x) :
differentiable_at ℂ (λx, complex.cos (f x)) x | hc.has_fderiv_at.ccos.differentiable_at | lemma | differentiable_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",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.ccos (hc : differentiable_on ℂ f s) :
differentiable_on ℂ (λx, complex.cos (f x)) s | λx h, (hc x h).ccos | lemma | differentiable_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",
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.ccos (hc : differentiable ℂ f) :
differentiable ℂ (λx, complex.cos (f x)) | λx, (hc x).ccos | lemma | differentiable.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",
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_ccos (hf : differentiable_within_at ℂ f s x)
(hxs : unique_diff_within_at ℂ s x) :
fderiv_within ℂ (λx, complex.cos (f x)) s x = - complex.sin (f x) • (fderiv_within ℂ f s x) | hf.has_fderiv_within_at.ccos.fderiv_within hxs | lemma | fderiv_within_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_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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