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
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
sin_sub_pi_div_two (θ : angle) : sin (θ - ↑(π / 2)) = -cos θ | begin
induction θ using real.angle.induction_on,
exact sin_sub_pi_div_two _
end | lemma | real.angle.sin_sub_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_pi_div_two_sub (θ : angle) : sin (↑(π / 2) - θ) = cos θ | begin
induction θ using real.angle.induction_on,
exact sin_pi_div_two_sub _
end | lemma | real.angle.sin_pi_div_two_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_add_pi_div_two (θ : angle) : cos (θ + ↑(π / 2)) = -sin θ | begin
induction θ using real.angle.induction_on,
exact cos_add_pi_div_two _
end | lemma | real.angle.cos_add_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_sub_pi_div_two (θ : angle) : cos (θ - ↑(π / 2)) = sin θ | begin
induction θ using real.angle.induction_on,
exact cos_sub_pi_div_two _
end | lemma | real.angle.cos_sub_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_pi_div_two_sub (θ : angle) : cos (↑(π / 2) - θ) = sin θ | begin
induction θ using real.angle.induction_on,
exact cos_pi_div_two_sub _
end | lemma | real.angle.cos_pi_div_two_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sin_eq_of_two_nsmul_eq {θ ψ : angle} (h : (2 : ℕ) • θ = (2 : ℕ) • ψ) : | |sin θ| = |sin ψ| :=
begin
rw two_nsmul_eq_iff at h,
rcases h with rfl | rfl,
{ refl },
{ rw [sin_add_pi, abs_neg] }
end | lemma | real.angle.abs_sin_eq_of_two_nsmul_eq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"abs_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_sin_eq_of_two_zsmul_eq {θ ψ : angle} (h : (2 : ℤ) • θ = (2 : ℤ) • ψ) : | |sin θ| = |sin ψ| :=
begin
simp_rw [two_zsmul, ←two_nsmul] at h,
exact abs_sin_eq_of_two_nsmul_eq h
end | lemma | real.angle.abs_sin_eq_of_two_zsmul_eq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_cos_eq_of_two_nsmul_eq {θ ψ : angle} (h : (2 : ℕ) • θ = (2 : ℕ) • ψ) : | |cos θ| = |cos ψ| :=
begin
rw two_nsmul_eq_iff at h,
rcases h with rfl | rfl,
{ refl },
{ rw [cos_add_pi, abs_neg] }
end | lemma | real.angle.abs_cos_eq_of_two_nsmul_eq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"abs_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_cos_eq_of_two_zsmul_eq {θ ψ : angle} (h : (2 : ℤ) • θ = (2 : ℤ) • ψ) : | |cos θ| = |cos ψ| :=
begin
simp_rw [two_zsmul, ←two_nsmul] at h,
exact abs_cos_eq_of_two_nsmul_eq h
end | lemma | real.angle.abs_cos_eq_of_two_zsmul_eq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_Ico_mod (θ ψ : ℝ) : ↑(to_Ico_mod two_pi_pos ψ θ) = (θ : angle) | begin
rw angle_eq_iff_two_pi_dvd_sub,
refine ⟨-to_Ico_div two_pi_pos ψ θ, _⟩,
rw [to_Ico_mod_sub_self, zsmul_eq_mul, mul_comm]
end | lemma | real.angle.coe_to_Ico_mod | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"mul_comm",
"to_Ico_div",
"to_Ico_mod",
"to_Ico_mod_sub_self",
"zsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_Ioc_mod (θ ψ : ℝ) : ↑(to_Ioc_mod two_pi_pos ψ θ) = (θ : angle) | begin
rw angle_eq_iff_two_pi_dvd_sub,
refine ⟨-to_Ioc_div two_pi_pos ψ θ, _⟩,
rw [to_Ioc_mod_sub_self, zsmul_eq_mul, mul_comm]
end | lemma | real.angle.coe_to_Ioc_mod | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"mul_comm",
"to_Ioc_div",
"to_Ioc_mod",
"to_Ioc_mod_sub_self",
"zsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real (θ : angle) : ℝ | (to_Ioc_mod_periodic two_pi_pos (-π)).lift θ | def | real.angle.to_real | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"lift",
"to_Ioc_mod_periodic"
] | Convert a `real.angle` to a real number in the interval `Ioc (-π) π`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_real_coe (θ : ℝ) : (θ : angle).to_real = to_Ioc_mod two_pi_pos (-π) θ | rfl | lemma | real.angle.to_real_coe | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_coe_eq_self_iff {θ : ℝ} : (θ : angle).to_real = θ ↔ -π < θ ∧ θ ≤ π | begin
rw [to_real_coe, to_Ioc_mod_eq_self two_pi_pos],
ring_nf
end | lemma | real.angle.to_real_coe_eq_self_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"to_Ioc_mod_eq_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_coe_eq_self_iff_mem_Ioc {θ : ℝ} : (θ : angle).to_real = θ ↔ θ ∈ set.Ioc (-π) π | by rw [to_real_coe_eq_self_iff, ←set.mem_Ioc] | lemma | real.angle.to_real_coe_eq_self_iff_mem_Ioc | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"set.Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_injective : function.injective to_real | begin
intros θ ψ h,
induction θ using real.angle.induction_on,
induction ψ using real.angle.induction_on,
simpa [to_real_coe, to_Ioc_mod_eq_to_Ioc_mod, zsmul_eq_mul, mul_comm _ (2 * π),
←angle_eq_iff_two_pi_dvd_sub, eq_comm] using h,
end | lemma | real.angle.to_real_injective | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"mul_comm",
"real.angle.induction_on",
"to_Ioc_mod_eq_to_Ioc_mod",
"zsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_inj {θ ψ : angle} : θ.to_real = ψ.to_real ↔ θ = ψ | to_real_injective.eq_iff | lemma | real.angle.to_real_inj | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_to_real (θ : angle): (θ.to_real : angle) = θ | begin
induction θ using real.angle.induction_on,
exact coe_to_Ioc_mod _ _
end | lemma | real.angle.coe_to_real | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_pi_lt_to_real (θ : angle) : -π < θ.to_real | begin
induction θ using real.angle.induction_on,
exact left_lt_to_Ioc_mod _ _ _
end | lemma | real.angle.neg_pi_lt_to_real | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"left_lt_to_Ioc_mod",
"real.angle.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_le_pi (θ : angle) : θ.to_real ≤ π | begin
induction θ using real.angle.induction_on,
convert to_Ioc_mod_le_right two_pi_pos _ _,
ring
end | lemma | real.angle.to_real_le_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle.induction_on",
"ring",
"to_Ioc_mod_le_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_to_real_le_pi (θ : angle) : |θ.to_real| ≤ π | abs_le.2 ⟨(neg_pi_lt_to_real _).le, to_real_le_pi _⟩ | lemma | real.angle.abs_to_real_le_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_mem_Ioc (θ : angle) : θ.to_real ∈ set.Ioc (-π) π | ⟨neg_pi_lt_to_real _, to_real_le_pi _⟩ | lemma | real.angle.to_real_mem_Ioc | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"set.Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_Ioc_mod_to_real (θ : angle): to_Ioc_mod two_pi_pos (-π) θ.to_real = θ.to_real | begin
induction θ using real.angle.induction_on,
rw to_real_coe,
exact to_Ioc_mod_to_Ioc_mod _ _ _ _
end | lemma | real.angle.to_Ioc_mod_to_real | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle.induction_on",
"to_Ioc_mod",
"to_Ioc_mod_to_Ioc_mod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_zero : (0 : angle).to_real = 0 | begin
rw [←coe_zero, to_real_coe_eq_self_iff],
exact ⟨(left.neg_neg_iff.2 real.pi_pos), real.pi_pos.le⟩
end | lemma | real.angle.to_real_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.pi_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_eq_zero_iff {θ : angle} : θ.to_real = 0 ↔ θ = 0 | begin
nth_rewrite 0 ←to_real_zero,
exact to_real_inj
end | lemma | real.angle.to_real_eq_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_pi : (π : angle).to_real = π | begin
rw [to_real_coe_eq_self_iff],
exact ⟨left.neg_lt_self real.pi_pos, le_refl _⟩
end | lemma | real.angle.to_real_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.pi_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_eq_pi_iff {θ : angle} : θ.to_real = π ↔ θ = π | by rw [← to_real_inj, to_real_pi] | lemma | real.angle.to_real_eq_pi_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_ne_zero : (π : angle) ≠ 0 | begin
rw [←to_real_injective.ne_iff, to_real_pi, to_real_zero],
exact pi_ne_zero
end | lemma | real.angle.pi_ne_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_pi_div_two : ((π / 2 : ℝ) : angle).to_real = π / 2 | to_real_coe_eq_self_iff.2 $ by split; linarith [pi_pos] | lemma | real.angle.to_real_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_eq_pi_div_two_iff {θ : angle} : θ.to_real = π / 2 ↔ θ = (π / 2 : ℝ) | by rw [← to_real_inj, to_real_pi_div_two] | lemma | real.angle.to_real_eq_pi_div_two_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_neg_pi_div_two : ((-π / 2 : ℝ) : angle).to_real = -π / 2 | to_real_coe_eq_self_iff.2 $ by split; linarith [pi_pos] | lemma | real.angle.to_real_neg_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_eq_neg_pi_div_two_iff {θ : angle} : θ.to_real = -π / 2 ↔ θ = (-π / 2 : ℝ) | by rw [← to_real_inj, to_real_neg_pi_div_two] | lemma | real.angle.to_real_eq_neg_pi_div_two_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_div_two_ne_zero : ((π / 2 : ℝ) : angle) ≠ 0 | begin
rw [←to_real_injective.ne_iff, to_real_pi_div_two, to_real_zero],
exact div_ne_zero real.pi_ne_zero two_ne_zero
end | lemma | real.angle.pi_div_two_ne_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"div_ne_zero",
"real.pi_ne_zero",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_pi_div_two_ne_zero : ((-π / 2 : ℝ) : angle) ≠ 0 | begin
rw [←to_real_injective.ne_iff, to_real_neg_pi_div_two, to_real_zero],
exact div_ne_zero (neg_ne_zero.2 real.pi_ne_zero) two_ne_zero
end | lemma | real.angle.neg_pi_div_two_ne_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"div_ne_zero",
"real.pi_ne_zero",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_to_real_coe_eq_self_iff {θ : ℝ} : |(θ : angle).to_real| = θ ↔ 0 ≤ θ ∧ θ ≤ π | ⟨λ h, h ▸ ⟨abs_nonneg _, abs_to_real_le_pi _⟩, λ h,
(to_real_coe_eq_self_iff.2 ⟨(left.neg_neg_iff.2 real.pi_pos).trans_le h.1, h.2⟩).symm ▸
abs_eq_self.2 h.1⟩ | lemma | real.angle.abs_to_real_coe_eq_self_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.pi_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_to_real_neg_coe_eq_self_iff {θ : ℝ} : |(-θ : angle).to_real| = θ ↔ 0 ≤ θ ∧ θ ≤ π | begin
refine ⟨λ h, h ▸ ⟨abs_nonneg _, abs_to_real_le_pi _⟩, λ h, _⟩,
by_cases hnegpi : θ = π, { simp [hnegpi, real.pi_pos.le] },
rw [←coe_neg, to_real_coe_eq_self_iff.2 ⟨neg_lt_neg (lt_of_le_of_ne h.2 hnegpi),
(neg_nonpos.2 h.1).trans real.pi_pos.le⟩, abs_neg,
abs_... | lemma | real.angle.abs_to_real_neg_coe_eq_self_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"abs_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_to_real_eq_pi_div_two_iff {θ : angle} : | |θ.to_real| = π / 2 ↔ (θ = (π / 2 : ℝ) ∨ θ = (-π / 2 : ℝ)) :=
by rw [abs_eq (div_nonneg real.pi_pos.le two_pos.le), ←neg_div, to_real_eq_pi_div_two_iff,
to_real_eq_neg_pi_div_two_iff] | lemma | real.angle.abs_to_real_eq_pi_div_two_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"abs_eq",
"div_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_to_real_eq_mul {n : ℕ} (h : n ≠ 0) {θ : angle} :
(n • θ).to_real = n * θ.to_real ↔ θ.to_real ∈ set.Ioc (-π / n) (π / n) | begin
nth_rewrite 0 ←coe_to_real θ,
have h' : 0 < (n : ℝ), { exact_mod_cast nat.pos_of_ne_zero h },
rw [←coe_nsmul, nsmul_eq_mul, to_real_coe_eq_self_iff, set.mem_Ioc, div_lt_iff' h',
le_div_iff' h']
end | lemma | real.angle.nsmul_to_real_eq_mul | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"div_lt_iff'",
"le_div_iff'",
"nsmul_eq_mul",
"set.Ioc",
"set.mem_Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_nsmul_to_real_eq_two_mul {θ : angle} :
((2 : ℕ) • θ).to_real = 2 * θ.to_real ↔ θ.to_real ∈ set.Ioc (-π / 2) (π / 2) | by exact_mod_cast nsmul_to_real_eq_mul two_ne_zero | lemma | real.angle.two_nsmul_to_real_eq_two_mul | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"set.Ioc",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_zsmul_to_real_eq_two_mul {θ : angle} :
((2 : ℤ) • θ).to_real = 2 * θ.to_real ↔ θ.to_real ∈ set.Ioc (-π / 2) (π / 2) | by rw [two_zsmul, ←two_nsmul, two_nsmul_to_real_eq_two_mul] | lemma | real.angle.two_zsmul_to_real_eq_two_mul | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"set.Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_coe_eq_self_sub_two_mul_int_mul_pi_iff {θ : ℝ} {k : ℤ} :
(θ : angle).to_real = θ - 2 * k * π ↔ θ ∈ set.Ioc ((2 * k - 1 : ℝ) * π) ((2 * k + 1) * π) | begin
rw [←sub_zero (θ : angle), ←zsmul_zero k, ←coe_two_pi, ←coe_zsmul, ←coe_sub,
zsmul_eq_mul, ←mul_assoc, mul_comm (k : ℝ), to_real_coe_eq_self_iff, set.mem_Ioc],
exact ⟨λ h, ⟨by linarith, by linarith⟩, λ h, ⟨by linarith, by linarith⟩⟩
end | lemma | real.angle.to_real_coe_eq_self_sub_two_mul_int_mul_pi_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"mul_comm",
"set.Ioc",
"set.mem_Ioc",
"zsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_coe_eq_self_sub_two_pi_iff {θ : ℝ} :
(θ : angle).to_real = θ - 2 * π ↔ θ ∈ set.Ioc π (3 * π) | by { convert @to_real_coe_eq_self_sub_two_mul_int_mul_pi_iff θ 1; norm_num } | lemma | real.angle.to_real_coe_eq_self_sub_two_pi_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"set.Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_coe_eq_self_add_two_pi_iff {θ : ℝ} :
(θ : angle).to_real = θ + 2 * π ↔ θ ∈ set.Ioc (-3 * π) (-π) | by { convert @to_real_coe_eq_self_sub_two_mul_int_mul_pi_iff θ (-1); norm_num } | lemma | real.angle.to_real_coe_eq_self_add_two_pi_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"set.Ioc"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_nsmul_to_real_eq_two_mul_sub_two_pi {θ : angle} :
((2 : ℕ) • θ).to_real = 2 * θ.to_real - 2 * π ↔ π / 2 < θ.to_real | begin
nth_rewrite 0 ←coe_to_real θ,
rw [←coe_nsmul, two_nsmul, ←two_mul, to_real_coe_eq_self_sub_two_pi_iff, set.mem_Ioc],
exact ⟨λ h, by linarith,
λ h, ⟨(div_lt_iff' (zero_lt_two' ℝ)).1 h, by linarith [pi_pos, to_real_le_pi θ]⟩⟩
end | lemma | real.angle.two_nsmul_to_real_eq_two_mul_sub_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"div_lt_iff'",
"set.mem_Ioc",
"zero_lt_two'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_zsmul_to_real_eq_two_mul_sub_two_pi {θ : angle} :
((2 : ℤ) • θ).to_real = 2 * θ.to_real - 2 * π ↔ π / 2 < θ.to_real | by rw [two_zsmul, ←two_nsmul, two_nsmul_to_real_eq_two_mul_sub_two_pi] | lemma | real.angle.two_zsmul_to_real_eq_two_mul_sub_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_nsmul_to_real_eq_two_mul_add_two_pi {θ : angle} :
((2 : ℕ) • θ).to_real = 2 * θ.to_real + 2 * π ↔ θ.to_real ≤ -π / 2 | begin
nth_rewrite 0 ←coe_to_real θ,
rw [←coe_nsmul, two_nsmul, ←two_mul, to_real_coe_eq_self_add_two_pi_iff, set.mem_Ioc],
refine ⟨λ h, by linarith,
λ h, ⟨by linarith [pi_pos, neg_pi_lt_to_real θ], (le_div_iff' (zero_lt_two' ℝ)).1 h⟩⟩
end | lemma | real.angle.two_nsmul_to_real_eq_two_mul_add_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"le_div_iff'",
"set.mem_Ioc",
"zero_lt_two'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_zsmul_to_real_eq_two_mul_add_two_pi {θ : angle} :
((2 : ℤ) • θ).to_real = 2 * θ.to_real + 2 * π ↔ θ.to_real ≤ -π / 2 | by rw [two_zsmul, ←two_nsmul, two_nsmul_to_real_eq_two_mul_add_two_pi] | lemma | real.angle.two_zsmul_to_real_eq_two_mul_add_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_to_real (θ : angle) : real.sin θ.to_real = sin θ | by conv_rhs { rw [← coe_to_real θ, sin_coe] } | lemma | real.angle.sin_to_real | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_to_real (θ : angle) : real.cos θ.to_real = cos θ | by conv_rhs { rw [← coe_to_real θ, cos_coe] } | lemma | real.angle.cos_to_real | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_nonneg_iff_abs_to_real_le_pi_div_two {θ : angle} : 0 ≤ cos θ ↔ |θ.to_real| ≤ π / 2 | begin
nth_rewrite 0 ←coe_to_real θ,
rw [abs_le, cos_coe],
refine ⟨λ h, _, cos_nonneg_of_mem_Icc⟩,
by_contra hn,
rw [not_and_distrib, not_le, not_le] at hn,
refine (not_lt.2 h) _,
rcases hn with hn | hn,
{ rw ←real.cos_neg,
refine cos_neg_of_pi_div_two_lt_of_lt (by linarith) _,
linarith [neg_pi_l... | lemma | real.angle.cos_nonneg_iff_abs_to_real_le_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"abs_le",
"by_contra",
"not_and_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_pos_iff_abs_to_real_lt_pi_div_two {θ : angle} : 0 < cos θ ↔ |θ.to_real| < π / 2 | begin
rw [lt_iff_le_and_ne, lt_iff_le_and_ne, cos_nonneg_iff_abs_to_real_le_pi_div_two,
←and_congr_right],
rintro -,
rw [ne.def, ne.def, not_iff_not, @eq_comm ℝ 0, abs_to_real_eq_pi_div_two_iff, cos_eq_zero_iff]
end | lemma | real.angle.cos_pos_iff_abs_to_real_lt_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"lt_iff_le_and_ne",
"not_iff_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_neg_iff_pi_div_two_lt_abs_to_real {θ : angle} : cos θ < 0 ↔ π / 2 < |θ.to_real| | by rw [←not_le, ←not_le, not_iff_not, cos_nonneg_iff_abs_to_real_le_pi_div_two] | lemma | real.angle.cos_neg_iff_pi_div_two_lt_abs_to_real | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"not_iff_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_cos_eq_abs_sin_of_two_nsmul_add_two_nsmul_eq_pi {θ ψ : angle}
(h : (2 : ℕ) • θ + (2 : ℕ) • ψ = π) : |cos θ| = |sin ψ| | begin
rw [←eq_sub_iff_add_eq, ←two_nsmul_coe_div_two, ←nsmul_sub, two_nsmul_eq_iff] at h,
rcases h with rfl | rfl;
simp [cos_pi_div_two_sub]
end | lemma | real.angle.abs_cos_eq_abs_sin_of_two_nsmul_add_two_nsmul_eq_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_cos_eq_abs_sin_of_two_zsmul_add_two_zsmul_eq_pi {θ ψ : angle}
(h : (2 : ℤ) • θ + (2 : ℤ) • ψ = π) : |cos θ| = |sin ψ| | begin
simp_rw [two_zsmul, ←two_nsmul] at h,
exact abs_cos_eq_abs_sin_of_two_nsmul_add_two_nsmul_eq_pi h
end | lemma | real.angle.abs_cos_eq_abs_sin_of_two_zsmul_add_two_zsmul_eq_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan (θ : angle) : ℝ | sin θ / cos θ | def | real.angle.tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | The tangent of a `real.angle`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tan_eq_sin_div_cos (θ : angle) : tan θ = sin θ / cos θ | rfl | lemma | real.angle.tan_eq_sin_div_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_coe (x : ℝ) : tan (x : angle) = real.tan x | by rw [tan, sin_coe, cos_coe, real.tan_eq_sin_div_cos] | lemma | real.angle.tan_coe | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.tan",
"real.tan_eq_sin_div_cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_zero : tan (0 : angle) = 0 | by rw [←coe_zero, tan_coe, real.tan_zero] | lemma | real.angle.tan_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.tan_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_coe_pi : tan (π : angle) = 0 | by rw [tan_eq_sin_div_cos, sin_coe_pi, zero_div] | lemma | real.angle.tan_coe_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"zero_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_periodic : function.periodic tan (π : angle) | begin
intro θ,
induction θ using real.angle.induction_on,
rw [←coe_add, tan_coe, tan_coe],
exact real.tan_periodic θ
end | lemma | real.angle.tan_periodic | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"function.periodic",
"real.angle.induction_on",
"real.tan_periodic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_add_pi (θ : angle) : tan (θ + π) = tan θ | tan_periodic θ | lemma | real.angle.tan_add_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_sub_pi (θ : angle) : tan (θ - π) = tan θ | tan_periodic.sub_eq θ | lemma | real.angle.tan_sub_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_to_real (θ : angle) : real.tan θ.to_real = tan θ | by conv_rhs { rw [←coe_to_real θ, tan_coe] } | lemma | real.angle.tan_to_real | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.tan"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_eq_of_two_nsmul_eq {θ ψ : angle} (h : (2 : ℕ) • θ = (2 : ℕ) • ψ) : tan θ = tan ψ | begin
rw two_nsmul_eq_iff at h,
rcases h with rfl | rfl,
{ refl },
{ exact tan_add_pi _ }
end | lemma | real.angle.tan_eq_of_two_nsmul_eq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_eq_of_two_zsmul_eq {θ ψ : angle} (h : (2 : ℤ) • θ = (2 : ℤ) • ψ) : tan θ = tan ψ | begin
simp_rw [two_zsmul, ←two_nsmul] at h,
exact tan_eq_of_two_nsmul_eq h
end | lemma | real.angle.tan_eq_of_two_zsmul_eq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_eq_inv_of_two_nsmul_add_two_nsmul_eq_pi {θ ψ : angle}
(h : (2 : ℕ) • θ + (2 : ℕ) • ψ = π) : tan ψ = (tan θ)⁻¹ | begin
induction θ using real.angle.induction_on,
induction ψ using real.angle.induction_on,
rw [←smul_add, ←coe_add, ←coe_nsmul, two_nsmul, ←two_mul, angle_eq_iff_two_pi_dvd_sub] at h,
rcases h with ⟨k, h⟩,
rw [sub_eq_iff_eq_add, ←mul_inv_cancel_left₀ two_ne_zero π, mul_assoc, ←mul_add,
mul_right_inj' (... | lemma | real.angle.tan_eq_inv_of_two_nsmul_add_two_nsmul_eq_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"inv_mul_eq_div",
"mul_assoc",
"mul_comm",
"mul_inv_cancel_left₀",
"mul_right_inj'",
"real.angle.induction_on",
"two_ne_zero",
"two_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_eq_inv_of_two_zsmul_add_two_zsmul_eq_pi {θ ψ : angle}
(h : (2 : ℤ) • θ + (2 : ℤ) • ψ = π) : tan ψ = (tan θ)⁻¹ | begin
simp_rw [two_zsmul, ←two_nsmul] at h,
exact tan_eq_inv_of_two_nsmul_add_two_nsmul_eq_pi h
end | lemma | real.angle.tan_eq_inv_of_two_zsmul_add_two_zsmul_eq_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign (θ : angle) : sign_type | sign (sin θ) | def | real.angle.sign | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"sign",
"sign_type"
] | The sign of a `real.angle` is `0` if the angle is `0` or `π`, `1` if the angle is strictly
between `0` and `π` and `-1` is the angle is strictly between `-π` and `0`. It is defined as the
sign of the sine of the angle. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sign_zero : (0 : angle).sign = 0 | by rw [sign, sin_zero, sign_zero] | lemma | real.angle.sign_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"sign",
"sign_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign_coe_pi : (π : angle).sign = 0 | by rw [sign, sin_coe_pi, _root_.sign_zero] | lemma | real.angle.sign_coe_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"sign"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign_neg (θ : angle) : (-θ).sign = - θ.sign | by simp_rw [sign, sin_neg, left.sign_neg] | lemma | real.angle.sign_neg | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"left.sign_neg",
"sign",
"sign_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign_antiperiodic : function.antiperiodic sign (π : angle) | λ θ, by rw [sign, sign, sin_add_pi, left.sign_neg] | lemma | real.angle.sign_antiperiodic | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"function.antiperiodic",
"left.sign_neg",
"sign"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign_add_pi (θ : angle) : (θ + π).sign = -θ.sign | sign_antiperiodic θ | lemma | real.angle.sign_add_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"sign"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign_pi_add (θ : angle) : ((π : angle) + θ).sign = -θ.sign | by rw [add_comm, sign_add_pi] | lemma | real.angle.sign_pi_add | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"sign"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign_sub_pi (θ : angle) : (θ - π).sign = -θ.sign | sign_antiperiodic.sub_eq θ | lemma | real.angle.sign_sub_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"sign"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign_pi_sub (θ : angle) : ((π : angle) - θ).sign = θ.sign | by simp [sign_antiperiodic.sub_eq'] | lemma | real.angle.sign_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"sign"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign_eq_zero_iff {θ : angle} : θ.sign = 0 ↔ θ = 0 ∨ θ = π | by rw [sign, sign_eq_zero_iff, sin_eq_zero_iff] | lemma | real.angle.sign_eq_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"sign",
"sign_eq_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign_ne_zero_iff {θ : angle} : θ.sign ≠ 0 ↔ θ ≠ 0 ∧ θ ≠ π | by rw [←not_or_distrib, ←sign_eq_zero_iff] | lemma | real.angle.sign_ne_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_neg_iff_sign_neg {θ : angle} : θ.to_real < 0 ↔ θ.sign = -1 | begin
rw [sign, ←sin_to_real, sign_eq_neg_one_iff],
rcases lt_trichotomy θ.to_real 0 with (h|h|h),
{ exact ⟨λ _, real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_to_real θ), λ _, h⟩ },
{ simp [h] },
{ exact ⟨λ hn, false.elim (h.asymm hn),
λ hn, false.elim (hn.not_le (sin_nonneg_of_nonneg_of_le_pi h.le... | lemma | real.angle.to_real_neg_iff_sign_neg | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.sin_neg_of_neg_of_neg_pi_lt",
"sign",
"sign_eq_neg_one_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_real_nonneg_iff_sign_nonneg {θ : angle} : 0 ≤ θ.to_real ↔ 0 ≤ θ.sign | begin
rcases lt_trichotomy θ.to_real 0 with (h|h|h),
{ refine ⟨λ hn, false.elim (h.not_le hn), λ hn, _⟩,
rw [to_real_neg_iff_sign_neg.1 h] at hn,
exact false.elim (hn.not_lt dec_trivial) },
{ simp [h, sign, ←sin_to_real] },
{ refine ⟨λ _, _, λ _, h.le⟩,
rw [sign, ←sin_to_real, sign_nonneg_iff],
... | lemma | real.angle.to_real_nonneg_iff_sign_nonneg | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"sign",
"sign_nonneg_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign_to_real {θ : angle} (h : θ ≠ π) : _root_.sign θ.to_real = θ.sign | begin
rcases lt_trichotomy θ.to_real 0 with (ht|ht|ht),
{ simp [ht, to_real_neg_iff_sign_neg.1 ht] },
{ simp [sign, ht, ←sin_to_real] },
{ rw [sign, ←sin_to_real, sign_pos ht,
sign_pos (sin_pos_of_pos_of_lt_pi ht
((to_real_le_pi θ).lt_of_ne (to_real_eq_pi_iff.not.2 h)))] }
end | lemma | real.angle.sign_to_real | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"sign",
"sign_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_abs_to_real_of_sign_nonneg {θ : angle} (h : 0 ≤ θ.sign) : ↑|θ.to_real| = θ | by rw [abs_eq_self.2 (to_real_nonneg_iff_sign_nonneg.2 h), coe_to_real] | lemma | real.angle.coe_abs_to_real_of_sign_nonneg | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_coe_abs_to_real_of_sign_nonpos {θ : angle} (h : θ.sign ≤ 0) : -↑|θ.to_real| = θ | begin
rw sign_type.nonpos_iff at h,
rcases h with h|h,
{ rw [abs_of_neg (to_real_neg_iff_sign_neg.2 h), coe_neg, neg_neg, coe_to_real] },
{ rw sign_eq_zero_iff at h,
rcases h with rfl|rfl;
simp [abs_of_pos real.pi_pos] }
end | lemma | real.angle.neg_coe_abs_to_real_of_sign_nonpos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"abs_of_neg",
"abs_of_pos",
"real.pi_pos",
"sign_eq_zero_iff",
"sign_type.nonpos_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_iff_sign_eq_and_abs_to_real_eq {θ ψ : angle} :
θ = ψ ↔ θ.sign = ψ.sign ∧ |θ.to_real| = |ψ.to_real| | begin
refine ⟨_, λ h, _⟩, { rintro rfl, exact ⟨rfl, rfl⟩ },
rcases h with ⟨hs, hr⟩,
rw abs_eq_abs at hr,
rcases hr with (hr|hr),
{ exact to_real_injective hr },
{ by_cases h : θ = π,
{ rw [h, to_real_pi, ← neg_eq_iff_eq_neg] at hr,
exact false.elim ((neg_pi_lt_to_real ψ).ne hr) },
{ by_cases h... | lemma | real.angle.eq_iff_sign_eq_and_abs_to_real_eq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"abs_eq_abs",
"left.sign_neg",
"sign_type.neg_eq_self_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_iff_abs_to_real_eq_of_sign_eq {θ ψ : angle} (h : θ.sign = ψ.sign) :
θ = ψ ↔ |θ.to_real| = |ψ.to_real| | by simpa [h] using @eq_iff_sign_eq_and_abs_to_real_eq θ ψ | lemma | real.angle.eq_iff_abs_to_real_eq_of_sign_eq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign_coe_pi_div_two : (↑(π / 2) : angle).sign = 1 | by rw [sign, sin_coe, sin_pi_div_two, sign_one] | lemma | real.angle.sign_coe_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"sign",
"sign_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign_coe_neg_pi_div_two : (↑(-π / 2) : angle).sign = -1 | by rw [sign, sin_coe, neg_div, real.sin_neg, sin_pi_div_two, left.sign_neg, sign_one] | lemma | real.angle.sign_coe_neg_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"left.sign_neg",
"neg_div",
"real.sin_neg",
"sign",
"sign_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign_coe_nonneg_of_nonneg_of_le_pi {θ : ℝ} (h0 : 0 ≤ θ) (hpi : θ ≤ π) :
0 ≤ (θ : angle).sign | begin
rw [sign, sign_nonneg_iff],
exact sin_nonneg_of_nonneg_of_le_pi h0 hpi
end | lemma | real.angle.sign_coe_nonneg_of_nonneg_of_le_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"sign",
"sign_nonneg_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign_neg_coe_nonpos_of_nonneg_of_le_pi {θ : ℝ} (h0 : 0 ≤ θ) (hpi : θ ≤ π) :
(-θ : angle).sign ≤ 0 | begin
rw [sign, sign_nonpos_iff, sin_neg, left.neg_nonpos_iff],
exact sin_nonneg_of_nonneg_of_le_pi h0 hpi
end | lemma | real.angle.sign_neg_coe_nonpos_of_nonneg_of_le_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"sign",
"sign_nonpos_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign_two_nsmul_eq_sign_iff {θ : angle} :
((2 : ℕ) • θ).sign = θ.sign ↔ (θ = π ∨ |θ.to_real| < π / 2) | begin
by_cases hpi : θ = π, { simp [hpi] },
rw or_iff_right hpi,
refine ⟨λ h, _, λ h, _⟩,
{ by_contra hle,
rw [not_lt, le_abs, le_neg] at hle,
have hpi' : θ.to_real ≠ π, { simpa using hpi },
rcases hle with hle | hle; rcases hle.eq_or_lt with heq | hlt,
{ rw [←coe_to_real θ, ←heq] at h, simpa us... | lemma | real.angle.sign_two_nsmul_eq_sign_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"abs_lt",
"abs_of_pos",
"by_contra",
"div_pos",
"le_abs",
"mul_neg_of_pos_of_neg",
"neg_div",
"not_or_distrib",
"one_mul",
"or_iff_right",
"sign",
"sign_mul",
"sign_pos",
"zero_lt_two'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign_two_zsmul_eq_sign_iff {θ : angle} :
((2 : ℤ) • θ).sign = θ.sign ↔ (θ = π ∨ |θ.to_real| < π / 2) | by rw [two_zsmul, ←two_nsmul, sign_two_nsmul_eq_sign_iff] | lemma | real.angle.sign_two_zsmul_eq_sign_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"sign"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_sign {θ : angle} (h0 : θ ≠ 0) (hpi : θ ≠ π) : continuous_at sign θ | (continuous_at_sign_of_ne_zero (sin_ne_zero_iff.2 ⟨h0, hpi⟩)).comp continuous_sin.continuous_at | lemma | real.angle.continuous_at_sign | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"continuous_at",
"continuous_at_sign_of_ne_zero",
"sign"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.continuous_on.angle_sign_comp {α : Type*} [topological_space α] {f : α → angle}
{s : set α} (hf : continuous_on f s) (hs : ∀ z ∈ s, f z ≠ 0 ∧ f z ≠ π) :
continuous_on (sign ∘ f) s | begin
refine (continuous_at.continuous_on (λ θ hθ, _)).comp hf (set.maps_to_image f s),
obtain ⟨z, hz, rfl⟩ := hθ,
exact continuous_at_sign (hs _ hz).1 (hs _ hz).2
end | lemma | continuous_on.angle_sign_comp | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"continuous_at.continuous_on",
"continuous_on",
"set.maps_to_image",
"sign",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sign_eq_of_continuous_on {α : Type*} [topological_space α] {f : α → angle} {s : set α}
{x y : α} (hc : is_connected s) (hf : continuous_on f s) (hs : ∀ z ∈ s, f z ≠ 0 ∧ f z ≠ π)
(hx : x ∈ s) (hy : y ∈ s) : (f y).sign = (f x).sign | (hc.image _ (hf.angle_sign_comp hs)).is_preconnected.subsingleton
(set.mem_image_of_mem _ hy) (set.mem_image_of_mem _ hx) | lemma | real.angle.sign_eq_of_continuous_on | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"continuous_on",
"is_connected",
"is_preconnected.subsingleton",
"set.mem_image_of_mem",
"sign",
"topological_space"
] | Suppose a function to angles is continuous on a connected set and never takes the values `0`
or `π` on that set. Then the values of the function on that set all have the same sign. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tan_add {x y : ℝ}
(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) | by simpa only [← complex.of_real_inj, complex.of_real_sub, complex.of_real_add, complex.of_real_div,
complex.of_real_mul, complex.of_real_tan]
using @complex.tan_add (x:ℂ) (y:ℂ) (by convert h; norm_cast) | lemma | real.tan_add | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"complex.of_real_add",
"complex.of_real_div",
"complex.of_real_inj",
"complex.of_real_mul",
"complex.of_real_sub",
"complex.of_real_tan",
"complex.tan_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_add' {x y : ℝ}
(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 | real.tan_add' | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_two_mul {x:ℝ} : tan (2 * x) = 2 * tan x / (1 - tan x ^ 2) | by simpa only [← complex.of_real_inj, complex.of_real_sub, complex.of_real_div, complex.of_real_pow,
complex.of_real_mul, complex.of_real_tan, complex.of_real_bit0, complex.of_real_one]
using complex.tan_two_mul | lemma | real.tan_two_mul | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"complex.of_real_bit0",
"complex.of_real_div",
"complex.of_real_inj",
"complex.of_real_mul",
"complex.of_real_one",
"complex.of_real_pow",
"complex.of_real_sub",
"complex.of_real_tan",
"complex.tan_two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_ne_zero_iff {θ : ℝ} : tan θ ≠ 0 ↔ ∀ k : ℤ, θ ≠ k * π / 2 | by rw [← complex.of_real_ne_zero, complex.of_real_tan, complex.tan_ne_zero_iff]; norm_cast | lemma | real.tan_ne_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"complex.of_real_ne_zero",
"complex.of_real_tan",
"complex.tan_ne_zero_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_eq_zero_iff {θ : ℝ} : tan θ = 0 ↔ ∃ k : ℤ, θ = k * π / 2 | by rw [← not_iff_not, not_exists, ← ne, tan_ne_zero_iff] | lemma | real.tan_eq_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"not_exists",
"not_iff_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_int_mul_pi_div_two (n : ℤ) : tan (n * π/2) = 0 | tan_eq_zero_iff.mpr (by use n) | lemma | real.tan_int_mul_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.