statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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lt_arcsin_iff_sin_lt' {x y : ℝ} (hx : x ∈ Ico (-(π / 2)) (π / 2)) :
x < arcsin y ↔ sin x < y | not_le.symm.trans $ (not_congr $ arcsin_le_iff_le_sin' hx).trans not_le | lemma | real.lt_arcsin_iff_sin_lt' | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_eq_iff_eq_sin {x y : ℝ} (hy : y ∈ Ioo (-(π / 2)) (π / 2)) :
arcsin x = y ↔ x = sin y | by simp only [le_antisymm_iff, arcsin_le_iff_le_sin' (mem_Ico_of_Ioo hy),
le_arcsin_iff_sin_le' (mem_Ioc_of_Ioo hy)] | lemma | real.arcsin_eq_iff_eq_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_nonneg {x : ℝ} : 0 ≤ arcsin x ↔ 0 ≤ x | (le_arcsin_iff_sin_le' ⟨neg_lt_zero.2 pi_div_two_pos, pi_div_two_pos.le⟩).trans $ by rw [sin_zero] | lemma | real.arcsin_nonneg | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_nonpos {x : ℝ} : arcsin x ≤ 0 ↔ x ≤ 0 | neg_nonneg.symm.trans $ arcsin_neg x ▸ arcsin_nonneg.trans neg_nonneg | lemma | real.arcsin_nonpos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_eq_zero_iff {x : ℝ} : arcsin x = 0 ↔ x = 0 | by simp [le_antisymm_iff] | lemma | real.arcsin_eq_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zero_eq_arcsin_iff {x} : 0 = arcsin x ↔ x = 0 | eq_comm.trans arcsin_eq_zero_iff | lemma | real.zero_eq_arcsin_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_pos {x : ℝ} : 0 < arcsin x ↔ 0 < x | lt_iff_lt_of_le_iff_le arcsin_nonpos | lemma | real.arcsin_pos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_lt_zero {x : ℝ} : arcsin x < 0 ↔ x < 0 | lt_iff_lt_of_le_iff_le arcsin_nonneg | lemma | real.arcsin_lt_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_lt_pi_div_two {x : ℝ} : arcsin x < π / 2 ↔ x < 1 | (arcsin_lt_iff_lt_sin' (right_mem_Ioc.2 $ neg_lt_self pi_div_two_pos)).trans $
by rw sin_pi_div_two | lemma | real.arcsin_lt_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_pi_div_two_lt_arcsin {x : ℝ} : -(π / 2) < arcsin x ↔ -1 < x | (lt_arcsin_iff_sin_lt' $ left_mem_Ico.2 $ neg_lt_self pi_div_two_pos).trans $
by rw [sin_neg, sin_pi_div_two] | lemma | real.neg_pi_div_two_lt_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_eq_pi_div_two {x : ℝ} : arcsin x = π / 2 ↔ 1 ≤ x | ⟨λ h, not_lt.1 $ λ h', (arcsin_lt_pi_div_two.2 h').ne h, arcsin_of_one_le⟩ | lemma | real.arcsin_eq_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_div_two_eq_arcsin {x} : π / 2 = arcsin x ↔ 1 ≤ x | eq_comm.trans arcsin_eq_pi_div_two | lemma | real.pi_div_two_eq_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_div_two_le_arcsin {x} : π / 2 ≤ arcsin x ↔ 1 ≤ x | (arcsin_le_pi_div_two x).le_iff_eq.trans pi_div_two_eq_arcsin | lemma | real.pi_div_two_le_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_eq_neg_pi_div_two {x : ℝ} : arcsin x = -(π / 2) ↔ x ≤ -1 | ⟨λ h, not_lt.1 $ λ h', (neg_pi_div_two_lt_arcsin.2 h').ne' h, arcsin_of_le_neg_one⟩ | lemma | real.arcsin_eq_neg_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_pi_div_two_eq_arcsin {x} : -(π / 2) = arcsin x ↔ x ≤ -1 | eq_comm.trans arcsin_eq_neg_pi_div_two | lemma | real.neg_pi_div_two_eq_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_le_neg_pi_div_two {x} : arcsin x ≤ -(π / 2) ↔ x ≤ -1 | (neg_pi_div_two_le_arcsin x).le_iff_eq.trans arcsin_eq_neg_pi_div_two | lemma | real.arcsin_le_neg_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi_div_four_le_arcsin {x} : π / 4 ≤ arcsin x ↔ sqrt 2 / 2 ≤ x | by { rw [← sin_pi_div_four, le_arcsin_iff_sin_le'], have := pi_pos, split; linarith } | lemma | real.pi_div_four_le_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maps_to_sin_Ioo : maps_to sin (Ioo (-(π / 2)) (π / 2)) (Ioo (-1) 1) | λ x h, by rwa [mem_Ioo, ← arcsin_lt_pi_div_two, ← neg_pi_div_two_lt_arcsin,
arcsin_sin h.1.le h.2.le] | lemma | real.maps_to_sin_Ioo | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_local_homeomorph : local_homeomorph ℝ ℝ | { to_fun := sin,
inv_fun := arcsin,
source := Ioo (-(π / 2)) (π / 2),
target := Ioo (-1) 1,
map_source' := maps_to_sin_Ioo,
map_target' := λ y hy, ⟨neg_pi_div_two_lt_arcsin.2 hy.1, arcsin_lt_pi_div_two.2 hy.2⟩,
left_inv' := λ x hx, arcsin_sin hx.1.le hx.2.le,
right_inv' := λ y hy, sin_arcsin hy.1.le hy.2.... | def | real.sin_local_homeomorph | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"inv_fun",
"is_open_Ioo",
"local_homeomorph"
] | `real.sin` as a `local_homeomorph` between `(-π / 2, π / 2)` and `(-1, 1)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cos_arcsin_nonneg (x : ℝ) : 0 ≤ cos (arcsin x) | cos_nonneg_of_mem_Icc ⟨neg_pi_div_two_le_arcsin _, arcsin_le_pi_div_two _⟩ | lemma | real.cos_arcsin_nonneg | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_arcsin (x : ℝ) : cos (arcsin x) = sqrt (1 - x ^ 2) | begin
by_cases hx₁ : -1 ≤ x, swap,
{ rw not_le at hx₁,
rw [arcsin_of_le_neg_one hx₁.le, cos_neg, cos_pi_div_two, sqrt_eq_zero_of_nonpos],
nlinarith },
by_cases hx₂ : x ≤ 1, swap,
{ rw not_le at hx₂,
rw [arcsin_of_one_le hx₂.le, cos_pi_div_two, sqrt_eq_zero_of_nonpos],
nlinarith },
have : sin (... | lemma | real.cos_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"sq_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_arcsin (x : ℝ) : tan (arcsin x) = x / sqrt (1 - x ^ 2) | begin
rw [tan_eq_sin_div_cos, cos_arcsin],
by_cases hx₁ : -1 ≤ x, swap,
{ have h : sqrt (1 - x ^ 2) = 0, { exact sqrt_eq_zero_of_nonpos (by nlinarith) }, rw h, simp },
by_cases hx₂ : x ≤ 1, swap,
{ have h : sqrt (1 - x ^ 2) = 0, { exact sqrt_eq_zero_of_nonpos (by nlinarith) }, rw h, simp },
rw sin_arcsin hx... | lemma | real.tan_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos (x : ℝ) : ℝ | π / 2 - arcsin x | def | real.arccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | Inverse of the `cos` function, returns values in the range `0 ≤ arccos x` and `arccos x ≤ π`.
It defaults to `π` on `(-∞, -1)` and to `0` to `(1, ∞)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arccos_eq_pi_div_two_sub_arcsin (x : ℝ) : arccos x = π / 2 - arcsin x | rfl | lemma | real.arccos_eq_pi_div_two_sub_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_eq_pi_div_two_sub_arccos (x : ℝ) : arcsin x = π / 2 - arccos x | by simp [arccos] | lemma | real.arcsin_eq_pi_div_two_sub_arccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_le_pi (x : ℝ) : arccos x ≤ π | by unfold arccos; linarith [neg_pi_div_two_le_arcsin x] | lemma | real.arccos_le_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_nonneg (x : ℝ) : 0 ≤ arccos x | by unfold arccos; linarith [arcsin_le_pi_div_two x] | lemma | real.arccos_nonneg | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_pos {x : ℝ} : 0 < arccos x ↔ x < 1 | by simp [arccos] | lemma | real.arccos_pos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_arccos {x : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) : cos (arccos x) = x | by rw [arccos, cos_pi_div_two_sub, sin_arcsin hx₁ hx₂] | lemma | real.cos_arccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_cos {x : ℝ} (hx₁ : 0 ≤ x) (hx₂ : x ≤ π) : arccos (cos x) = x | by rw [arccos, ← sin_pi_div_two_sub, arcsin_sin]; simp [sub_eq_add_neg]; linarith | lemma | real.arccos_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_on_arccos : strict_anti_on arccos (Icc (-1) 1) | λ x hx y hy h, sub_lt_sub_left (strict_mono_on_arcsin hx hy h) _ | lemma | real.strict_anti_on_arccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"strict_anti_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_inj_on : inj_on arccos (Icc (-1) 1) | strict_anti_on_arccos.inj_on | lemma | real.arccos_inj_on | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_inj {x y : ℝ} (hx₁ : -1 ≤ x) (hx₂ : x ≤ 1) (hy₁ : -1 ≤ y) (hy₂ : y ≤ 1) :
arccos x = arccos y ↔ x = y | arccos_inj_on.eq_iff ⟨hx₁, hx₂⟩ ⟨hy₁, hy₂⟩ | lemma | real.arccos_inj | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_zero : arccos 0 = π / 2 | by simp [arccos] | lemma | real.arccos_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_one : arccos 1 = 0 | by simp [arccos] | lemma | real.arccos_one | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_neg_one : arccos (-1) = π | by simp [arccos, add_halves] | lemma | real.arccos_neg_one | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"add_halves"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_eq_zero {x} : arccos x = 0 ↔ 1 ≤ x | by simp [arccos, sub_eq_zero] | lemma | real.arccos_eq_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_eq_pi_div_two {x} : arccos x = π / 2 ↔ x = 0 | by simp [arccos] | lemma | real.arccos_eq_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_eq_pi {x} : arccos x = π ↔ x ≤ -1 | by rw [arccos, sub_eq_iff_eq_add, ← sub_eq_iff_eq_add', div_two_sub_self, neg_pi_div_two_eq_arcsin] | lemma | real.arccos_eq_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"div_two_sub_self"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_neg (x : ℝ) : arccos (-x) = π - arccos x | by rw [← add_halves π, arccos, arcsin_neg, arccos, add_sub_assoc, sub_sub_self, sub_neg_eq_add] | lemma | real.arccos_neg | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"add_halves"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_of_one_le {x : ℝ} (hx : 1 ≤ x) : arccos x = 0 | by rw [arccos, arcsin_of_one_le hx, sub_self] | lemma | real.arccos_of_one_le | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_of_le_neg_one {x : ℝ} (hx : x ≤ -1) : arccos x = π | by rw [arccos, arcsin_of_le_neg_one hx, sub_neg_eq_add, add_halves'] | lemma | real.arccos_of_le_neg_one | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"add_halves'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_arccos (x : ℝ) : sin (arccos x) = sqrt (1 - x ^ 2) | begin
by_cases hx₁ : -1 ≤ x, swap,
{ rw not_le at hx₁,
rw [arccos_of_le_neg_one hx₁.le, sin_pi, sqrt_eq_zero_of_nonpos],
nlinarith },
by_cases hx₂ : x ≤ 1, swap,
{ rw not_le at hx₂,
rw [arccos_of_one_le hx₂.le, sin_zero, sqrt_eq_zero_of_nonpos],
nlinarith },
rw [arccos_eq_pi_div_two_sub_arcsin... | lemma | real.sin_arccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_le_pi_div_two {x} : arccos x ≤ π / 2 ↔ 0 ≤ x | by simp [arccos] | lemma | real.arccos_le_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_lt_pi_div_two {x : ℝ} : arccos x < π / 2 ↔ 0 < x | by simp [arccos] | lemma | real.arccos_lt_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_le_pi_div_four {x} : arccos x ≤ π / 4 ↔ sqrt 2 / 2 ≤ x | by { rw [arccos, ← pi_div_four_le_arcsin], split; { intro, linarith } } | lemma | real.arccos_le_pi_div_four | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_arccos : continuous arccos | continuous_const.sub continuous_arcsin | lemma | real.continuous_arccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_arccos (x : ℝ) : tan (arccos x) = sqrt (1 - x ^ 2) / x | by rw [arccos, tan_pi_div_two_sub, tan_arcsin, inv_div] | lemma | real.tan_arccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"inv_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arccos_eq_arcsin {x : ℝ} (h : 0 ≤ x) :
arccos x = arcsin (sqrt (1 - x ^ 2)) | (arcsin_eq_of_sin_eq (sin_arccos _)
⟨(left.neg_nonpos_iff.2 (div_nonneg pi_pos.le (by norm_num))).trans (arccos_nonneg _),
arccos_le_pi_div_two.2 h⟩).symm | lemma | real.arccos_eq_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"div_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arcsin_eq_arccos {x : ℝ} (h : 0 ≤ x) :
arcsin x = arccos (sqrt (1 - x ^ 2)) | begin
rw [eq_comm, ← cos_arcsin],
exact arccos_cos (arcsin_nonneg.2 h)
((arcsin_le_pi_div_two _).trans (div_le_self pi_pos.le one_le_two))
end | lemma | real.arcsin_eq_arccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse.lean | [
"analysis.special_functions.trigonometric.basic",
"topology.algebra.order.proj_Icc"
] | [
"div_le_self",
"one_le_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_arcsin_aux {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
has_strict_deriv_at arcsin (1 / sqrt (1 - x ^ 2)) x ∧ cont_diff_at ℝ ⊤ arcsin x | begin
cases h₁.lt_or_lt with h₁ h₁,
{ have : 1 - x ^ 2 < 0, by nlinarith [h₁],
rw [sqrt_eq_zero'.2 this.le, div_zero],
have : arcsin =ᶠ[𝓝 x] λ _, -(π / 2) :=
(gt_mem_nhds h₁).mono (λ y hy, arcsin_of_le_neg_one hy.le),
exact ⟨(has_strict_deriv_at_const _ _).congr_of_eventually_eq this.symm,
... | lemma | real.deriv_arcsin_aux | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_at",
"div_zero",
"gt_mem_nhds",
"has_strict_deriv_at",
"has_strict_deriv_at_const",
"lt_mem_nhds",
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_arcsin {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
has_strict_deriv_at arcsin (1 / sqrt (1 - x ^ 2)) x | (deriv_arcsin_aux h₁ h₂).1 | lemma | real.has_strict_deriv_at_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_arcsin {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
has_deriv_at arcsin (1 / sqrt (1 - x ^ 2)) x | (has_strict_deriv_at_arcsin h₁ h₂).has_deriv_at | lemma | real.has_deriv_at_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_arcsin {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) {n : ℕ∞} :
cont_diff_at ℝ n arcsin x | (deriv_arcsin_aux h₁ h₂).2.of_le le_top | lemma | real.cont_diff_at_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_at",
"le_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_arcsin_Ici {x : ℝ} (h : x ≠ -1) :
has_deriv_within_at arcsin (1 / sqrt (1 - x ^ 2)) (Ici x) x | begin
rcases em (x = 1) with (rfl|h'),
{ convert (has_deriv_within_at_const _ _ (π / 2)).congr _ _;
simp [arcsin_of_one_le] { contextual := tt } },
{ exact (has_deriv_at_arcsin h h').has_deriv_within_at }
end | lemma | real.has_deriv_within_at_arcsin_Ici | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"em",
"has_deriv_within_at",
"has_deriv_within_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_arcsin_Iic {x : ℝ} (h : x ≠ 1) :
has_deriv_within_at arcsin (1 / sqrt (1 - x ^ 2)) (Iic x) x | begin
rcases em (x = -1) with (rfl|h'),
{ convert (has_deriv_within_at_const _ _ (-(π / 2))).congr _ _;
simp [arcsin_of_le_neg_one] { contextual := tt } },
{ exact (has_deriv_at_arcsin h' h).has_deriv_within_at }
end | lemma | real.has_deriv_within_at_arcsin_Iic | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"em",
"has_deriv_within_at",
"has_deriv_within_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_arcsin_Ici {x : ℝ} :
differentiable_within_at ℝ arcsin (Ici x) x ↔ x ≠ -1 | begin
refine ⟨_, λ h, (has_deriv_within_at_arcsin_Ici h).differentiable_within_at⟩,
rintro h rfl,
have : sin ∘ arcsin =ᶠ[𝓝[≥] (-1 : ℝ)] id,
{ filter_upwards [Icc_mem_nhds_within_Ici ⟨le_rfl, neg_lt_self (zero_lt_one' ℝ)⟩]
with x using sin_arcsin', },
have := h.has_deriv_within_at.sin.congr_of_eventuall... | lemma | real.differentiable_within_at_arcsin_Ici | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"Icc_mem_nhds_within_Ici",
"differentiable_within_at",
"has_deriv_within_at_id",
"unique_diff_on_Ici",
"zero_lt_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_arcsin_Iic {x : ℝ} :
differentiable_within_at ℝ arcsin (Iic x) x ↔ x ≠ 1 | begin
refine ⟨λ h, _, λ h, (has_deriv_within_at_arcsin_Iic h).differentiable_within_at⟩,
rw [← neg_neg x, ← image_neg_Ici] at h,
have := (h.comp (-x) differentiable_within_at_id.neg (maps_to_image _ _)).neg,
simpa [(∘), differentiable_within_at_arcsin_Ici] using this
end | lemma | real.differentiable_within_at_arcsin_Iic | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_arcsin {x : ℝ} :
differentiable_at ℝ arcsin x ↔ x ≠ -1 ∧ x ≠ 1 | ⟨λ h, ⟨differentiable_within_at_arcsin_Ici.1 h.differentiable_within_at,
differentiable_within_at_arcsin_Iic.1 h.differentiable_within_at⟩,
λ h, (has_deriv_at_arcsin h.1 h.2).differentiable_at⟩ | lemma | real.differentiable_at_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_arcsin : deriv arcsin = λ x, 1 / sqrt (1 - x ^ 2) | begin
funext x,
by_cases h : x ≠ -1 ∧ x ≠ 1,
{ exact (has_deriv_at_arcsin h.1 h.2).deriv },
{ rw [deriv_zero_of_not_differentiable_at (mt differentiable_at_arcsin.1 h)],
simp only [not_and_distrib, ne.def, not_not] at h,
rcases h with (rfl|rfl); simp }
end | lemma | real.deriv_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"deriv",
"deriv_zero_of_not_differentiable_at",
"not_and_distrib",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_arcsin : differentiable_on ℝ arcsin {-1, 1}ᶜ | λ x hx, (differentiable_at_arcsin.2
⟨λ h, hx (or.inl h), λ h, hx (or.inr h)⟩).differentiable_within_at | lemma | real.differentiable_on_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable_on",
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_arcsin {n : ℕ∞} :
cont_diff_on ℝ n arcsin {-1, 1}ᶜ | λ x hx, (cont_diff_at_arcsin (mt or.inl hx) (mt or.inr hx)).cont_diff_within_at | lemma | real.cont_diff_on_arcsin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_on",
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_arcsin_iff {x : ℝ} {n : ℕ∞} :
cont_diff_at ℝ n arcsin x ↔ n = 0 ∨ (x ≠ -1 ∧ x ≠ 1) | ⟨λ h, or_iff_not_imp_left.2 $ λ hn, differentiable_at_arcsin.1 $ h.differentiable_at $
enat.one_le_iff_ne_zero.2 hn,
λ h, h.elim (λ hn, hn.symm ▸ (cont_diff_zero.2 continuous_arcsin).cont_diff_at) $
λ hx, cont_diff_at_arcsin hx.1 hx.2⟩ | lemma | real.cont_diff_at_arcsin_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_arccos {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
has_strict_deriv_at arccos (-(1 / sqrt (1 - x ^ 2))) x | (has_strict_deriv_at_arcsin h₁ h₂).const_sub (π / 2) | lemma | real.has_strict_deriv_at_arccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_arccos {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) :
has_deriv_at arccos (-(1 / sqrt (1 - x ^ 2))) x | (has_deriv_at_arcsin h₁ h₂).const_sub (π / 2) | lemma | real.has_deriv_at_arccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_arccos {x : ℝ} (h₁ : x ≠ -1) (h₂ : x ≠ 1) {n : ℕ∞} :
cont_diff_at ℝ n arccos x | cont_diff_at_const.sub (cont_diff_at_arcsin h₁ h₂) | lemma | real.cont_diff_at_arccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_arccos_Ici {x : ℝ} (h : x ≠ -1) :
has_deriv_within_at arccos (-(1 / sqrt (1 - x ^ 2))) (Ici x) x | (has_deriv_within_at_arcsin_Ici h).const_sub _ | lemma | real.has_deriv_within_at_arccos_Ici | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at_arccos_Iic {x : ℝ} (h : x ≠ 1) :
has_deriv_within_at arccos (-(1 / sqrt (1 - x ^ 2))) (Iic x) x | (has_deriv_within_at_arcsin_Iic h).const_sub _ | lemma | real.has_deriv_within_at_arccos_Iic | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"has_deriv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_arccos_Ici {x : ℝ} :
differentiable_within_at ℝ arccos (Ici x) x ↔ x ≠ -1 | (differentiable_within_at_const_sub_iff _).trans differentiable_within_at_arcsin_Ici | lemma | real.differentiable_within_at_arccos_Ici | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable_within_at",
"differentiable_within_at_const_sub_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at_arccos_Iic {x : ℝ} :
differentiable_within_at ℝ arccos (Iic x) x ↔ x ≠ 1 | (differentiable_within_at_const_sub_iff _).trans differentiable_within_at_arcsin_Iic | lemma | real.differentiable_within_at_arccos_Iic | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable_within_at",
"differentiable_within_at_const_sub_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at_arccos {x : ℝ} :
differentiable_at ℝ arccos x ↔ x ≠ -1 ∧ x ≠ 1 | (differentiable_at_const_sub_iff _).trans differentiable_at_arcsin | lemma | real.differentiable_at_arccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable_at",
"differentiable_at_const_sub_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_arccos : deriv arccos = λ x, -(1 / sqrt (1 - x ^ 2)) | funext $ λ x, (deriv_const_sub _).trans $ by simp only [deriv_arcsin] | lemma | real.deriv_arccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"deriv",
"deriv_const_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on_arccos : differentiable_on ℝ arccos {-1, 1}ᶜ | differentiable_on_arcsin.const_sub _ | lemma | real.differentiable_on_arccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on_arccos {n : ℕ∞} :
cont_diff_on ℝ n arccos {-1, 1}ᶜ | cont_diff_on_const.sub cont_diff_on_arcsin | lemma | real.cont_diff_on_arccos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_arccos_iff {x : ℝ} {n : ℕ∞} :
cont_diff_at ℝ n arccos x ↔ n = 0 ∨ (x ≠ -1 ∧ x ≠ 1) | by refine iff.trans ⟨λ h, _, λ h, _⟩ cont_diff_at_arcsin_iff;
simpa [arccos] using (@cont_diff_at_const _ _ _ _ _ _ _ _ _ _ (π / 2)).sub h | lemma | real.cont_diff_at_arccos_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/inverse_deriv.lean | [
"analysis.special_functions.trigonometric.inverse",
"analysis.special_functions.trigonometric.deriv"
] | [
"cont_diff_at",
"cont_diff_at_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complex.has_sum_cos' (z : ℂ) :
has_sum (λ n : ℕ, (z * complex.I) ^ (2 * n) / ↑(2 * n)!) (complex.cos z) | begin
rw [complex.cos, complex.exp_eq_exp_ℂ],
have := ((exp_series_div_has_sum_exp ℂ (z * complex.I)).add
(exp_series_div_has_sum_exp ℂ (-z * complex.I))).div_const 2,
replace := ((nat.div_mod_equiv 2)).symm.has_sum_iff.mpr this,
dsimp [function.comp] at this,
simp_rw [←mul_comm 2 _] at this,
refi... | lemma | complex.has_sum_cos' | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/series.lean | [
"analysis.special_functions.exponential"
] | [
"complex.I",
"complex.cos",
"complex.exp_eq_exp_ℂ",
"exp_series_div_has_sum_exp",
"fin.coe_one",
"fin.coe_zero",
"has_sum",
"has_sum_fintype",
"mul_div_cancel_left",
"mul_neg",
"mul_pow",
"nat.div_mod_equiv",
"neg_div",
"neg_mul",
"neg_sq",
"pow_mul",
"pow_succ'",
"two_ne_zero",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complex.has_sum_sin' (z : ℂ) :
has_sum (λ n : ℕ, (z * complex.I) ^ (2 * n + 1) / ↑(2 * n + 1)! / complex.I) (complex.sin z) | begin
rw [complex.sin, complex.exp_eq_exp_ℂ],
have := (((exp_series_div_has_sum_exp ℂ (-z * complex.I)).sub
(exp_series_div_has_sum_exp ℂ (z * complex.I))).mul_right complex.I).div_const 2,
replace := ((nat.div_mod_equiv 2)).symm.has_sum_iff.mpr this,
dsimp [function.comp] at this,
simp_rw [←mul_com... | lemma | complex.has_sum_sin' | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/series.lean | [
"analysis.special_functions.exponential"
] | [
"complex.I",
"complex.div_I",
"complex.exp_eq_exp_ℂ",
"complex.sin",
"exp_series_div_has_sum_exp",
"fin.coe_one",
"fin.coe_zero",
"has_sum",
"has_sum_fintype",
"mul_assoc",
"mul_div_cancel_left",
"mul_neg",
"mul_pow",
"nat.div_mod_equiv",
"neg_div",
"neg_mul",
"neg_sq",
"pow_mul",
... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complex.has_sum_cos (z : ℂ) :
has_sum (λ n : ℕ, ((-1) ^ n) * z ^ (2 * n) / ↑(2 * n)!) (complex.cos z) | begin
convert complex.has_sum_cos' z using 1,
simp_rw [mul_pow, pow_mul, complex.I_sq, mul_comm]
end | lemma | complex.has_sum_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/series.lean | [
"analysis.special_functions.exponential"
] | [
"complex.I_sq",
"complex.cos",
"complex.has_sum_cos'",
"has_sum",
"mul_comm",
"mul_pow",
"pow_mul"
] | The power series expansion of `complex.cos`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complex.has_sum_sin (z : ℂ) :
has_sum (λ n : ℕ, ((-1) ^ n) * z ^ (2 * n + 1) / ↑(2 * n + 1)!) (complex.sin z) | begin
convert complex.has_sum_sin' z using 1,
simp_rw [mul_pow, pow_succ', pow_mul, complex.I_sq, ←mul_assoc,
mul_div_assoc, div_right_comm, div_self complex.I_ne_zero, mul_comm _ ((-1 : ℂ)^_), mul_one_div,
mul_div_assoc, mul_assoc]
end | lemma | complex.has_sum_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/series.lean | [
"analysis.special_functions.exponential"
] | [
"complex.I_ne_zero",
"complex.I_sq",
"complex.has_sum_sin'",
"complex.sin",
"div_right_comm",
"div_self",
"has_sum",
"mul_assoc",
"mul_comm",
"mul_div_assoc",
"mul_one_div",
"mul_pow",
"pow_mul",
"pow_succ'"
] | The power series expansion of `complex.sin`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
complex.cos_eq_tsum' (z : ℂ) :
complex.cos z = ∑' n : ℕ, (z * complex.I) ^ (2 * n) / ↑(2 * n)! | (complex.has_sum_cos' z).tsum_eq.symm | lemma | complex.cos_eq_tsum' | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/series.lean | [
"analysis.special_functions.exponential"
] | [
"complex.I",
"complex.cos",
"complex.has_sum_cos'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complex.sin_eq_tsum' (z : ℂ) :
complex.sin z = ∑' n : ℕ, (z * complex.I) ^ (2 * n + 1) / ↑(2 * n + 1)! / complex.I | (complex.has_sum_sin' z).tsum_eq.symm | lemma | complex.sin_eq_tsum' | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/series.lean | [
"analysis.special_functions.exponential"
] | [
"complex.I",
"complex.has_sum_sin'",
"complex.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complex.cos_eq_tsum (z : ℂ) :
complex.cos z = ∑' n : ℕ, ((-1) ^ n) * z ^ (2 * n) / ↑(2 * n)! | (complex.has_sum_cos z).tsum_eq.symm | lemma | complex.cos_eq_tsum | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/series.lean | [
"analysis.special_functions.exponential"
] | [
"complex.cos",
"complex.has_sum_cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complex.sin_eq_tsum (z : ℂ) :
complex.sin z = ∑' n : ℕ, ((-1) ^ n) * z ^ (2 * n + 1) / ↑(2 * n + 1)! | (complex.has_sum_sin z).tsum_eq.symm | lemma | complex.sin_eq_tsum | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/series.lean | [
"analysis.special_functions.exponential"
] | [
"complex.has_sum_sin",
"complex.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.has_sum_cos (r : ℝ) :
has_sum (λ n : ℕ, ((-1) ^ n) * r ^ (2 * n) / ↑(2 * n)!) (real.cos r) | by exact_mod_cast complex.has_sum_cos r | lemma | real.has_sum_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/series.lean | [
"analysis.special_functions.exponential"
] | [
"complex.has_sum_cos",
"has_sum",
"real.cos"
] | The power series expansion of `real.cos`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.has_sum_sin (r : ℝ) :
has_sum (λ n : ℕ, ((-1) ^ n) * r ^ (2 * n + 1) / ↑(2 * n + 1)!) (real.sin r) | by exact_mod_cast complex.has_sum_sin r | lemma | real.has_sum_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/series.lean | [
"analysis.special_functions.exponential"
] | [
"complex.has_sum_sin",
"has_sum",
"real.sin"
] | The power series expansion of `real.sin`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.cos_eq_tsum (r : ℝ) :
real.cos r = ∑' n : ℕ, ((-1) ^ n) * r ^ (2 * n) / ↑(2 * n)! | (real.has_sum_cos r).tsum_eq.symm | lemma | real.cos_eq_tsum | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/series.lean | [
"analysis.special_functions.exponential"
] | [
"real.cos",
"real.has_sum_cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.sin_eq_tsum (r : ℝ) :
real.sin r = ∑' n : ℕ, ((-1) ^ n) * r ^ (2 * n + 1) / ↑(2 * n + 1)! | (real.has_sum_sin r).tsum_eq.symm | lemma | real.sin_eq_tsum | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/series.lean | [
"analysis.special_functions.exponential"
] | [
"real.has_sum_sin",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_inverse_at_top_nhds_0_nat : tendsto (λ n : ℕ, (n : ℝ)⁻¹) at_top (𝓝 0) | tendsto_inv_at_top_zero.comp tendsto_coe_nat_at_top_at_top | lemma | tendsto_inverse_at_top_nhds_0_nat | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"tendsto_coe_nat_at_top_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_const_div_at_top_nhds_0_nat (C : ℝ) : tendsto (λ n : ℕ, C / n) at_top (𝓝 0) | by simpa only [mul_zero] using tendsto_const_nhds.mul tendsto_inverse_at_top_nhds_0_nat | lemma | tendsto_const_div_at_top_nhds_0_nat | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"mul_zero",
"tendsto_inverse_at_top_nhds_0_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnreal.tendsto_inverse_at_top_nhds_0_nat : tendsto (λ n : ℕ, (n : ℝ≥0)⁻¹) at_top (𝓝 0) | by { rw ← nnreal.tendsto_coe, exact tendsto_inverse_at_top_nhds_0_nat } | lemma | nnreal.tendsto_inverse_at_top_nhds_0_nat | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"nnreal.tendsto_coe",
"tendsto_inverse_at_top_nhds_0_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnreal.tendsto_const_div_at_top_nhds_0_nat (C : ℝ≥0) :
tendsto (λ n : ℕ, C / n) at_top (𝓝 0) | by simpa using tendsto_const_nhds.mul nnreal.tendsto_inverse_at_top_nhds_0_nat | lemma | nnreal.tendsto_const_div_at_top_nhds_0_nat | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"nnreal.tendsto_inverse_at_top_nhds_0_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_one_div_add_at_top_nhds_0_nat :
tendsto (λ n : ℕ, 1 / ((n : ℝ) + 1)) at_top (𝓝 0) | suffices tendsto (λ n : ℕ, 1 / (↑(n + 1) : ℝ)) at_top (𝓝 0), by simpa,
(tendsto_add_at_top_iff_nat 1).2 (tendsto_const_div_at_top_nhds_0_nat 1) | lemma | tendsto_one_div_add_at_top_nhds_0_nat | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"tendsto_const_div_at_top_nhds_0_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_coe_nat_div_add_at_top
{𝕜 : Type*} [division_ring 𝕜] [topological_space 𝕜] [char_zero 𝕜] [algebra ℝ 𝕜]
[has_continuous_smul ℝ 𝕜] [topological_division_ring 𝕜]
(x : 𝕜) :
tendsto (λ n:ℕ, (n:𝕜) / (n + x)) at_top (𝓝 1) | begin
refine tendsto.congr' ((eventually_ne_at_top 0).mp (eventually_of_forall (λ n hn, _))) _,
{ exact λ n:ℕ, 1 / (1 + x / n) },
{ field_simp [nat.cast_ne_zero.mpr hn] },
{ have : 𝓝 (1:𝕜) = 𝓝 (1 / (1 + x * ↑(0:ℝ))),
by rw [algebra_map.coe_zero, mul_zero, add_zero, div_one],
rw this,
refine tends... | lemma | tendsto_coe_nat_div_add_at_top | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"algebra",
"algebra_map",
"algebra_map.coe_zero",
"char_zero",
"continuous_algebra_map",
"div_eq_mul_inv",
"div_one",
"division_ring",
"has_continuous_smul",
"map_nat_cast",
"mul_zero",
"tendsto_inverse_at_top_nhds_0_nat",
"topological_division_ring",
"topological_space"
] | The limit of `n / (n + x)` is 1, for any constant `x` (valid in `ℝ` or any topological division
algebra over `ℝ`, e.g., `ℂ`).
TODO: introduce a typeclass saying that `1 / n` tends to 0 at top, making it possible to get this
statement simultaneously on `ℚ`, `ℝ` and `ℂ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_add_one_pow_at_top_at_top_of_pos [linear_ordered_semiring α] [archimedean α] {r : α}
(h : 0 < r) :
tendsto (λ n:ℕ, (r + 1)^n) at_top at_top | tendsto_at_top_at_top_of_monotone' (λ n m, pow_le_pow (le_add_of_nonneg_left (le_of_lt h))) $
not_bdd_above_iff.2 $ λ x, set.exists_range_iff.2 $ add_one_pow_unbounded_of_pos _ h | lemma | tendsto_add_one_pow_at_top_at_top_of_pos | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"add_one_pow_unbounded_of_pos",
"archimedean",
"linear_ordered_semiring",
"pow_le_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_pow_at_top_at_top_of_one_lt [linear_ordered_ring α] [archimedean α]
{r : α} (h : 1 < r) :
tendsto (λn:ℕ, r ^ n) at_top at_top | sub_add_cancel r 1 ▸ tendsto_add_one_pow_at_top_at_top_of_pos (sub_pos.2 h) | lemma | tendsto_pow_at_top_at_top_of_one_lt | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"archimedean",
"linear_ordered_ring",
"tendsto_add_one_pow_at_top_at_top_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nat.tendsto_pow_at_top_at_top_of_one_lt {m : ℕ} (h : 1 < m) :
tendsto (λn:ℕ, m ^ n) at_top at_top | tsub_add_cancel_of_le (le_of_lt h) ▸
tendsto_add_one_pow_at_top_at_top_of_pos (tsub_pos_of_lt h) | lemma | nat.tendsto_pow_at_top_at_top_of_one_lt | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"tendsto_add_one_pow_at_top_at_top_of_pos",
"tsub_add_cancel_of_le",
"tsub_pos_of_lt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_pow_at_top_nhds_0_of_lt_1 {𝕜 : Type*} [linear_ordered_field 𝕜] [archimedean 𝕜]
[topological_space 𝕜] [order_topology 𝕜] {r : 𝕜} (h₁ : 0 ≤ r) (h₂ : r < 1) :
tendsto (λn:ℕ, r^n) at_top (𝓝 0) | h₁.eq_or_lt.elim
(assume : 0 = r,
(tendsto_add_at_top_iff_nat 1).mp $ by simp [pow_succ, ← this, tendsto_const_nhds])
(assume : 0 < r,
have tendsto (λn, (r⁻¹ ^ n)⁻¹) at_top (𝓝 0),
from tendsto_inv_at_top_zero.comp
(tendsto_pow_at_top_at_top_of_one_lt $ one_lt_inv this h₂),
this.congr (λ n... | lemma | tendsto_pow_at_top_nhds_0_of_lt_1 | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"archimedean",
"linear_ordered_field",
"one_lt_inv",
"order_topology",
"pow_succ",
"tendsto_const_nhds",
"tendsto_pow_at_top_at_top_of_one_lt",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_pow_at_top_nhds_within_0_of_lt_1 {𝕜 : Type*} [linear_ordered_field 𝕜] [archimedean 𝕜]
[topological_space 𝕜] [order_topology 𝕜] {r : 𝕜} (h₁ : 0 < r) (h₂ : r < 1) :
tendsto (λn:ℕ, r^n) at_top (𝓝[>] 0) | tendsto_inf.2 ⟨tendsto_pow_at_top_nhds_0_of_lt_1 h₁.le h₂,
tendsto_principal.2 $ eventually_of_forall $ λ n, pow_pos h₁ _⟩ | lemma | tendsto_pow_at_top_nhds_within_0_of_lt_1 | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"archimedean",
"linear_ordered_field",
"order_topology",
"pow_pos",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity_basis_dist_pow_of_lt_1 {α : Type*} [pseudo_metric_space α]
{r : ℝ} (h₀ : 0 < r) (h₁ : r < 1) :
(𝓤 α).has_basis (λ k : ℕ, true) (λ k, {p : α × α | dist p.1 p.2 < r ^ k}) | metric.mk_uniformity_basis (λ i _, pow_pos h₀ _) $ λ ε ε0,
(exists_pow_lt_of_lt_one ε0 h₁).imp $ λ k hk, ⟨trivial, hk.le⟩ | lemma | uniformity_basis_dist_pow_of_lt_1 | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"exists_pow_lt_of_lt_one",
"metric.mk_uniformity_basis",
"pow_pos",
"pseudo_metric_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
geom_lt {u : ℕ → ℝ} {c : ℝ} (hc : 0 ≤ c) {n : ℕ} (hn : 0 < n)
(h : ∀ k < n, c * u k < u (k + 1)) :
c ^ n * u 0 < u n | begin
refine (monotone_mul_left_of_nonneg hc).seq_pos_lt_seq_of_le_of_lt hn _ _ h,
{ simp },
{ simp [pow_succ, mul_assoc, le_refl] }
end | lemma | geom_lt | analysis.specific_limits | src/analysis/specific_limits/basic.lean | [
"algebra.geom_sum",
"order.filter.archimedean",
"order.iterate",
"topology.instances.ennreal",
"topology.algebra.algebra"
] | [
"monotone_mul_left_of_nonneg",
"mul_assoc",
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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