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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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sin_sub_pi (x : ℝ) : sin (x - π) = -sin x | sin_antiperiodic.sub_eq x | lemma | real.sin_sub_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_sub_two_pi (x : ℝ) : sin (x - 2 * π) = sin x | sin_periodic.sub_eq x | lemma | real.sin_sub_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_pi_sub (x : ℝ) : sin (π - x) = sin x | neg_neg (sin x) ▸ sin_neg x ▸ sin_antiperiodic.sub_eq' | lemma | real.sin_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_two_pi_sub (x : ℝ) : sin (2 * π - x) = -sin x | sin_neg x ▸ sin_periodic.sub_eq' | lemma | real.sin_two_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_nat_mul_pi (n : ℕ) : sin (n * π) = 0 | sin_antiperiodic.nat_mul_eq_of_eq_zero sin_zero n | lemma | real.sin_nat_mul_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_int_mul_pi (n : ℤ) : sin (n * π) = 0 | sin_antiperiodic.int_mul_eq_of_eq_zero sin_zero n | lemma | real.sin_int_mul_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_add_nat_mul_two_pi (x : ℝ) (n : ℕ) : sin (x + n * (2 * π)) = sin x | sin_periodic.nat_mul n x | lemma | real.sin_add_nat_mul_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_add_int_mul_two_pi (x : ℝ) (n : ℤ) : sin (x + n * (2 * π)) = sin x | sin_periodic.int_mul n x | lemma | real.sin_add_int_mul_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_sub_nat_mul_two_pi (x : ℝ) (n : ℕ) : sin (x - n * (2 * π)) = sin x | sin_periodic.sub_nat_mul_eq n | lemma | real.sin_sub_nat_mul_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_sub_int_mul_two_pi (x : ℝ) (n : ℤ) : sin (x - n * (2 * π)) = sin x | sin_periodic.sub_int_mul_eq n | lemma | real.sin_sub_int_mul_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_nat_mul_two_pi_sub (x : ℝ) (n : ℕ) : sin (n * (2 * π) - x) = -sin x | sin_neg x ▸ sin_periodic.nat_mul_sub_eq n | lemma | real.sin_nat_mul_two_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_int_mul_two_pi_sub (x : ℝ) (n : ℤ) : sin (n * (2 * π) - x) = -sin x | sin_neg x ▸ sin_periodic.int_mul_sub_eq n | lemma | real.sin_int_mul_two_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_antiperiodic : function.antiperiodic cos π | by simp [cos_add] | lemma | real.cos_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 | |
cos_periodic : function.periodic cos (2 * π) | cos_antiperiodic.periodic | lemma | real.cos_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 | |
cos_add_pi (x : ℝ) : cos (x + π) = -cos x | cos_antiperiodic x | lemma | real.cos_add_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_add_two_pi (x : ℝ) : cos (x + 2 * π) = cos x | cos_periodic x | lemma | real.cos_add_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_sub_pi (x : ℝ) : cos (x - π) = -cos x | cos_antiperiodic.sub_eq x | lemma | real.cos_sub_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_sub_two_pi (x : ℝ) : cos (x - 2 * π) = cos x | cos_periodic.sub_eq x | lemma | real.cos_sub_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_pi_sub (x : ℝ) : cos (π - x) = -cos x | cos_neg x ▸ cos_antiperiodic.sub_eq' | lemma | real.cos_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_two_pi_sub (x : ℝ) : cos (2 * π - x) = cos x | cos_neg x ▸ cos_periodic.sub_eq' | lemma | real.cos_two_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_nat_mul_two_pi (n : ℕ) : cos (n * (2 * π)) = 1 | (cos_periodic.nat_mul_eq n).trans cos_zero | lemma | real.cos_nat_mul_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_int_mul_two_pi (n : ℤ) : cos (n * (2 * π)) = 1 | (cos_periodic.int_mul_eq n).trans cos_zero | lemma | real.cos_int_mul_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_add_nat_mul_two_pi (x : ℝ) (n : ℕ) : cos (x + n * (2 * π)) = cos x | cos_periodic.nat_mul n x | lemma | real.cos_add_nat_mul_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_add_int_mul_two_pi (x : ℝ) (n : ℤ) : cos (x + n * (2 * π)) = cos x | cos_periodic.int_mul n x | lemma | real.cos_add_int_mul_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_sub_nat_mul_two_pi (x : ℝ) (n : ℕ) : cos (x - n * (2 * π)) = cos x | cos_periodic.sub_nat_mul_eq n | lemma | real.cos_sub_nat_mul_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_sub_int_mul_two_pi (x : ℝ) (n : ℤ) : cos (x - n * (2 * π)) = cos x | cos_periodic.sub_int_mul_eq n | lemma | real.cos_sub_int_mul_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_nat_mul_two_pi_sub (x : ℝ) (n : ℕ) : cos (n * (2 * π) - x) = cos x | cos_neg x ▸ cos_periodic.nat_mul_sub_eq n | lemma | real.cos_nat_mul_two_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_int_mul_two_pi_sub (x : ℝ) (n : ℤ) : cos (n * (2 * π) - x) = cos x | cos_neg x ▸ cos_periodic.int_mul_sub_eq n | lemma | real.cos_int_mul_two_pi_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_nat_mul_two_pi_add_pi (n : ℕ) : cos (n * (2 * π) + π) = -1 | by simpa only [cos_zero] using (cos_periodic.nat_mul n).add_antiperiod_eq cos_antiperiodic | lemma | real.cos_nat_mul_two_pi_add_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_int_mul_two_pi_add_pi (n : ℤ) : cos (n * (2 * π) + π) = -1 | by simpa only [cos_zero] using (cos_periodic.int_mul n).add_antiperiod_eq cos_antiperiodic | lemma | real.cos_int_mul_two_pi_add_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_nat_mul_two_pi_sub_pi (n : ℕ) : cos (n * (2 * π) - π) = -1 | by simpa only [cos_zero] using (cos_periodic.nat_mul n).sub_antiperiod_eq cos_antiperiodic | lemma | real.cos_nat_mul_two_pi_sub_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_int_mul_two_pi_sub_pi (n : ℤ) : cos (n * (2 * π) - π) = -1 | by simpa only [cos_zero] using (cos_periodic.int_mul n).sub_antiperiod_eq cos_antiperiodic | lemma | real.cos_int_mul_two_pi_sub_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_pos_of_pos_of_lt_pi {x : ℝ} (h0x : 0 < x) (hxp : x < π) : 0 < sin x | if hx2 : x ≤ 2 then sin_pos_of_pos_of_le_two h0x hx2
else
have (2 : ℝ) + 2 = 4, from rfl,
have π - x ≤ 2, from sub_le_iff_le_add.2
(le_trans pi_le_four (this ▸ add_le_add_left (le_of_not_ge hx2) _)),
sin_pi_sub x ▸ sin_pos_of_pos_of_le_two (sub_pos.2 hxp) this | lemma | real.sin_pos_of_pos_of_lt_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_pos_of_mem_Ioo {x : ℝ} (hx : x ∈ Ioo 0 π) : 0 < sin x | sin_pos_of_pos_of_lt_pi hx.1 hx.2 | lemma | real.sin_pos_of_mem_Ioo | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_nonneg_of_mem_Icc {x : ℝ} (hx : x ∈ Icc 0 π) : 0 ≤ sin x | begin
rw ← closure_Ioo pi_ne_zero.symm at hx,
exact closure_lt_subset_le continuous_const continuous_sin
(closure_mono (λ y, sin_pos_of_mem_Ioo) hx)
end | lemma | real.sin_nonneg_of_mem_Icc | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"closure_Ioo",
"closure_lt_subset_le",
"closure_mono",
"continuous_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_nonneg_of_nonneg_of_le_pi {x : ℝ} (h0x : 0 ≤ x) (hxp : x ≤ π) : 0 ≤ sin x | sin_nonneg_of_mem_Icc ⟨h0x, hxp⟩ | lemma | real.sin_nonneg_of_nonneg_of_le_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_neg_of_neg_of_neg_pi_lt {x : ℝ} (hx0 : x < 0) (hpx : -π < x) : sin x < 0 | neg_pos.1 $ sin_neg x ▸ sin_pos_of_pos_of_lt_pi (neg_pos.2 hx0) (neg_lt.1 hpx) | lemma | real.sin_neg_of_neg_of_neg_pi_lt | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_nonpos_of_nonnpos_of_neg_pi_le {x : ℝ} (hx0 : x ≤ 0) (hpx : -π ≤ x) : sin x ≤ 0 | neg_nonneg.1 $ sin_neg x ▸ sin_nonneg_of_nonneg_of_le_pi (neg_nonneg.2 hx0) (neg_le.1 hpx) | lemma | real.sin_nonpos_of_nonnpos_of_neg_pi_le | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_pi_div_two : sin (π / 2) = 1 | have sin (π / 2) = 1 ∨ sin (π / 2) = -1 :=
by simpa [sq, mul_self_eq_one_iff] using sin_sq_add_cos_sq (π / 2),
this.resolve_right
(λ h, (show ¬(0 : ℝ) < -1, by norm_num) $
h ▸ sin_pos_of_pos_of_lt_pi pi_div_two_pos (half_lt_self pi_pos)) | lemma | real.sin_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"mul_self_eq_one_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_add_pi_div_two (x : ℝ) : sin (x + π / 2) = cos x | by simp [sin_add] | lemma | real.sin_add_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_sub_pi_div_two (x : ℝ) : sin (x - π / 2) = -cos x | by simp [sub_eq_add_neg, sin_add] | lemma | real.sin_sub_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_pi_div_two_sub (x : ℝ) : sin (π / 2 - x) = cos x | by simp [sub_eq_add_neg, sin_add] | lemma | real.sin_pi_div_two_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_add_pi_div_two (x : ℝ) : cos (x + π / 2) = -sin x | by simp [cos_add] | lemma | real.cos_add_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_sub_pi_div_two (x : ℝ) : cos (x - π / 2) = sin x | by simp [sub_eq_add_neg, cos_add] | lemma | real.cos_sub_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_pi_div_two_sub (x : ℝ) : cos (π / 2 - x) = sin x | by rw [← cos_neg, neg_sub, cos_sub_pi_div_two] | lemma | real.cos_pi_div_two_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_pos_of_mem_Ioo {x : ℝ} (hx : x ∈ Ioo (-(π / 2)) (π / 2)) : 0 < cos x | sin_add_pi_div_two x ▸ sin_pos_of_mem_Ioo ⟨by linarith [hx.1], by linarith [hx.2]⟩ | lemma | real.cos_pos_of_mem_Ioo | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_nonneg_of_mem_Icc {x : ℝ} (hx : x ∈ Icc (-(π / 2)) (π / 2)) : 0 ≤ cos x | sin_add_pi_div_two x ▸ sin_nonneg_of_mem_Icc ⟨by linarith [hx.1], by linarith [hx.2]⟩ | lemma | real.cos_nonneg_of_mem_Icc | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_nonneg_of_neg_pi_div_two_le_of_le {x : ℝ} (hl : -(π / 2) ≤ x) (hu : x ≤ π / 2) :
0 ≤ cos x | cos_nonneg_of_mem_Icc ⟨hl, hu⟩ | lemma | real.cos_nonneg_of_neg_pi_div_two_le_of_le | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_neg_of_pi_div_two_lt_of_lt {x : ℝ} (hx₁ : π / 2 < x) (hx₂ : x < π + π / 2) : cos x < 0 | neg_pos.1 $ cos_pi_sub x ▸ cos_pos_of_mem_Ioo ⟨by linarith, by linarith⟩ | lemma | real.cos_neg_of_pi_div_two_lt_of_lt | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_nonpos_of_pi_div_two_le_of_le {x : ℝ} (hx₁ : π / 2 ≤ x) (hx₂ : x ≤ π + π / 2) :
cos x ≤ 0 | neg_nonneg.1 $ cos_pi_sub x ▸ cos_nonneg_of_mem_Icc ⟨by linarith, by linarith⟩ | lemma | real.cos_nonpos_of_pi_div_two_le_of_le | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_eq_sqrt_one_sub_cos_sq {x : ℝ} (hl : 0 ≤ x) (hu : x ≤ π) :
sin x = sqrt (1 - cos x ^ 2) | by rw [← abs_sin_eq_sqrt_one_sub_cos_sq, abs_of_nonneg (sin_nonneg_of_nonneg_of_le_pi hl hu)] | lemma | real.sin_eq_sqrt_one_sub_cos_sq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"abs_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_eq_sqrt_one_sub_sin_sq {x : ℝ} (hl : -(π / 2) ≤ x) (hu : x ≤ π / 2) :
cos x = sqrt (1 - sin x ^ 2) | by rw [← abs_cos_eq_sqrt_one_sub_sin_sq, abs_of_nonneg (cos_nonneg_of_mem_Icc ⟨hl, hu⟩)] | lemma | real.cos_eq_sqrt_one_sub_sin_sq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"abs_of_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_eq_zero_iff_of_lt_of_lt {x : ℝ} (hx₁ : -π < x) (hx₂ : x < π) :
sin x = 0 ↔ x = 0 | ⟨λ h, le_antisymm
(le_of_not_gt (λ h0, lt_irrefl (0 : ℝ) $
calc 0 < sin x : sin_pos_of_pos_of_lt_pi h0 hx₂
... = 0 : h))
(le_of_not_gt (λ h0, lt_irrefl (0 : ℝ) $
calc 0 = sin x : h.symm
... < 0 : sin_neg_of_neg_of_neg_pi_lt h0 hx₁)),
λ h, by simp [h]⟩ | lemma | real.sin_eq_zero_iff_of_lt_of_lt | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_eq_zero_iff {x : ℝ} : sin x = 0 ↔ ∃ n : ℤ, (n : ℝ) * π = x | ⟨λ h, ⟨⌊x / π⌋, le_antisymm (sub_nonneg.1 (int.sub_floor_div_mul_nonneg _ pi_pos))
(sub_nonpos.1 $ le_of_not_gt $ λ h₃,
(sin_pos_of_pos_of_lt_pi h₃ (int.sub_floor_div_mul_lt _ pi_pos)).ne
(by simp [sub_eq_add_neg, sin_add, h, sin_int_mul_pi]))⟩,
λ ⟨n, hn⟩, hn ▸ sin_int_mul_pi _⟩ | lemma | real.sin_eq_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"int.sub_floor_div_mul_lt",
"int.sub_floor_div_mul_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_ne_zero_iff {x : ℝ} : sin x ≠ 0 ↔ ∀ n : ℤ, (n : ℝ) * π ≠ x | by rw [← not_exists, not_iff_not, sin_eq_zero_iff] | lemma | real.sin_ne_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"not_exists",
"not_iff_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_eq_zero_iff_cos_eq {x : ℝ} : sin x = 0 ↔ cos x = 1 ∨ cos x = -1 | by rw [← mul_self_eq_one_iff, ← sin_sq_add_cos_sq x,
sq, sq, ← sub_eq_iff_eq_add, sub_self];
exact ⟨λ h, by rw [h, mul_zero], eq_zero_of_mul_self_eq_zero ∘ eq.symm⟩ | lemma | real.sin_eq_zero_iff_cos_eq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"eq_zero_of_mul_self_eq_zero",
"mul_self_eq_one_iff",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_eq_one_iff (x : ℝ) : cos x = 1 ↔ ∃ n : ℤ, (n : ℝ) * (2 * π) = x | ⟨λ h, let ⟨n, hn⟩ := sin_eq_zero_iff.1 (sin_eq_zero_iff_cos_eq.2 (or.inl h)) in
⟨n / 2, (int.mod_two_eq_zero_or_one n).elim
(λ hn0, by rwa [← mul_assoc, ← @int.cast_two ℝ, ← int.cast_mul, int.div_mul_cancel
((int.dvd_iff_mod_eq_zero _ _).2 hn0)])
(λ hn1, by rw [← int.mod_add_div n 2, hn1, int.ca... | lemma | real.cos_eq_one_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"int.cast_add",
"int.cast_mul",
"int.cast_one",
"int.cast_two",
"int.div_mul_cancel",
"int.dvd_iff_mod_eq_zero",
"int.mod_add_div",
"int.mod_two_eq_zero_or_one",
"mul_assoc",
"mul_comm",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_eq_one_iff_of_lt_of_lt {x : ℝ} (hx₁ : -(2 * π) < x) (hx₂ : x < 2 * π) :
cos x = 1 ↔ x = 0 | ⟨λ h,
begin
rcases (cos_eq_one_iff _).1 h with ⟨n, rfl⟩,
rw [mul_lt_iff_lt_one_left two_pi_pos] at hx₂,
rw [neg_lt, neg_mul_eq_neg_mul, mul_lt_iff_lt_one_left two_pi_pos] at hx₁,
norm_cast at hx₁ hx₂,
obtain rfl : n = 0 := le_antisymm (by linarith) (by linarith),
simp
end,
... | lemma | real.cos_eq_one_iff_of_lt_of_lt | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"mul_lt_iff_lt_one_left",
"neg_mul_eq_neg_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_lt_cos_of_nonneg_of_le_pi_div_two {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y ≤ π / 2)
(hxy : x < y) :
cos y < cos x | begin
rw [← sub_lt_zero, cos_sub_cos],
have : 0 < sin ((y + x) / 2),
{ refine sin_pos_of_pos_of_lt_pi _ _; linarith },
have : 0 < sin ((y - x) / 2),
{ refine sin_pos_of_pos_of_lt_pi _ _; linarith },
nlinarith,
end | lemma | real.cos_lt_cos_of_nonneg_of_le_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_lt_cos_of_nonneg_of_le_pi {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y ≤ π) (hxy : x < y) :
cos y < cos x | match (le_total x (π / 2) : x ≤ π / 2 ∨ π / 2 ≤ x), le_total y (π / 2) with
| or.inl hx, or.inl hy := cos_lt_cos_of_nonneg_of_le_pi_div_two hx₁ hy hxy
| or.inl hx, or.inr hy := (lt_or_eq_of_le hx).elim
(λ hx, calc cos y ≤ 0 : cos_nonpos_of_pi_div_two_le_of_le hy (by linarith [pi_pos])
... < cos x : cos_pos_of_mem... | lemma | real.cos_lt_cos_of_nonneg_of_le_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_anti_on_cos : strict_anti_on cos (Icc 0 π) | λ x hx y hy hxy, cos_lt_cos_of_nonneg_of_le_pi hx.1 hy.2 hxy | lemma | real.strict_anti_on_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"strict_anti_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_le_cos_of_nonneg_of_le_pi {x y : ℝ} (hx₁ : 0 ≤ x) (hy₂ : y ≤ π) (hxy : x ≤ y) :
cos y ≤ cos x | (strict_anti_on_cos.le_iff_le ⟨hx₁.trans hxy, hy₂⟩ ⟨hx₁, hxy.trans hy₂⟩).2 hxy | lemma | real.cos_le_cos_of_nonneg_of_le_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_lt_sin_of_lt_of_le_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) ≤ x)
(hy₂ : y ≤ π / 2) (hxy : x < y) : sin x < sin y | by rw [← cos_sub_pi_div_two, ← cos_sub_pi_div_two, ← cos_neg (x - _), ← cos_neg (y - _)];
apply cos_lt_cos_of_nonneg_of_le_pi; linarith | lemma | real.sin_lt_sin_of_lt_of_le_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
strict_mono_on_sin : strict_mono_on sin (Icc (-(π / 2)) (π / 2)) | λ x hx y hy hxy, sin_lt_sin_of_lt_of_le_pi_div_two hx.1 hy.2 hxy | lemma | real.strict_mono_on_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"strict_mono_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_le_sin_of_le_of_le_pi_div_two {x y : ℝ} (hx₁ : -(π / 2) ≤ x)
(hy₂ : y ≤ π / 2) (hxy : x ≤ y) : sin x ≤ sin y | (strict_mono_on_sin.le_iff_le ⟨hx₁, hxy.trans hy₂⟩ ⟨hx₁.trans hxy, hy₂⟩).2 hxy | lemma | real.sin_le_sin_of_le_of_le_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inj_on_sin : inj_on sin (Icc (-(π / 2)) (π / 2)) | strict_mono_on_sin.inj_on | lemma | real.inj_on_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inj_on_cos : inj_on cos (Icc 0 π) | strict_anti_on_cos.inj_on | lemma | real.inj_on_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
surj_on_sin : surj_on sin (Icc (-(π / 2)) (π / 2)) (Icc (-1) 1) | by simpa only [sin_neg, sin_pi_div_two]
using intermediate_value_Icc (neg_le_self pi_div_two_pos.le) continuous_sin.continuous_on | lemma | real.surj_on_sin | 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 | |
surj_on_cos : surj_on cos (Icc 0 π) (Icc (-1) 1) | by simpa only [cos_zero, cos_pi]
using intermediate_value_Icc' pi_pos.le continuous_cos.continuous_on | lemma | real.surj_on_cos | 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 | |
sin_mem_Icc (x : ℝ) : sin x ∈ Icc (-1 : ℝ) 1 | ⟨neg_one_le_sin x, sin_le_one x⟩ | lemma | real.sin_mem_Icc | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_mem_Icc (x : ℝ) : cos x ∈ Icc (-1 : ℝ) 1 | ⟨neg_one_le_cos x, cos_le_one x⟩ | lemma | real.cos_mem_Icc | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maps_to_sin (s : set ℝ) : maps_to sin s (Icc (-1 : ℝ) 1) | λ x _, sin_mem_Icc x | lemma | real.maps_to_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
maps_to_cos (s : set ℝ) : maps_to cos s (Icc (-1 : ℝ) 1) | λ x _, cos_mem_Icc x | lemma | real.maps_to_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bij_on_sin : bij_on sin (Icc (-(π / 2)) (π / 2)) (Icc (-1) 1) | ⟨maps_to_sin _, inj_on_sin, surj_on_sin⟩ | lemma | real.bij_on_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bij_on_cos : bij_on cos (Icc 0 π) (Icc (-1) 1) | ⟨maps_to_cos _, inj_on_cos, surj_on_cos⟩ | lemma | real.bij_on_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_cos : range cos = (Icc (-1) 1 : set ℝ) | subset.antisymm (range_subset_iff.2 cos_mem_Icc) surj_on_cos.subset_range | lemma | real.range_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_sin : range sin = (Icc (-1) 1 : set ℝ) | subset.antisymm (range_subset_iff.2 sin_mem_Icc) surj_on_sin.subset_range | lemma | real.range_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_cos_infinite : (range real.cos).infinite | by { rw real.range_cos, exact Icc_infinite (by norm_num) } | lemma | real.range_cos_infinite | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"infinite",
"real.cos",
"real.range_cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_sin_infinite : (range real.sin).infinite | by { rw real.range_sin, exact Icc_infinite (by norm_num) } | lemma | real.range_sin_infinite | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"infinite",
"real.range_sin",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sqrt_two_add_series (x : ℝ) : ℕ → ℝ | | 0 := x
| (n+1) := sqrt (2 + sqrt_two_add_series n) | def | real.sqrt_two_add_series | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | the series `sqrt_two_add_series x n` is `sqrt(2 + sqrt(2 + ... ))` with `n` square roots,
starting with `x`. We define it here because `cos (pi / 2 ^ (n+1)) = sqrt_two_add_series 0 n / 2` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sqrt_two_add_series_zero : sqrt_two_add_series x 0 = x | by simp | lemma | real.sqrt_two_add_series_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sqrt_two_add_series_one : sqrt_two_add_series 0 1 = sqrt 2 | by simp | lemma | real.sqrt_two_add_series_one | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sqrt_two_add_series_two : sqrt_two_add_series 0 2 = sqrt (2 + sqrt 2) | by simp | lemma | real.sqrt_two_add_series_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sqrt_two_add_series_zero_nonneg : ∀(n : ℕ), 0 ≤ sqrt_two_add_series 0 n | | 0 := le_refl 0
| (n+1) := sqrt_nonneg _ | lemma | real.sqrt_two_add_series_zero_nonneg | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sqrt_two_add_series_nonneg {x : ℝ} (h : 0 ≤ x) : ∀(n : ℕ), 0 ≤ sqrt_two_add_series x n | | 0 := h
| (n+1) := sqrt_nonneg _ | lemma | real.sqrt_two_add_series_nonneg | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sqrt_two_add_series_lt_two : ∀(n : ℕ), sqrt_two_add_series 0 n < 2 | | 0 := by norm_num
| (n+1) :=
begin
refine lt_of_lt_of_le _ (sqrt_sq zero_lt_two.le).le,
rw [sqrt_two_add_series, sqrt_lt_sqrt_iff, ← lt_sub_iff_add_lt'],
{ refine (sqrt_two_add_series_lt_two n).trans_le _, norm_num },
{ exact add_nonneg zero_le_two (sqrt_two_add_series_zero_nonneg n) }
end | lemma | real.sqrt_two_add_series_lt_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"zero_le_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sqrt_two_add_series_succ (x : ℝ) :
∀(n : ℕ), sqrt_two_add_series x (n+1) = sqrt_two_add_series (sqrt (2 + x)) n | | 0 := rfl
| (n+1) := by rw [sqrt_two_add_series, sqrt_two_add_series_succ, sqrt_two_add_series] | lemma | real.sqrt_two_add_series_succ | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sqrt_two_add_series_monotone_left {x y : ℝ} (h : x ≤ y) :
∀(n : ℕ), sqrt_two_add_series x n ≤ sqrt_two_add_series y n | | 0 := h
| (n+1) :=
begin
rw [sqrt_two_add_series, sqrt_two_add_series],
exact sqrt_le_sqrt (add_le_add_left (sqrt_two_add_series_monotone_left _) _)
end | lemma | real.sqrt_two_add_series_monotone_left | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_pi_over_two_pow : ∀(n : ℕ), cos (π / 2 ^ (n+1)) = sqrt_two_add_series 0 n / 2 | | 0 := by simp
| (n+1) :=
begin
have : (2 : ℝ) ≠ 0 := two_ne_zero,
symmetry, rw [div_eq_iff_mul_eq this], symmetry,
rw [sqrt_two_add_series, sqrt_eq_iff_sq_eq, mul_pow, cos_sq, ←mul_div_assoc,
nat.add_succ, pow_succ, mul_div_mul_left _ _ this, cos_pi_over_two_pow, add_mul],
congr, { norm_num... | lemma | real.cos_pi_over_two_pow | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"div_eq_iff_mul_eq",
"div_lt_div'",
"div_pos",
"mul_assoc",
"mul_comm",
"mul_div_cancel_left",
"mul_div_mul_left",
"mul_pow",
"pow_lt_pow",
"pow_one",
"pow_pos",
"pow_succ",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_sq_pi_over_two_pow (n : ℕ) :
sin (π / 2 ^ (n+1)) ^ 2 = 1 - (sqrt_two_add_series 0 n / 2) ^ 2 | by rw [sin_sq, cos_pi_over_two_pow] | lemma | real.sin_sq_pi_over_two_pow | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_sq_pi_over_two_pow_succ (n : ℕ) :
sin (π / 2 ^ (n+2)) ^ 2 = 1 / 2 - sqrt_two_add_series 0 n / 4 | begin
rw [sin_sq_pi_over_two_pow, sqrt_two_add_series, div_pow, sq_sqrt, add_div, ←sub_sub],
congr, norm_num, norm_num, apply add_nonneg, norm_num, apply sqrt_two_add_series_zero_nonneg,
end | lemma | real.sin_sq_pi_over_two_pow_succ | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"add_div",
"div_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_pi_over_two_pow_succ (n : ℕ) :
sin (π / 2 ^ (n+2)) = sqrt (2 - sqrt_two_add_series 0 n) / 2 | begin
symmetry, rw [div_eq_iff_mul_eq], symmetry,
rw [sqrt_eq_iff_sq_eq, mul_pow, sin_sq_pi_over_two_pow_succ, sub_mul],
{ congr, norm_num, rw [mul_comm], convert mul_div_cancel' _ _, norm_num, norm_num },
{ rw [sub_nonneg], apply le_of_lt, apply sqrt_two_add_series_lt_two },
apply le_of_lt, apply mul_pos, ap... | lemma | real.sin_pi_over_two_pow_succ | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [
"div_eq_iff_mul_eq",
"div_lt_div_left",
"div_one",
"div_pos",
"mul_comm",
"mul_div_cancel'",
"mul_pow",
"pow_lt_pow",
"pow_pos",
"pow_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_pi_div_four : cos (π / 4) = sqrt 2 / 2 | by { transitivity cos (π / 2 ^ 2), congr, norm_num, simp } | lemma | real.cos_pi_div_four | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_pi_div_four : sin (π / 4) = sqrt 2 / 2 | by { transitivity sin (π / 2 ^ 2), congr, norm_num, simp } | lemma | real.sin_pi_div_four | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_pi_div_eight : cos (π / 8) = sqrt (2 + sqrt 2) / 2 | by { transitivity cos (π / 2 ^ 3), congr, norm_num, simp } | lemma | real.cos_pi_div_eight | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_pi_div_eight : sin (π / 8) = sqrt (2 - sqrt 2) / 2 | by { transitivity sin (π / 2 ^ 3), congr, norm_num, simp } | lemma | real.sin_pi_div_eight | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_pi_div_sixteen : cos (π / 16) = sqrt (2 + sqrt (2 + sqrt 2)) / 2 | by { transitivity cos (π / 2 ^ 4), congr, norm_num, simp } | lemma | real.cos_pi_div_sixteen | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_pi_div_sixteen : sin (π / 16) = sqrt (2 - sqrt (2 + sqrt 2)) / 2 | by { transitivity sin (π / 2 ^ 4), congr, norm_num, simp } | lemma | real.sin_pi_div_sixteen | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_pi_div_thirty_two : cos (π / 32) = sqrt (2 + sqrt (2 + sqrt (2 + sqrt 2))) / 2 | by { transitivity cos (π / 2 ^ 5), congr, norm_num, simp } | lemma | real.cos_pi_div_thirty_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/basic.lean | [
"analysis.special_functions.exp"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_pi_div_thirty_two : sin (π / 32) = sqrt (2 - sqrt (2 + sqrt (2 + sqrt 2))) / 2 | by { transitivity sin (π / 2 ^ 5), congr, norm_num, simp } | lemma | real.sin_pi_div_thirty_two | 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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