statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
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cos_pi_div_three : cos (π / 3) = 1 / 2 | begin
have h₁ : (2 * cos (π / 3) - 1) ^ 2 * (2 * cos (π / 3) + 2) = 0,
{ have : cos (3 * (π / 3)) = cos π := by { congr' 1, ring },
linarith [cos_pi, cos_three_mul (π / 3)] },
cases mul_eq_zero.mp h₁ with h h,
{ linarith [pow_eq_zero h] },
{ have : cos π < cos (π / 3),
{ refine cos_lt_cos_of_nonneg_of... | lemma | real.cos_pi_div_three | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"pow_eq_zero",
"ring"
] | The cosine of `π / 3` is `1 / 2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sq_cos_pi_div_six : cos (π / 6) ^ 2 = 3 / 4 | begin
have h1 : cos (π / 6) ^ 2 = 1 / 2 + 1 / 2 / 2,
{ convert cos_sq (π / 6),
have h2 : 2 * (π / 6) = π / 3 := by cancel_denoms,
rw [h2, cos_pi_div_three] },
rw ← sub_eq_zero at h1 ⊢,
convert h1 using 1,
ring
end | lemma | real.sq_cos_pi_div_six | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"ring"
] | The square of the cosine of `π / 6` is `3 / 4` (this is sometimes more convenient than the
result for cosine itself). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cos_pi_div_six : cos (π / 6) = (sqrt 3) / 2 | begin
suffices : sqrt 3 = cos (π / 6) * 2,
{ field_simp [(by norm_num : 0 ≠ 2)], exact this.symm },
rw sqrt_eq_iff_sq_eq,
{ have h1 := (mul_right_inj' (by norm_num : (4:ℝ) ≠ 0)).mpr sq_cos_pi_div_six,
rw ← sub_eq_zero at h1 ⊢,
convert h1 using 1,
ring },
{ norm_num },
{ have : 0 < cos (π / 6) :=... | lemma | real.cos_pi_div_six | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"mul_right_inj'",
"ring"
] | The cosine of `π / 6` is `√3 / 2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sin_pi_div_six : sin (π / 6) = 1 / 2 | begin
rw [← cos_pi_div_two_sub, ← cos_pi_div_three],
congr,
ring
end | lemma | real.sin_pi_div_six | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"ring"
] | The sine of `π / 6` is `1 / 2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sq_sin_pi_div_three : sin (π / 3) ^ 2 = 3 / 4 | begin
rw [← cos_pi_div_two_sub, ← sq_cos_pi_div_six],
congr,
ring
end | lemma | real.sq_sin_pi_div_three | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"ring"
] | The square of the sine of `π / 3` is `3 / 4` (this is sometimes more convenient than the
result for cosine itself). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sin_pi_div_three : sin (π / 3) = (sqrt 3) / 2 | begin
rw [← cos_pi_div_two_sub, ← cos_pi_div_six],
congr,
ring
end | lemma | real.sin_pi_div_three | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"ring"
] | The sine of `π / 3` is `√3 / 2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sin_order_iso : Icc (-(π / 2)) (π / 2) ≃o Icc (-1:ℝ) 1 | (strict_mono_on_sin.order_iso _ _).trans $ order_iso.set_congr _ _ bij_on_sin.image_eq | def | real.sin_order_iso | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"order_iso.set_congr"
] | `real.sin` as an `order_iso` between `[-(π / 2), π / 2]` and `[-1, 1]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_sin_order_iso_apply (x : Icc (-(π / 2)) (π / 2)) :
(sin_order_iso x : ℝ) = sin x | rfl | lemma | real.coe_sin_order_iso_apply | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_order_iso_apply (x : Icc (-(π / 2)) (π / 2)) :
sin_order_iso x = ⟨sin x, sin_mem_Icc x⟩ | rfl | lemma | real.sin_order_iso_apply | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_pi_div_four : tan (π / 4) = 1 | begin
rw [tan_eq_sin_div_cos, cos_pi_div_four, sin_pi_div_four],
have h : (sqrt 2) / 2 > 0 := by cancel_denoms,
exact div_self (ne_of_gt h),
end | lemma | real.tan_pi_div_four | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"div_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_pi_div_two : tan (π / 2) = 0 | by simp [tan_eq_sin_div_cos] | lemma | real.tan_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_pos_of_pos_of_lt_pi_div_two {x : ℝ} (h0x : 0 < x) (hxp : x < π / 2) : 0 < tan x | by rw tan_eq_sin_div_cos; exact div_pos (sin_pos_of_pos_of_lt_pi h0x (by linarith))
(cos_pos_of_mem_Ioo ⟨by linarith, hxp⟩) | lemma | real.tan_pos_of_pos_of_lt_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"div_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_nonneg_of_nonneg_of_le_pi_div_two {x : ℝ} (h0x : 0 ≤ x) (hxp : x ≤ π / 2) : 0 ≤ tan x | match lt_or_eq_of_le h0x, lt_or_eq_of_le hxp with
| or.inl hx0, or.inl hxp := le_of_lt (tan_pos_of_pos_of_lt_pi_div_two hx0 hxp)
| or.inl hx0, or.inr hxp := by simp [hxp, tan_eq_sin_div_cos]
| or.inr hx0, _ := by simp [hx0.symm]
end | lemma | real.tan_nonneg_of_nonneg_of_le_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_neg_of_neg_of_pi_div_two_lt {x : ℝ} (hx0 : x < 0) (hpx : -(π / 2) < x) : tan x < 0 | neg_pos.1 (tan_neg x ▸ tan_pos_of_pos_of_lt_pi_div_two (by linarith) (by linarith [pi_pos])) | lemma | real.tan_neg_of_neg_of_pi_div_two_lt | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_nonpos_of_nonpos_of_neg_pi_div_two_le {x : ℝ} (hx0 : x ≤ 0) (hpx : -(π / 2) ≤ x) :
tan x ≤ 0 | neg_nonneg.1 (tan_neg x ▸ tan_nonneg_of_nonneg_of_le_pi_div_two (by linarith) (by linarith)) | lemma | real.tan_nonpos_of_nonpos_of_neg_pi_div_two_le | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_lt_tan_of_nonneg_of_lt_pi_div_two {x y : ℝ}
(hx₁ : 0 ≤ x) (hy₂ : y < π / 2) (hxy : x < y) :
tan x < tan y | begin
rw [tan_eq_sin_div_cos, tan_eq_sin_div_cos],
exact div_lt_div
(sin_lt_sin_of_lt_of_le_pi_div_two (by linarith) (le_of_lt hy₂) hxy)
(cos_le_cos_of_nonneg_of_le_pi hx₁ (by linarith) (le_of_lt hxy))
(sin_nonneg_of_nonneg_of_le_pi (by linarith) (by linarith))
(cos_pos_of_mem_Ioo ⟨by linarith, hy₂⟩... | lemma | real.tan_lt_tan_of_nonneg_of_lt_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"div_lt_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_lt_tan_of_lt_of_lt_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) < x)
(hy₂ : y < π / 2) (hxy : x < y) : tan x < tan y | match le_total x 0, le_total y 0 with
| or.inl hx0, or.inl hy0 := neg_lt_neg_iff.1 $ by rw [← tan_neg, ← tan_neg]; exact
tan_lt_tan_of_nonneg_of_lt_pi_div_two (neg_nonneg.2 hy0)
(neg_lt.2 hx₁) (neg_lt_neg hxy)
| or.inl hx0, or.inr hy0 := (lt_or_eq_of_le hy0).elim
(λ hy0, calc tan x ≤ 0 : tan_nonpos_of_nonpos_of... | lemma | real.tan_lt_tan_of_lt_of_lt_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on_tan : strict_mono_on tan (Ioo (-(π / 2)) (π / 2)) | λ x hx y hy, tan_lt_tan_of_lt_of_lt_pi_div_two hx.1 hy.2 | lemma | real.strict_mono_on_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inj_on_tan : inj_on tan (Ioo (-(π / 2)) (π / 2)) | strict_mono_on_tan.inj_on | lemma | real.inj_on_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_inj_of_lt_of_lt_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) < x) (hx₂ : x < π / 2)
(hy₁ : -(π / 2) < y) (hy₂ : y < π / 2) (hxy : tan x = tan y) : x = y | inj_on_tan ⟨hx₁, hx₂⟩ ⟨hy₁, hy₂⟩ hxy | lemma | real.tan_inj_of_lt_of_lt_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_periodic : function.periodic tan π | by simpa only [function.periodic, tan_eq_sin_div_cos] using sin_antiperiodic.div cos_antiperiodic | lemma | real.tan_periodic | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"function.periodic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_add_pi (x : ℝ) : tan (x + π) = tan x | tan_periodic x | lemma | real.tan_add_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_sub_pi (x : ℝ) : tan (x - π) = tan x | tan_periodic.sub_eq x | lemma | real.tan_sub_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_pi_sub (x : ℝ) : tan (π - x) = -tan x | tan_neg x ▸ tan_periodic.sub_eq' | lemma | real.tan_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_pi_div_two_sub (x : ℝ) : tan (π / 2 - x) = (tan x)⁻¹ | by rw [tan_eq_sin_div_cos, tan_eq_sin_div_cos, inv_div, sin_pi_div_two_sub, cos_pi_div_two_sub] | lemma | real.tan_pi_div_two_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"inv_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_nat_mul_pi (n : ℕ) : tan (n * π) = 0 | tan_zero ▸ tan_periodic.nat_mul_eq n | lemma | real.tan_nat_mul_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_int_mul_pi (n : ℤ) : tan (n * π) = 0 | tan_zero ▸ tan_periodic.int_mul_eq n | lemma | real.tan_int_mul_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_add_nat_mul_pi (x : ℝ) (n : ℕ) : tan (x + n * π) = tan x | tan_periodic.nat_mul n x | lemma | real.tan_add_nat_mul_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_add_int_mul_pi (x : ℝ) (n : ℤ) : tan (x + n * π) = tan x | tan_periodic.int_mul n x | lemma | real.tan_add_int_mul_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_sub_nat_mul_pi (x : ℝ) (n : ℕ) : tan (x - n * π) = tan x | tan_periodic.sub_nat_mul_eq n | lemma | real.tan_sub_nat_mul_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_sub_int_mul_pi (x : ℝ) (n : ℤ) : tan (x - n * π) = tan x | tan_periodic.sub_int_mul_eq n | lemma | real.tan_sub_int_mul_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_nat_mul_pi_sub (x : ℝ) (n : ℕ) : tan (n * π - x) = -tan x | tan_neg x ▸ tan_periodic.nat_mul_sub_eq n | lemma | real.tan_nat_mul_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_int_mul_pi_sub (x : ℝ) (n : ℤ) : tan (n * π - x) = -tan x | tan_neg x ▸ tan_periodic.int_mul_sub_eq n | lemma | real.tan_int_mul_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_sin_pi_div_two : tendsto sin (𝓝[<] (π/2)) (𝓝 1) | by { convert continuous_sin.continuous_within_at, simp } | lemma | real.tendsto_sin_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_cos_pi_div_two : tendsto cos (𝓝[<] (π/2)) (𝓝[>] 0) | begin
apply tendsto_nhds_within_of_tendsto_nhds_of_eventually_within,
{ convert continuous_cos.continuous_within_at, simp },
{ filter_upwards [Ioo_mem_nhds_within_Iio (right_mem_Ioc.mpr (neg_lt_self pi_div_two_pos))]
with x hx using cos_pos_of_mem_Ioo hx },
end | lemma | real.tendsto_cos_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"Ioo_mem_nhds_within_Iio",
"tendsto_nhds_within_of_tendsto_nhds_of_eventually_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_tan_pi_div_two : tendsto tan (𝓝[<] (π/2)) at_top | begin
convert tendsto_cos_pi_div_two.inv_tendsto_zero.at_top_mul zero_lt_one
tendsto_sin_pi_div_two,
simp only [pi.inv_apply, ← div_eq_inv_mul, ← tan_eq_sin_div_cos]
end | lemma | real.tendsto_tan_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"div_eq_inv_mul",
"pi.inv_apply",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_sin_neg_pi_div_two : tendsto sin (𝓝[>] (-(π/2))) (𝓝 (-1)) | by { convert continuous_sin.continuous_within_at, simp } | lemma | real.tendsto_sin_neg_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_cos_neg_pi_div_two : tendsto cos (𝓝[>] (-(π/2))) (𝓝[>] 0) | begin
apply tendsto_nhds_within_of_tendsto_nhds_of_eventually_within,
{ convert continuous_cos.continuous_within_at, simp },
{ filter_upwards [Ioo_mem_nhds_within_Ioi (left_mem_Ico.mpr (neg_lt_self pi_div_two_pos))]
with x hx using cos_pos_of_mem_Ioo hx },
end | lemma | real.tendsto_cos_neg_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"Ioo_mem_nhds_within_Ioi",
"tendsto_nhds_within_of_tendsto_nhds_of_eventually_within"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_tan_neg_pi_div_two : tendsto tan (𝓝[>] (-(π/2))) at_bot | begin
convert tendsto_cos_neg_pi_div_two.inv_tendsto_zero.at_top_mul_neg (by norm_num)
tendsto_sin_neg_pi_div_two,
simp only [pi.inv_apply, ← div_eq_inv_mul, ← tan_eq_sin_div_cos]
end | lemma | real.tendsto_tan_neg_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"div_eq_inv_mul",
"pi.inv_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_eq_zero_iff_cos_eq {z : ℂ} : sin z = 0 ↔ cos z = 1 ∨ cos z = -1 | by rw [← mul_self_eq_one_iff, ← sin_sq_add_cos_sq, sq, sq, ← sub_eq_iff_eq_add, sub_self];
exact ⟨λ h, by rw [h, mul_zero], eq_zero_of_mul_self_eq_zero ∘ eq.symm⟩ | lemma | complex.sin_eq_zero_iff_cos_eq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"eq_zero_of_mul_self_eq_zero",
"mul_self_eq_one_iff",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_pi_div_two : cos (π / 2) = 0 | calc cos (π / 2) = real.cos (π / 2) : by rw [of_real_cos]; simp
... = 0 : by simp | lemma | complex.cos_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"real.cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_pi_div_two : sin (π / 2) = 1 | calc sin (π / 2) = real.sin (π / 2) : by rw [of_real_sin]; simp
... = 1 : by simp | lemma | complex.sin_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_pi : sin π = 0 | by rw [← of_real_sin, real.sin_pi]; simp | lemma | complex.sin_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"real.sin_pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_pi : cos π = -1 | by rw [← of_real_cos, real.cos_pi]; simp | lemma | complex.cos_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"real.cos_pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_add_pi (x : ℂ) : sin (x + π) = -sin x | sin_antiperiodic x | lemma | complex.sin_add_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_add_two_pi (x : ℂ) : sin (x + 2 * π) = sin x | sin_periodic x | lemma | complex.sin_add_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_sub_pi (x : ℂ) : sin (x - π) = -sin x | sin_antiperiodic.sub_eq x | lemma | complex.sin_sub_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_sub_two_pi (x : ℂ) : sin (x - 2 * π) = sin x | sin_periodic.sub_eq x | lemma | complex.sin_sub_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_pi_sub (x : ℂ) : sin (π - x) = sin x | neg_neg (sin x) ▸ sin_neg x ▸ sin_antiperiodic.sub_eq' | lemma | complex.sin_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_two_pi_sub (x : ℂ) : sin (2 * π - x) = -sin x | sin_neg x ▸ sin_periodic.sub_eq' | lemma | complex.sin_two_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_add_nat_mul_two_pi (x : ℂ) (n : ℕ) : sin (x + n * (2 * π)) = sin x | sin_periodic.nat_mul n x | lemma | complex.sin_add_nat_mul_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_add_int_mul_two_pi (x : ℂ) (n : ℤ) : sin (x + n * (2 * π)) = sin x | sin_periodic.int_mul n x | lemma | complex.sin_add_int_mul_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_sub_nat_mul_two_pi (x : ℂ) (n : ℕ) : sin (x - n * (2 * π)) = sin x | sin_periodic.sub_nat_mul_eq n | lemma | complex.sin_sub_nat_mul_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_sub_int_mul_two_pi (x : ℂ) (n : ℤ) : sin (x - n * (2 * π)) = sin x | sin_periodic.sub_int_mul_eq n | lemma | complex.sin_sub_int_mul_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_nat_mul_two_pi_sub (x : ℂ) (n : ℕ) : sin (n * (2 * π) - x) = -sin x | sin_neg x ▸ sin_periodic.nat_mul_sub_eq n | lemma | complex.sin_nat_mul_two_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_int_mul_two_pi_sub (x : ℂ) (n : ℤ) : sin (n * (2 * π) - x) = -sin x | sin_neg x ▸ sin_periodic.int_mul_sub_eq n | lemma | complex.sin_int_mul_two_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_add_pi (x : ℂ) : cos (x + π) = -cos x | cos_antiperiodic x | lemma | complex.cos_add_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_add_two_pi (x : ℂ) : cos (x + 2 * π) = cos x | cos_periodic x | lemma | complex.cos_add_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_sub_pi (x : ℂ) : cos (x - π) = -cos x | cos_antiperiodic.sub_eq x | lemma | complex.cos_sub_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_sub_two_pi (x : ℂ) : cos (x - 2 * π) = cos x | cos_periodic.sub_eq x | lemma | complex.cos_sub_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_pi_sub (x : ℂ) : cos (π - x) = -cos x | cos_neg x ▸ cos_antiperiodic.sub_eq' | lemma | complex.cos_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_two_pi_sub (x : ℂ) : cos (2 * π - x) = cos x | cos_neg x ▸ cos_periodic.sub_eq' | lemma | complex.cos_two_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_add_nat_mul_two_pi (x : ℂ) (n : ℕ) : cos (x + n * (2 * π)) = cos x | cos_periodic.nat_mul n x | lemma | complex.cos_add_nat_mul_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_add_int_mul_two_pi (x : ℂ) (n : ℤ) : cos (x + n * (2 * π)) = cos x | cos_periodic.int_mul n x | lemma | complex.cos_add_int_mul_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_sub_nat_mul_two_pi (x : ℂ) (n : ℕ) : cos (x - n * (2 * π)) = cos x | cos_periodic.sub_nat_mul_eq n | lemma | complex.cos_sub_nat_mul_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_sub_int_mul_two_pi (x : ℂ) (n : ℤ) : cos (x - n * (2 * π)) = cos x | cos_periodic.sub_int_mul_eq n | lemma | complex.cos_sub_int_mul_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_nat_mul_two_pi_sub (x : ℂ) (n : ℕ) : cos (n * (2 * π) - x) = cos x | cos_neg x ▸ cos_periodic.nat_mul_sub_eq n | lemma | complex.cos_nat_mul_two_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_int_mul_two_pi_sub (x : ℂ) (n : ℤ) : cos (n * (2 * π) - x) = cos x | cos_neg x ▸ cos_periodic.int_mul_sub_eq n | lemma | complex.cos_int_mul_two_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_add_pi_div_two (x : ℂ) : sin (x + π / 2) = cos x | by simp [sin_add] | lemma | complex.sin_add_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_sub_pi_div_two (x : ℂ) : sin (x - π / 2) = -cos x | by simp [sub_eq_add_neg, sin_add] | lemma | complex.sin_sub_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_pi_div_two_sub (x : ℂ) : sin (π / 2 - x) = cos x | by simp [sub_eq_add_neg, sin_add] | lemma | complex.sin_pi_div_two_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_add_pi_div_two (x : ℂ) : cos (x + π / 2) = -sin x | by simp [cos_add] | lemma | complex.cos_add_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_sub_pi_div_two (x : ℂ) : cos (x - π / 2) = sin x | by simp [sub_eq_add_neg, cos_add] | lemma | complex.cos_sub_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_pi_div_two_sub (x : ℂ) : cos (π / 2 - x) = sin x | by rw [← cos_neg, neg_sub, cos_sub_pi_div_two] | lemma | complex.cos_pi_div_two_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_periodic : function.periodic tan π | by simpa only [tan_eq_sin_div_cos] using sin_antiperiodic.div cos_antiperiodic | lemma | complex.tan_periodic | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"function.periodic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_add_pi (x : ℂ) : tan (x + π) = tan x | tan_periodic x | lemma | complex.tan_add_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_sub_pi (x : ℂ) : tan (x - π) = tan x | tan_periodic.sub_eq x | lemma | complex.tan_sub_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_pi_sub (x : ℂ) : tan (π - x) = -tan x | tan_neg x ▸ tan_periodic.sub_eq' | lemma | complex.tan_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_pi_div_two_sub (x : ℂ) : tan (π / 2 - x) = (tan x)⁻¹ | by rw [tan_eq_sin_div_cos, tan_eq_sin_div_cos, inv_div, sin_pi_div_two_sub, cos_pi_div_two_sub] | lemma | complex.tan_pi_div_two_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"inv_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_add_nat_mul_pi (x : ℂ) (n : ℕ) : tan (x + n * π) = tan x | tan_periodic.nat_mul n x | lemma | complex.tan_add_nat_mul_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_add_int_mul_pi (x : ℂ) (n : ℤ) : tan (x + n * π) = tan x | tan_periodic.int_mul n x | lemma | complex.tan_add_int_mul_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_sub_nat_mul_pi (x : ℂ) (n : ℕ) : tan (x - n * π) = tan x | tan_periodic.sub_nat_mul_eq n | lemma | complex.tan_sub_nat_mul_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_sub_int_mul_pi (x : ℂ) (n : ℤ) : tan (x - n * π) = tan x | tan_periodic.sub_int_mul_eq n | lemma | complex.tan_sub_int_mul_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_nat_mul_pi_sub (x : ℂ) (n : ℕ) : tan (n * π - x) = -tan x | tan_neg x ▸ tan_periodic.nat_mul_sub_eq n | lemma | complex.tan_nat_mul_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_int_mul_pi_sub (x : ℂ) (n : ℤ) : tan (n * π - x) = -tan x | tan_neg x ▸ tan_periodic.int_mul_sub_eq n | lemma | complex.tan_int_mul_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_antiperiodic : function.antiperiodic exp (π * I) | by simp [exp_add, exp_mul_I] | lemma | complex.exp_antiperiodic | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"exp",
"exp_add",
"function.antiperiodic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_periodic : function.periodic exp (2 * π * I) | (mul_assoc (2:ℂ) π I).symm ▸ exp_antiperiodic.periodic | lemma | complex.exp_periodic | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"exp",
"function.periodic",
"mul_assoc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_mul_I_antiperiodic : function.antiperiodic (λ x, exp (x * I)) π | by simpa only [mul_inv_cancel_right₀ I_ne_zero] using exp_antiperiodic.mul_const I_ne_zero | lemma | complex.exp_mul_I_antiperiodic | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"exp",
"function.antiperiodic",
"mul_inv_cancel_right₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_mul_I_periodic : function.periodic (λ x, exp (x * I)) (2 * π) | exp_mul_I_antiperiodic.periodic | lemma | complex.exp_mul_I_periodic | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"exp",
"function.periodic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_pi_mul_I : exp (π * I) = -1 | exp_zero ▸ exp_antiperiodic.eq | lemma | complex.exp_pi_mul_I | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"exp",
"exp_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_two_pi_mul_I : exp (2 * π * I) = 1 | exp_periodic.eq.trans exp_zero | lemma | complex.exp_two_pi_mul_I | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"exp",
"exp_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_nat_mul_two_pi_mul_I (n : ℕ) : exp (n * (2 * π * I)) = 1 | (exp_periodic.nat_mul_eq n).trans exp_zero | lemma | complex.exp_nat_mul_two_pi_mul_I | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"exp",
"exp_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_int_mul_two_pi_mul_I (n : ℤ) : exp (n * (2 * π * I)) = 1 | (exp_periodic.int_mul_eq n).trans exp_zero | lemma | complex.exp_int_mul_two_pi_mul_I | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"exp",
"exp_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_add_pi_mul_I (z : ℂ) : exp (z + π * I) = -exp z | exp_antiperiodic z | lemma | complex.exp_add_pi_mul_I | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exp_sub_pi_mul_I (z : ℂ) : exp (z - π * I) = -exp z | exp_antiperiodic.sub_eq z | lemma | complex.exp_sub_pi_mul_I | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_exp_mul_exp_add_exp_neg_le_of_abs_im_le {a b : ℝ} (ha : a ≤ 0)
{z : ℂ} (hz : |z.im| ≤ b) (hb : b ≤ π / 2) :
abs (exp (a * (exp z + exp (-z)))) ≤ real.exp (a * real.cos b * real.exp (|z.re|)) | begin
simp only [abs_exp, real.exp_le_exp, of_real_mul_re, add_re, exp_re, neg_im, real.cos_neg,
← add_mul, mul_assoc, mul_comm (real.cos b), neg_re, ← real.cos_abs z.im],
have : real.exp (|z.re|) ≤ real.exp z.re + real.exp (-z.re),
from apply_abs_le_add_of_nonneg (λ x, (real.exp_pos x).le) z.re,
refine m... | lemma | complex.abs_exp_mul_exp_add_exp_neg_le_of_abs_im_le | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"exp",
"mul_assoc",
"mul_comm",
"mul_le_mul",
"mul_le_mul_of_nonpos_left",
"real.cos",
"real.cos_abs",
"real.cos_le_cos_of_nonneg_of_le_pi",
"real.cos_neg",
"real.cos_nonneg_of_mem_Icc",
"real.exp",
"real.exp_le_exp",
"real.exp_pos"
] | A supporting lemma for the **Phragmen-Lindelöf principle** in a horizontal strip. If `z : ℂ`
belongs to a horizontal strip `|complex.im z| ≤ b`, `b ≤ π / 2`, and `a ≤ 0`, then
$$\left|exp^{a\left(e^{z}+e^{-z}\right)}\right| \le e^{a\cos b \exp^{|re z|}}.$$ | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sin_lt {x : ℝ} (h : 0 < x) : sin x < x | begin
cases lt_or_le 1 x with h' h',
{ exact (sin_le_one x).trans_lt h' },
have hx : |x| = x := abs_of_nonneg h.le,
have := le_of_abs_le (sin_bound $ show |x| ≤ 1, by rwa [hx]),
rw [sub_le_iff_le_add', hx] at this,
apply this.trans_lt,
rw [sub_add, sub_lt_self_iff, sub_pos, div_eq_mul_inv (x ^ 3)],
refi... | lemma | real.sin_lt | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/bounds.lean | [
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"abs_of_nonneg",
"div_eq_mul_inv",
"le_of_abs_le",
"mul_lt_mul'",
"pow_le_pow_of_le_one",
"pow_pos"
] | For 0 < x, we have sin x < x. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sin_gt_sub_cube {x : ℝ} (h : 0 < x) (h' : x ≤ 1) : x - x ^ 3 / 4 < sin x | begin
have hx : |x| = x := abs_of_nonneg h.le,
have := neg_le_of_abs_le (sin_bound $ show |x| ≤ 1, by rwa [hx]),
rw [le_sub_iff_add_le, hx] at this,
refine lt_of_lt_of_le _ this,
have : x ^ 3 / 4 - x ^ 3 / 6 = x ^ 3 * 12⁻¹ := by norm_num [div_eq_mul_inv, ← mul_sub],
rw [add_comm, sub_add, sub_neg_eq_add, su... | lemma | real.sin_gt_sub_cube | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/bounds.lean | [
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"abs_of_nonneg",
"div_eq_mul_inv",
"mul_lt_mul'",
"neg_le_of_abs_le",
"pow_le_pow_of_le_one",
"pow_pos"
] | For 0 < x ≤ 1 we have x - x ^ 3 / 4 < sin x.
This is also true for x > 1, but it's nontrivial for x just above 1. This inequality is not
tight; the tighter inequality is sin x > x - x ^ 3 / 6 for all x > 0, but this inequality has
a simpler proof. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
deriv_tan_sub_id (x : ℝ) (h : cos x ≠ 0) :
deriv (λ y : ℝ, tan y - y) x = 1 / cos x ^ 2 - 1 | has_deriv_at.deriv $ by simpa using (has_deriv_at_tan h).add (has_deriv_at_id x).neg | lemma | real.deriv_tan_sub_id | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/bounds.lean | [
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"deriv",
"has_deriv_at.deriv",
"has_deriv_at_id"
] | The derivative of `tan x - x` is `1/(cos x)^2 - 1` away from the zeroes of cos. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lt_tan {x : ℝ} (h1 : 0 < x) (h2 : x < π / 2) : x < tan x | begin
let U := Ico 0 (π / 2),
have intU : interior U = Ioo 0 (π / 2) := interior_Ico,
have half_pi_pos : 0 < π / 2 := div_pos pi_pos two_pos,
have cos_pos : ∀ {y : ℝ}, y ∈ U → 0 < cos y,
{ intros y hy,
exact cos_pos_of_mem_Ioo (Ico_subset_Ioo_left (neg_lt_zero.mpr half_pi_pos) hy) },
have sin_pos : ... | theorem | real.lt_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/bounds.lean | [
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"continuous_on",
"continuous_on.mono",
"continuous_on_id",
"convex.strict_mono_on_of_deriv_pos",
"convex_Ico",
"deriv",
"div_le_self",
"div_pos",
"gt_iff_lt",
"interior",
"interior_Ico",
"interior_subset",
"inv_one",
"lt_inv",
"mul_self_pos",
"nat.succ_pos'",
"one_div",
"one_le_two... | For all `0 < x < π/2` we have `x < tan x`.
This is proved by checking that the function `tan x - x` vanishes
at zero and has non-negative derivative. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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