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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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continuous_on_tan : continuous_on tan {x | cos x ≠ 0} | begin
suffices : continuous_on (λ x, sin x / cos x) {x | cos x ≠ 0},
{ have h_eq : (λ x, sin x / cos x) = tan, by {ext1 x, rw tan_eq_sin_div_cos, },
rwa h_eq at this, },
exact continuous_on_sin.div continuous_on_cos (λ x, id),
end | lemma | real.continuous_on_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_tan : continuous (λ x : {x | cos x ≠ 0}, tan x) | continuous_on_iff_continuous_restrict.1 continuous_on_tan | lemma | real.continuous_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_tan_Ioo : continuous_on tan (Ioo (-(π/2)) (π/2)) | begin
refine continuous_on.mono continuous_on_tan (λ x, _),
simp only [and_imp, mem_Ioo, mem_set_of_eq, ne.def],
rw cos_eq_zero_iff,
rintros hx_gt hx_lt ⟨r, hxr_eq⟩,
cases le_or_lt 0 r,
{ rw lt_iff_not_ge at hx_lt,
refine hx_lt _,
rw [hxr_eq, ← one_mul (π / 2), mul_div_assoc, ge_iff_le, mul_le_mul_r... | lemma | real.continuous_on_tan_Ioo | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"and_imp",
"continuous_on",
"continuous_on.mono",
"ge_iff_le",
"half_pos",
"int.cast_neg",
"int.cast_one",
"le_div_iff",
"mul_comm",
"mul_div_assoc",
"mul_le_mul_right",
"neg_mul_eq_neg_mul",
"one_mul",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surj_on_tan : surj_on tan (Ioo (-(π / 2)) (π / 2)) univ | have _ := neg_lt_self pi_div_two_pos,
continuous_on_tan_Ioo.surj_on_of_tendsto (nonempty_Ioo.2 this)
(by simp [tendsto_tan_neg_pi_div_two, this]) (by simp [tendsto_tan_pi_div_two, this]) | lemma | real.surj_on_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_surjective : function.surjective tan | λ x, surj_on_tan.subset_range trivial | lemma | real.tan_surjective | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
image_tan_Ioo : tan '' (Ioo (-(π / 2)) (π / 2)) = univ | univ_subset_iff.1 surj_on_tan | lemma | real.image_tan_Ioo | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_order_iso : Ioo (-(π / 2)) (π / 2) ≃o ℝ | (strict_mono_on_tan.order_iso _ _).trans $ (order_iso.set_congr _ _ image_tan_Ioo).trans
order_iso.set.univ | def | real.tan_order_iso | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"order_iso.set.univ",
"order_iso.set_congr"
] | `real.tan` as an `order_iso` between `(-(π / 2), π / 2)` and `ℝ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arctan (x : ℝ) : ℝ | tan_order_iso.symm x | def | real.arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [] | Inverse of the `tan` function, returns values in the range `-π / 2 < arctan x` and
`arctan x < π / 2` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tan_arctan (x : ℝ) : tan (arctan x) = x | tan_order_iso.apply_symm_apply x | lemma | real.tan_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arctan_mem_Ioo (x : ℝ) : arctan x ∈ Ioo (-(π / 2)) (π / 2) | subtype.coe_prop _ | lemma | real.arctan_mem_Ioo | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"subtype.coe_prop"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_arctan : range arctan = Ioo (-(π / 2)) (π / 2) | ((equiv_like.surjective _).range_comp _).trans subtype.range_coe | lemma | real.range_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"equiv_like.surjective",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arctan_tan {x : ℝ} (hx₁ : -(π / 2) < x) (hx₂ : x < π / 2) : arctan (tan x) = x | subtype.ext_iff.1 $ tan_order_iso.symm_apply_apply ⟨x, hx₁, hx₂⟩ | lemma | real.arctan_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_arctan_pos (x : ℝ) : 0 < cos (arctan x) | cos_pos_of_mem_Ioo $ arctan_mem_Ioo x | lemma | real.cos_arctan_pos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_sq_arctan (x : ℝ) : cos (arctan x) ^ 2 = 1 / (1 + x ^ 2) | by rw [one_div, ← inv_one_add_tan_sq (cos_arctan_pos x).ne', tan_arctan] | lemma | real.cos_sq_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_arctan (x : ℝ) : sin (arctan x) = x / sqrt (1 + x ^ 2) | by rw [← tan_div_sqrt_one_add_tan_sq (cos_arctan_pos x), tan_arctan] | lemma | real.sin_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_arctan (x : ℝ) : cos (arctan x) = 1 / sqrt (1 + x ^ 2) | by rw [one_div, ← inv_sqrt_one_add_tan_sq (cos_arctan_pos x), tan_arctan] | lemma | real.cos_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arctan_lt_pi_div_two (x : ℝ) : arctan x < π / 2 | (arctan_mem_Ioo x).2 | lemma | real.arctan_lt_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 | |
neg_pi_div_two_lt_arctan (x : ℝ) : -(π / 2) < arctan x | (arctan_mem_Ioo x).1 | lemma | real.neg_pi_div_two_lt_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arctan_eq_arcsin (x : ℝ) : arctan x = arcsin (x / sqrt (1 + x ^ 2)) | eq.symm $ arcsin_eq_of_sin_eq (sin_arctan x) (mem_Icc_of_Ioo $ arctan_mem_Ioo x) | lemma | real.arctan_eq_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_eq_arctan {x : ℝ} (h : x ∈ Ioo (-(1:ℝ)) 1) :
arcsin x = arctan (x / sqrt (1 - x ^ 2)) | begin
rw [arctan_eq_arcsin, div_pow, sq_sqrt, one_add_div, div_div,
← sqrt_mul, mul_div_cancel', sub_add_cancel, sqrt_one, div_one];
nlinarith [h.1, h.2],
end | lemma | real.arcsin_eq_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"div_div",
"div_one",
"div_pow",
"mul_div_cancel'",
"one_add_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arctan_zero : arctan 0 = 0 | by simp [arctan_eq_arcsin] | lemma | real.arctan_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arctan_eq_of_tan_eq {x y : ℝ} (h : tan x = y) (hx : x ∈ Ioo (-(π / 2)) (π / 2)) :
arctan y = x | inj_on_tan (arctan_mem_Ioo _) hx (by rw [tan_arctan, h]) | lemma | real.arctan_eq_of_tan_eq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arctan_one : arctan 1 = π / 4 | arctan_eq_of_tan_eq tan_pi_div_four $ by split; linarith [pi_pos] | lemma | real.arctan_one | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arctan_neg (x : ℝ) : arctan (-x) = - arctan x | by simp [arctan_eq_arcsin, neg_div] | lemma | real.arctan_neg | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"neg_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arctan_eq_arccos {x : ℝ} (h : 0 ≤ x) : arctan x = arccos ((sqrt (1 + x ^ 2))⁻¹) | begin
rw [arctan_eq_arcsin, arccos_eq_arcsin], swap, { exact inv_nonneg.2 (sqrt_nonneg _) },
congr' 1,
rw [←sqrt_inv, sq_sqrt, ←one_div, one_sub_div, add_sub_cancel', sqrt_div, sqrt_sq h],
all_goals { positivity }
end | lemma | real.arctan_eq_arccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"one_sub_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_eq_arctan {x : ℝ} (h : 0 < x) :
arccos x = arctan (sqrt (1 - x ^ 2) / x) | begin
rw [arccos, eq_comm],
refine arctan_eq_of_tan_eq _ ⟨_, _⟩,
{ rw [tan_pi_div_two_sub, tan_arcsin, inv_div] },
{ linarith only [arcsin_le_pi_div_two x, pi_pos] },
{ linarith only [arcsin_pos.2 h] }
end | lemma | real.arccos_eq_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"inv_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_arctan : continuous arctan | continuous_subtype_coe.comp tan_order_iso.to_homeomorph.continuous_inv_fun | lemma | real.continuous_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_arctan {x : ℝ} : continuous_at arctan x | continuous_arctan.continuous_at | lemma | real.continuous_at_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_local_homeomorph : local_homeomorph ℝ ℝ | { to_fun := tan,
inv_fun := arctan,
source := Ioo (-(π / 2)) (π / 2),
target := univ,
map_source' := maps_to_univ _ _,
map_target' := λ y hy, arctan_mem_Ioo y,
left_inv' := λ x hx, arctan_tan hx.1 hx.2,
right_inv' := λ y hy, tan_arctan y,
open_source := is_open_Ioo,
open_target := is_open_univ,
cont... | def | real.tan_local_homeomorph | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [
"inv_fun",
"is_open_Ioo",
"is_open_univ",
"local_homeomorph"
] | `real.tan` as a `local_homeomorph` between `(-(π / 2), π / 2)` and the whole line. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_tan_local_homeomorph : ⇑tan_local_homeomorph = tan | rfl | lemma | real.coe_tan_local_homeomorph | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_tan_local_homeomorph_symm : ⇑tan_local_homeomorph.symm = arctan | rfl | lemma | real.coe_tan_local_homeomorph_symm | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan.lean | [
"analysis.special_functions.trigonometric.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_tan {x : ℝ} (h : cos x ≠ 0) :
has_strict_deriv_at tan (1 / (cos x)^2) x | by exact_mod_cast (complex.has_strict_deriv_at_tan (by exact_mod_cast h)).real_of_complex | lemma | real.has_strict_deriv_at_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"complex.has_strict_deriv_at_tan",
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_tan {x : ℝ} (h : cos x ≠ 0) :
has_deriv_at tan (1 / (cos x)^2) x | by exact_mod_cast (complex.has_deriv_at_tan (by exact_mod_cast h)).real_of_complex | lemma | real.has_deriv_at_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"complex.has_deriv_at_tan",
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_abs_tan_of_cos_eq_zero {x : ℝ} (hx : cos x = 0) :
tendsto (λ x, abs (tan x)) (𝓝[≠] x) at_top | begin
have hx : complex.cos x = 0, by exact_mod_cast hx,
simp only [← complex.abs_of_real, complex.of_real_tan],
refine (complex.tendsto_abs_tan_of_cos_eq_zero hx).comp _,
refine tendsto.inf complex.continuous_of_real.continuous_at _,
exact tendsto_principal_principal.2 (λ y, mt complex.of_real_inj.1)
end | lemma | real.tendsto_abs_tan_of_cos_eq_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"complex.abs_of_real",
"complex.cos",
"complex.of_real_tan",
"complex.tendsto_abs_tan_of_cos_eq_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_abs_tan_at_top (k : ℤ) :
tendsto (λ x, abs (tan x)) (𝓝[≠] ((2 * k + 1) * π / 2)) at_top | tendsto_abs_tan_of_cos_eq_zero $ cos_eq_zero_iff.2 ⟨k, rfl⟩ | lemma | real.tendsto_abs_tan_at_top | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_at_tan {x : ℝ} : continuous_at tan x ↔ cos x ≠ 0 | begin
refine ⟨λ hc h₀, _, λ h, (has_deriv_at_tan h).continuous_at⟩,
exact not_tendsto_nhds_of_tendsto_at_top (tendsto_abs_tan_of_cos_eq_zero h₀) _
(hc.norm.tendsto.mono_left inf_le_left)
end | lemma | real.continuous_at_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"continuous_at",
"inf_le_left",
"not_tendsto_nhds_of_tendsto_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_tan {x : ℝ} : differentiable_at ℝ tan x ↔ cos x ≠ 0 | ⟨λ h, continuous_at_tan.1 h.continuous_at, λ h, (has_deriv_at_tan h).differentiable_at⟩ | lemma | real.differentiable_at_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_tan (x : ℝ) : deriv tan x = 1 / (cos x)^2 | if h : cos x = 0 then
have ¬differentiable_at ℝ tan x := mt differentiable_at_tan.1 (not_not.2 h),
by simp [deriv_zero_of_not_differentiable_at this, h, sq]
else (has_deriv_at_tan h).deriv | lemma | real.deriv_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"deriv",
"deriv_zero_of_not_differentiable_at",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_tan {n x} : cont_diff_at ℝ n tan x ↔ cos x ≠ 0 | ⟨λ h, continuous_at_tan.1 h.continuous_at,
λ h, (complex.cont_diff_at_tan.2 $ by exact_mod_cast h).real_of_complex⟩ | lemma | real.cont_diff_at_tan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_tan_of_mem_Ioo {x : ℝ} (h : x ∈ Ioo (-(π/2):ℝ) (π/2)) :
has_deriv_at tan (1 / (cos x)^2) x | has_deriv_at_tan (cos_pos_of_mem_Ioo h).ne' | lemma | real.has_deriv_at_tan_of_mem_Ioo | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_tan_of_mem_Ioo {x : ℝ} (h : x ∈ Ioo (-(π/2):ℝ) (π/2)) :
differentiable_at ℝ tan x | (has_deriv_at_tan_of_mem_Ioo h).differentiable_at | lemma | real.differentiable_at_tan_of_mem_Ioo | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_arctan (x : ℝ) : has_strict_deriv_at arctan (1 / (1 + x^2)) x | have A : cos (arctan x) ≠ 0 := (cos_arctan_pos x).ne',
by simpa [cos_sq_arctan]
using tan_local_homeomorph.has_strict_deriv_at_symm trivial (by simpa) (has_strict_deriv_at_tan A) | lemma | real.has_strict_deriv_at_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_arctan (x : ℝ) : has_deriv_at arctan (1 / (1 + x^2)) x | (has_strict_deriv_at_arctan x).has_deriv_at | lemma | real.has_deriv_at_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_arctan (x : ℝ) : differentiable_at ℝ arctan x | (has_deriv_at_arctan x).differentiable_at | lemma | real.differentiable_at_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_arctan : differentiable ℝ arctan | differentiable_at_arctan | lemma | real.differentiable_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_arctan : deriv arctan = (λ x, 1 / (1 + x^2)) | funext $ λ x, (has_deriv_at_arctan x).deriv | lemma | real.deriv_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"deriv",
"deriv_arctan"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_arctan {n : ℕ∞} : cont_diff ℝ n arctan | cont_diff_iff_cont_diff_at.2 $ λ x,
have cos (arctan x) ≠ 0 := (cos_arctan_pos x).ne',
tan_local_homeomorph.cont_diff_at_symm_deriv (by simpa) trivial (has_deriv_at_tan this)
(cont_diff_at_tan.2 this) | lemma | real.cont_diff_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.arctan (hf : has_strict_deriv_at f f' x) :
has_strict_deriv_at (λ x, arctan (f x)) ((1 / (1 + (f x)^2)) * f') x | (real.has_strict_deriv_at_arctan (f x)).comp x hf | lemma | has_strict_deriv_at.arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"has_strict_deriv_at",
"real.has_strict_deriv_at_arctan"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.arctan (hf : has_deriv_at f f' x) :
has_deriv_at (λ x, arctan (f x)) ((1 / (1 + (f x)^2)) * f') x | (real.has_deriv_at_arctan (f x)).comp x hf | lemma | has_deriv_at.arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"has_deriv_at",
"real.has_deriv_at_arctan"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.arctan (hf : has_deriv_within_at f f' s x) :
has_deriv_within_at (λ x, arctan (f x)) ((1 / (1 + (f x)^2)) * f') s x | (real.has_deriv_at_arctan (f x)).comp_has_deriv_within_at x hf | lemma | has_deriv_within_at.arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"has_deriv_within_at",
"real.has_deriv_at_arctan"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_arctan (hf : differentiable_within_at ℝ f s x)
(hxs : unique_diff_within_at ℝ s x) :
deriv_within (λ x, arctan (f x)) s x = (1 / (1 + (f x)^2)) * (deriv_within f s x) | hf.has_deriv_within_at.arctan.deriv_within hxs | lemma | deriv_within_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_arctan (hc : differentiable_at ℝ f x) :
deriv (λ x, arctan (f x)) x = (1 / (1 + (f x)^2)) * (deriv f x) | hc.has_deriv_at.arctan.deriv | lemma | deriv_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.arctan (hf : has_strict_fderiv_at f f' x) :
has_strict_fderiv_at (λ x, arctan (f x)) ((1 / (1 + (f x)^2)) • f') x | (has_strict_deriv_at_arctan (f x)).comp_has_strict_fderiv_at x hf | lemma | has_strict_fderiv_at.arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.arctan (hf : has_fderiv_at f f' x) :
has_fderiv_at (λ x, arctan (f x)) ((1 / (1 + (f x)^2)) • f') x | (has_deriv_at_arctan (f x)).comp_has_fderiv_at x hf | lemma | has_fderiv_at.arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.arctan (hf : has_fderiv_within_at f f' s x) :
has_fderiv_within_at (λ x, arctan (f x)) ((1 / (1 + (f x)^2)) • f') s x | (has_deriv_at_arctan (f x)).comp_has_fderiv_within_at x hf | lemma | has_fderiv_within_at.arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_arctan (hf : differentiable_within_at ℝ f s x)
(hxs : unique_diff_within_at ℝ s x) :
fderiv_within ℝ (λ x, arctan (f x)) s x = (1 / (1 + (f x)^2)) • (fderiv_within ℝ f s x) | hf.has_fderiv_within_at.arctan.fderiv_within hxs | lemma | fderiv_within_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_arctan (hc : differentiable_at ℝ f x) :
fderiv ℝ (λ x, arctan (f x)) x = (1 / (1 + (f x)^2)) • (fderiv ℝ f x) | hc.has_fderiv_at.arctan.fderiv | lemma | fderiv_arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.arctan (hf : differentiable_within_at ℝ f s x) :
differentiable_within_at ℝ (λ x, real.arctan (f x)) s x | hf.has_fderiv_within_at.arctan.differentiable_within_at | lemma | differentiable_within_at.arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"differentiable_within_at",
"real.arctan"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.arctan (hc : differentiable_at ℝ f x) :
differentiable_at ℝ (λ x, arctan (f x)) x | hc.has_fderiv_at.arctan.differentiable_at | lemma | differentiable_at.arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.arctan (hc : differentiable_on ℝ f s) :
differentiable_on ℝ (λ x, arctan (f x)) s | λ x h, (hc x h).arctan | lemma | differentiable_on.arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.arctan (hc : differentiable ℝ f) :
differentiable ℝ (λ x, arctan (f x)) | λ x, (hc x).arctan | lemma | differentiable.arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.arctan (h : cont_diff_at ℝ n f x) :
cont_diff_at ℝ n (λ x, arctan (f x)) x | cont_diff_arctan.cont_diff_at.comp x h | lemma | cont_diff_at.arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.arctan (h : cont_diff ℝ n f) :
cont_diff ℝ n (λ x, arctan (f x)) | cont_diff_arctan.comp h | lemma | cont_diff.arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.arctan (h : cont_diff_within_at ℝ n f s x) :
cont_diff_within_at ℝ n (λ x, arctan (f x)) s x | cont_diff_arctan.comp_cont_diff_within_at h | lemma | cont_diff_within_at.arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.arctan (h : cont_diff_on ℝ n f s) :
cont_diff_on ℝ n (λ x, arctan (f x)) s | cont_diff_arctan.comp_cont_diff_on h | lemma | cont_diff_on.arctan | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/arctan_deriv.lean | [
"analysis.special_functions.trigonometric.arctan",
"analysis.special_functions.trigonometric.complex_deriv"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_sin : continuous sin | by { change continuous (λ z, ((exp (-z * I) - exp (z * I)) * I) / 2), continuity, } | lemma | complex.continuous_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"continuity",
"continuous",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_sin {s : set ℂ} : continuous_on sin s | continuous_sin.continuous_on | lemma | complex.continuous_on_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_cos : continuous cos | by { change continuous (λ z, (exp (z * I) + exp (-z * I)) / 2), continuity, } | lemma | complex.continuous_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"continuity",
"continuous",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_cos {s : set ℂ} : continuous_on cos s | continuous_cos.continuous_on | lemma | complex.continuous_on_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_sinh : continuous sinh | by { change continuous (λ z, (exp z - exp (-z)) / 2), continuity, } | lemma | complex.continuous_sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"continuity",
"continuous",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_cosh : continuous cosh | by { change continuous (λ z, (exp z + exp (-z)) / 2), continuity, } | lemma | complex.continuous_cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"continuity",
"continuous",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_sin : continuous sin | complex.continuous_re.comp (complex.continuous_sin.comp complex.continuous_of_real) | lemma | real.continuous_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"complex.continuous_of_real",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_sin {s} : continuous_on sin s | continuous_sin.continuous_on | lemma | real.continuous_on_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_cos : continuous cos | complex.continuous_re.comp (complex.continuous_cos.comp complex.continuous_of_real) | lemma | real.continuous_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"complex.continuous_of_real",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_on_cos {s} : continuous_on cos s | continuous_cos.continuous_on | lemma | real.continuous_on_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"continuous_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_sinh : continuous sinh | complex.continuous_re.comp (complex.continuous_sinh.comp complex.continuous_of_real) | lemma | real.continuous_sinh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"complex.continuous_of_real",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_cosh : continuous cosh | complex.continuous_re.comp (complex.continuous_cosh.comp complex.continuous_of_real) | lemma | real.continuous_cosh | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"complex.continuous_of_real",
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_cos_eq_zero : 0 ∈ cos '' Icc (1:ℝ) 2 | intermediate_value_Icc' (by norm_num) continuous_on_cos
⟨le_of_lt cos_two_neg, le_of_lt cos_one_pos⟩ | lemma | real.exists_cos_eq_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"intermediate_value_Icc'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi : ℝ | 2 * classical.some exists_cos_eq_zero | def | real.pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | The number π = 3.14159265... Defined here using choice as twice a zero of cos in [1,2], from
which one can derive all its properties. For explicit bounds on π, see `data.real.pi.bounds`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cos_pi_div_two : cos (π / 2) = 0 | by rw [real.pi, mul_div_cancel_left _ (two_ne_zero' ℝ)];
exact (classical.some_spec exists_cos_eq_zero).2 | lemma | real.cos_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"mul_div_cancel_left",
"real.pi",
"two_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_pi_div_two : (1 : ℝ) ≤ π / 2 | by rw [real.pi, mul_div_cancel_left _ (two_ne_zero' ℝ)];
exact (classical.some_spec exists_cos_eq_zero).1.1 | lemma | real.one_le_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"mul_div_cancel_left",
"real.pi",
"two_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_div_two_le_two : π / 2 ≤ 2 | by rw [real.pi, mul_div_cancel_left _ (two_ne_zero' ℝ)];
exact (classical.some_spec exists_cos_eq_zero).1.2 | lemma | real.pi_div_two_le_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"mul_div_cancel_left",
"real.pi",
"two_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_le_pi : (2 : ℝ) ≤ π | (div_le_div_right (show (0 : ℝ) < 2, by norm_num)).1
(by rw div_self (two_ne_zero' ℝ); exact one_le_pi_div_two) | lemma | real.two_le_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"div_le_div_right",
"div_self",
"two_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_le_four : π ≤ 4 | (div_le_div_right (show (0 : ℝ) < 2, by norm_num)).1
(calc π / 2 ≤ 2 : pi_div_two_le_two
... = 4 / 2 : by norm_num) | lemma | real.pi_le_four | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"div_le_div_right"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_pos : 0 < π | lt_of_lt_of_le (by norm_num) two_le_pi | lemma | real.pi_pos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_ne_zero : π ≠ 0 | ne_of_gt pi_pos | lemma | real.pi_ne_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_div_two_pos : 0 < π / 2 | half_pos pi_pos | lemma | real.pi_div_two_pos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"half_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_pi_pos : 0 < 2 * π | by linarith [pi_pos] | lemma | real.two_pi_pos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi : ℝ≥0 | ⟨π, real.pi_pos.le⟩ | def | nnreal.pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | `π` considered as a nonnegative real. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_real_pi : (pi : ℝ) = π | rfl | lemma | nnreal.coe_real_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_pos : 0 < pi | by exact_mod_cast real.pi_pos | lemma | nnreal.pi_pos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"real.pi_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_ne_zero : pi ≠ 0 | pi_pos.ne' | lemma | nnreal.pi_ne_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_pi : sin π = 0 | by rw [← mul_div_cancel_left π (two_ne_zero' ℝ), two_mul, add_div,
sin_add, cos_pi_div_two]; simp | lemma | real.sin_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"add_div",
"mul_div_cancel_left",
"two_mul",
"two_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_pi : cos π = -1 | by rw [← mul_div_cancel_left π (two_ne_zero' ℝ), mul_div_assoc,
cos_two_mul, cos_pi_div_two];
simp [bit0, pow_add] | lemma | real.cos_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"mul_div_assoc",
"mul_div_cancel_left",
"pow_add",
"two_ne_zero'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_two_pi : sin (2 * π) = 0 | by simp [two_mul, sin_add] | lemma | real.sin_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_two_pi : cos (2 * π) = 1 | by simp [two_mul, cos_add] | lemma | real.cos_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_antiperiodic : function.antiperiodic sin π | by simp [sin_add] | lemma | real.sin_antiperiodic | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"function.antiperiodic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_periodic : function.periodic sin (2 * π) | sin_antiperiodic.periodic | lemma | real.sin_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 | |
sin_add_pi (x : ℝ) : sin (x + π) = -sin x | sin_antiperiodic x | lemma | real.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 : ℝ) : sin (x + 2 * π) = sin x | sin_periodic x | lemma | real.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 |
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