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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|---|---|---|---|---|---|---|---|---|---|---|
rpow_le_one_of_one_le_of_nonpos {x z : ℝ} (hx : 1 ≤ x) (hz : z ≤ 0) : x^z ≤ 1 | by { convert rpow_le_rpow_of_exponent_le hx hz, exact (rpow_zero x).symm } | lemma | real.rpow_le_one_of_one_le_of_nonpos | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_rpow {x z : ℝ} (hx : 1 < x) (hz : 0 < z) : 1 < x^z | by { rw ← one_rpow z, exact rpow_lt_rpow zero_le_one hx hz } | lemma | real.one_lt_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_rpow {x z : ℝ} (hx : 1 ≤ x) (hz : 0 ≤ z) : 1 ≤ x^z | by { rw ← one_rpow z, exact rpow_le_rpow zero_le_one hx hz } | lemma | real.one_le_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_rpow_of_pos_of_lt_one_of_neg (hx1 : 0 < x) (hx2 : x < 1) (hz : z < 0) :
1 < x^z | by { convert rpow_lt_rpow_of_exponent_gt hx1 hx2 hz, exact (rpow_zero x).symm } | lemma | real.one_lt_rpow_of_pos_of_lt_one_of_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_le_rpow_of_pos_of_le_one_of_nonpos (hx1 : 0 < x) (hx2 : x ≤ 1) (hz : z ≤ 0) :
1 ≤ x^z | by { convert rpow_le_rpow_of_exponent_ge hx1 hx2 hz, exact (rpow_zero x).symm } | lemma | real.one_le_rpow_of_pos_of_le_one_of_nonpos | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_one_iff_of_pos (hx : 0 < x) : x ^ y < 1 ↔ 1 < x ∧ y < 0 ∨ x < 1 ∧ 0 < y | by rw [rpow_def_of_pos hx, exp_lt_one_iff, mul_neg_iff, log_pos_iff hx, log_neg_iff hx] | lemma | real.rpow_lt_one_iff_of_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"mul_neg_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_lt_one_iff (hx : 0 ≤ x) : x ^ y < 1 ↔ x = 0 ∧ y ≠ 0 ∨ 1 < x ∧ y < 0 ∨ x < 1 ∧ 0 < y | begin
rcases hx.eq_or_lt with (rfl|hx),
{ rcases em (y = 0) with (rfl|hy); simp [*, lt_irrefl, zero_lt_one] },
{ simp [rpow_lt_one_iff_of_pos hx, hx.ne.symm] }
end | lemma | real.rpow_lt_one_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"em",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_rpow_iff_of_pos (hx : 0 < x) : 1 < x ^ y ↔ 1 < x ∧ 0 < y ∨ x < 1 ∧ y < 0 | by rw [rpow_def_of_pos hx, one_lt_exp_iff, mul_pos_iff, log_pos_iff hx, log_neg_iff hx] | lemma | real.one_lt_rpow_iff_of_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"mul_pos_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_lt_rpow_iff (hx : 0 ≤ x) : 1 < x ^ y ↔ 1 < x ∧ 0 < y ∨ 0 < x ∧ x < 1 ∧ y < 0 | begin
rcases hx.eq_or_lt with (rfl|hx),
{ rcases em (y = 0) with (rfl|hy); simp [*, lt_irrefl, (zero_lt_one' ℝ).not_lt] },
{ simp [one_lt_rpow_iff_of_pos hx, hx] }
end | lemma | real.one_lt_rpow_iff | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"em",
"zero_lt_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_rpow_of_exponent_ge' (hx0 : 0 ≤ x) (hx1 : x ≤ 1) (hz : 0 ≤ z) (hyz : z ≤ y) :
x^y ≤ x^z | begin
rcases eq_or_lt_of_le hx0 with rfl | hx0',
{ rcases eq_or_lt_of_le hz with rfl | hz',
{ exact (rpow_zero 0).symm ▸ (rpow_le_one hx0 hx1 hyz), },
rw [zero_rpow, zero_rpow]; linarith, },
{ exact rpow_le_rpow_of_exponent_ge hx0' hx1 hyz, },
end | lemma | real.rpow_le_rpow_of_exponent_ge' | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"eq_or_lt_of_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_left_inj_on {x : ℝ} (hx : x ≠ 0) :
inj_on (λ y : ℝ, y^x) {y : ℝ | 0 ≤ y} | begin
rintros y hy z hz (hyz : y ^ x = z ^ x),
rw [←rpow_one y, ←rpow_one z, ←_root_.mul_inv_cancel hx, rpow_mul hy, rpow_mul hz, hyz]
end | lemma | real.rpow_left_inj_on | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_rpow_iff_log_le (hx : 0 < x) (hy : 0 < y) :
x ≤ y^z ↔ real.log x ≤ z * real.log y | by rw [←real.log_le_log hx (real.rpow_pos_of_pos hy z), real.log_rpow hy] | lemma | real.le_rpow_iff_log_le | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"real.log",
"real.log_rpow",
"real.rpow_pos_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_rpow_of_log_le (hx : 0 ≤ x) (hy : 0 < y) (h : real.log x ≤ z * real.log y) :
x ≤ y^z | begin
obtain hx | rfl := hx.lt_or_eq,
{ exact (le_rpow_iff_log_le hx hy).2 h },
exact (real.rpow_pos_of_pos hy z).le,
end | lemma | real.le_rpow_of_log_le | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"real.log",
"real.rpow_pos_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_rpow_iff_log_lt (hx : 0 < x) (hy : 0 < y) :
x < y^z ↔ real.log x < z * real.log y | by rw [←real.log_lt_log_iff hx (real.rpow_pos_of_pos hy z), real.log_rpow hy] | lemma | real.lt_rpow_iff_log_lt | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"real.log",
"real.log_rpow",
"real.rpow_pos_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lt_rpow_of_log_lt (hx : 0 ≤ x) (hy : 0 < y) (h : real.log x < z * real.log y) :
x < y^z | begin
obtain hx | rfl := hx.lt_or_eq,
{ exact (lt_rpow_iff_log_lt hx hy).2 h },
exact real.rpow_pos_of_pos hy z,
end | lemma | real.lt_rpow_of_log_lt | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"real.log",
"real.rpow_pos_of_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_le_one_iff_of_pos (hx : 0 < x) : x ^ y ≤ 1 ↔ 1 ≤ x ∧ y ≤ 0 ∨ x ≤ 1 ∧ 0 ≤ y | by rw [rpow_def_of_pos hx, exp_le_one_iff, mul_nonpos_iff, log_nonneg_iff hx, log_nonpos_iff hx] | lemma | real.rpow_le_one_iff_of_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"mul_nonpos_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_log_mul_self_rpow_lt (x t : ℝ) (h1 : 0 < x) (h2 : x ≤ 1) (ht : 0 < t) : | |log x * x ^ t| < 1 / t :=
begin
rw lt_div_iff ht,
have := abs_log_mul_self_lt (x ^ t) (rpow_pos_of_pos h1 t) (rpow_le_one h1.le h2 ht.le),
rwa [log_rpow h1, mul_assoc, abs_mul, abs_of_pos ht, mul_comm] at this
end | lemma | real.abs_log_mul_self_rpow_lt | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"abs_mul",
"abs_of_pos",
"lt_div_iff",
"mul_assoc",
"mul_comm"
] | Bound for `|log x * x ^ t|` in the interval `(0, 1]`, for positive real `t`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_nat_rpow_nat_inv {x : ℝ} (hx : 0 ≤ x) {n : ℕ} (hn : n ≠ 0) :
(x ^ n) ^ (n⁻¹ : ℝ) = x | have hn0 : (n : ℝ) ≠ 0, from nat.cast_ne_zero.2 hn,
by rw [← rpow_nat_cast, ← rpow_mul hx, mul_inv_cancel hn0, rpow_one] | lemma | real.pow_nat_rpow_nat_inv | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"mul_inv_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_nat_inv_pow_nat {x : ℝ} (hx : 0 ≤ x) {n : ℕ} (hn : n ≠ 0) :
(x ^ (n⁻¹ : ℝ)) ^ n = x | have hn0 : (n : ℝ) ≠ 0, from nat.cast_ne_zero.2 hn,
by rw [← rpow_nat_cast, ← rpow_mul hx, inv_mul_cancel hn0, rpow_one] | lemma | real.rpow_nat_inv_pow_nat | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"inv_mul_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sqrt_eq_rpow (x : ℝ) : sqrt x = x ^ (1/(2:ℝ)) | begin
obtain h | h := le_or_lt 0 x,
{ rw [← mul_self_inj_of_nonneg (sqrt_nonneg _) (rpow_nonneg_of_nonneg h _), mul_self_sqrt h,
← sq, ← rpow_nat_cast, ← rpow_mul h],
norm_num },
{ have : 1 / (2:ℝ) * π = π / (2:ℝ), ring,
rw [sqrt_eq_zero_of_nonpos h.le, rpow_def_of_neg h, this, cos_pi_div_two, mul_z... | lemma | real.sqrt_eq_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"mul_self_inj_of_nonneg",
"mul_zero",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_div_two_eq_sqrt {x : ℝ} (r : ℝ) (hx : 0 ≤ x) : x ^ (r/2) = (sqrt x) ^ r | begin
rw [sqrt_eq_rpow, ← rpow_mul hx],
congr,
ring,
end | lemma | real.rpow_div_two_eq_sqrt | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_rat_pow_btwn_rat_aux (hn : n ≠ 0) (x y : ℝ) (h : x < y) (hy : 0 < y) :
∃ q : ℚ, 0 < q ∧ x < q^n ∧ ↑q^n < y | begin
have hn' : 0 < (n : ℝ) := by exact_mod_cast hn.bot_lt,
obtain ⟨q, hxq, hqy⟩ := exists_rat_btwn (rpow_lt_rpow (le_max_left 0 x) (max_lt hy h) $
inv_pos.mpr hn'),
have := rpow_nonneg_of_nonneg (le_max_left 0 x) n⁻¹,
have hq := this.trans_lt hxq,
replace hxq := rpow_lt_rpow this hxq hn',
replace hqy ... | lemma | real.exists_rat_pow_btwn_rat_aux | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"exists_rat_btwn"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_rat_pow_btwn_rat (hn : n ≠ 0) {x y : ℚ} (h : x < y) (hy : 0 < y) :
∃ q : ℚ, 0 < q ∧ x < q^n ∧ q^n < y | by apply_mod_cast exists_rat_pow_btwn_rat_aux hn x y; assumption | lemma | real.exists_rat_pow_btwn_rat | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_rat_pow_btwn {α : Type*} [linear_ordered_field α] [archimedean α] (hn : n ≠ 0)
{x y : α} (h : x < y) (hy : 0 < y) : ∃ q : ℚ, 0 < q ∧ x < q^n ∧ (q^n : α) < y | begin
obtain ⟨q₂, hx₂, hy₂⟩ := exists_rat_btwn (max_lt h hy),
obtain ⟨q₁, hx₁, hq₁₂⟩ := exists_rat_btwn hx₂,
have : (0 : α) < q₂ := (le_max_right _ _).trans_lt hx₂,
norm_cast at hq₁₂ this,
obtain ⟨q, hq, hq₁, hq₂⟩ := exists_rat_pow_btwn_rat hn hq₁₂ this,
refine ⟨q, hq, (le_max_left _ _).trans_lt $ hx₁.trans... | lemma | real.exists_rat_pow_btwn | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"archimedean",
"exists_rat_btwn",
"linear_ordered_field"
] | There is a rational power between any two positive elements of an archimedean ordered field. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rpow_pos (a b : ℝ) (b' : ℕ) (c : ℝ) (hb : (b':ℝ) = b) (h : a ^ b' = c) : a ^ b = c | by rw [← h, ← hb, real.rpow_nat_cast] | theorem | norm_num.rpow_pos | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"real.rpow_nat_cast"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_neg (a b : ℝ) (b' : ℕ) (c c' : ℝ)
(a0 : 0 ≤ a) (hb : (b':ℝ) = b) (h : a ^ b' = c) (hc : c⁻¹ = c') : a ^ -b = c' | by rw [← hc, ← h, ← hb, real.rpow_neg a0, real.rpow_nat_cast] | theorem | norm_num.rpow_neg | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"real.rpow_nat_cast",
"real.rpow_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prove_rpow (a b : expr) : tactic (expr × expr) | do
na ← a.to_rat,
ic ← mk_instance_cache `(ℝ),
match match_sign b with
| sum.inl b := do
(ic, a0) ← guard (na ≥ 0) >> prove_nonneg ic a,
nc ← mk_instance_cache `(ℕ),
(ic, nc, b', hb) ← prove_nat_uncast ic nc b,
(ic, c, h) ← prove_pow a na ic b',
cr ← c.to_rat,
(ic, c', hc) ← prove_inv ic... | def | norm_num.prove_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"real.rpow_zero"
] | Evaluate `real.rpow a b` where `a` is a rational numeral and `b` is an integer.
(This cannot go via the generalized version `prove_rpow'` because `rpow_pos` has a side condition;
we do not attempt to evaluate `a ^ b` where `a` and `b` are both negative because it comes
out to some garbage.) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eval_rpow : expr → tactic (expr × expr) | | `(@has_pow.pow _ _ real.has_pow %%a %%b) := b.to_int >> prove_rpow a b
| `(real.rpow %%a %%b) := b.to_int >> prove_rpow a b
| _ := tactic.failed | def | norm_num.eval_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | Evaluates expressions of the form `rpow a b` and `a ^ b` in the special case where
`b` is an integer and `a` is a positive rational (so it's really just a rational power). | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prove_rpow (a b : expr) : tactic strictness | do
strictness_a ← core a,
match strictness_a with
| nonnegative p := nonnegative <$> mk_app ``real.rpow_nonneg_of_nonneg [p, b]
| positive p := positive <$> mk_app ``real.rpow_pos_of_pos [p, b]
| _ := failed
end | def | tactic.positivity.prove_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [
"real.rpow_nonneg_of_nonneg",
"real.rpow_pos_of_pos"
] | Auxiliary definition for the `positivity` tactic to handle real powers of reals. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
positivity_rpow : expr → tactic strictness | | `(@has_pow.pow _ _ real.has_pow %%a %%b) := prove_rpow a b
| `(real.rpow %%a %%b) := prove_rpow a b
| _ := failed | def | tactic.positivity_rpow | analysis.special_functions.pow | src/analysis/special_functions/pow/real.lean | [
"analysis.special_functions.pow.complex"
] | [] | Extension for the `positivity` tactic: exponentiation by a real number is nonnegative when the
base is nonnegative and positive when the base is positive. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
angle : Type | add_circle (2 * π) | def | real.angle | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"add_circle"
] | The type of angles | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_coe : continuous (coe : ℝ → angle) | continuous_quotient_mk | lemma | real.angle.continuous_coe | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"continuous",
"continuous_quotient_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_hom : ℝ →+ angle | quotient_add_group.mk' _ | def | real.angle.coe_hom | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | Coercion `ℝ → angle` as an additive homomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_coe_hom : (coe_hom : ℝ → angle) = coe | rfl | lemma | real.angle.coe_coe_hom | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on {p : angle → Prop} (θ : angle) (h : ∀ x : ℝ, p x) : p θ | quotient.induction_on' θ h | lemma | real.angle.induction_on | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"quotient.induction_on'"
] | An induction principle to deduce results for `angle` from those for `ℝ`, used with
`induction θ using real.angle.induction_on`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_zero : ↑(0 : ℝ) = (0 : angle) | rfl | lemma | real.angle.coe_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_add (x y : ℝ) : ↑(x + y : ℝ) = (↑x + ↑y : angle) | rfl | lemma | real.angle.coe_add | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_neg (x : ℝ) : ↑(-x : ℝ) = -(↑x : angle) | rfl | lemma | real.angle.coe_neg | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sub (x y : ℝ) : ↑(x - y : ℝ) = (↑x - ↑y : angle) | rfl | lemma | real.angle.coe_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nsmul (n : ℕ) (x : ℝ) : ↑(n • x : ℝ) = (n • ↑x : angle) | rfl | lemma | real.angle.coe_nsmul | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_zsmul (z : ℤ) (x : ℝ) : ↑(z • x : ℝ) = (z • ↑x : angle) | rfl | lemma | real.angle.coe_zsmul | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_nat_mul_eq_nsmul (x : ℝ) (n : ℕ) :
↑((n : ℝ) * x) = n • (↑x : angle) | by simpa only [nsmul_eq_mul] using coe_hom.map_nsmul x n | lemma | real.angle.coe_nat_mul_eq_nsmul | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"nsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_int_mul_eq_zsmul (x : ℝ) (n : ℤ) :
↑((n : ℝ) * x : ℝ) = n • (↑x : angle) | by simpa only [zsmul_eq_mul] using coe_hom.map_zsmul x n | lemma | real.angle.coe_int_mul_eq_zsmul | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"zsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
angle_eq_iff_two_pi_dvd_sub {ψ θ : ℝ} : (θ : angle) = ψ ↔ ∃ k : ℤ, θ - ψ = 2 * π * k | by simp only [quotient_add_group.eq, add_subgroup.zmultiples_eq_closure,
add_subgroup.mem_closure_singleton, zsmul_eq_mul', (sub_eq_neg_add _ _).symm, eq_comm] | lemma | real.angle.angle_eq_iff_two_pi_dvd_sub | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"zsmul_eq_mul'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_two_pi : ↑(2 * π : ℝ) = (0 : angle) | angle_eq_iff_two_pi_dvd_sub.2 ⟨1, by rw [sub_zero, int.cast_one, mul_one]⟩ | lemma | real.angle.coe_two_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"int.cast_one",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_coe_pi : -(π : angle) = π | begin
rw [←coe_neg, angle_eq_iff_two_pi_dvd_sub],
use -1,
simp [two_mul, sub_eq_add_neg]
end | lemma | real.angle.neg_coe_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_nsmul_coe_div_two (θ : ℝ) : (2 : ℕ) • (↑(θ / 2) : angle) = θ | by rw [←coe_nsmul, two_nsmul, add_halves] | lemma | real.angle.two_nsmul_coe_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"add_halves"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_zsmul_coe_div_two (θ : ℝ) : (2 : ℤ) • (↑(θ / 2) : angle) = θ | by rw [←coe_zsmul, two_zsmul, add_halves] | lemma | real.angle.two_zsmul_coe_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"add_halves"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_nsmul_neg_pi_div_two : (2 : ℕ) • (↑(-π / 2) : angle) = π | by rw [two_nsmul_coe_div_two, coe_neg, neg_coe_pi] | lemma | real.angle.two_nsmul_neg_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_zsmul_neg_pi_div_two : (2 : ℤ) • (↑(-π / 2) : angle) = π | by rw [two_zsmul, ←two_nsmul, two_nsmul_neg_pi_div_two] | lemma | real.angle.two_zsmul_neg_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sub_coe_pi_eq_add_coe_pi (θ : angle) : θ - π = θ + π | by rw [sub_eq_add_neg, neg_coe_pi] | lemma | real.angle.sub_coe_pi_eq_add_coe_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_nsmul_coe_pi : (2 : ℕ) • (π : angle) = 0 | by simp [←coe_nat_mul_eq_nsmul] | lemma | real.angle.two_nsmul_coe_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_zsmul_coe_pi : (2 : ℤ) • (π : angle) = 0 | by simp [←coe_int_mul_eq_zsmul] | lemma | real.angle.two_zsmul_coe_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_pi_add_coe_pi : (π : real.angle) + π = 0 | by rw [←two_nsmul, two_nsmul_coe_pi] | lemma | real.angle.coe_pi_add_coe_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
zsmul_eq_iff {ψ θ : angle} {z : ℤ} (hz : z ≠ 0) :
z • ψ = z • θ ↔ (∃ k : fin z.nat_abs, ψ = θ + (k : ℕ) • (2 * π / z : ℝ)) | quotient_add_group.zmultiples_zsmul_eq_zsmul_iff hz | lemma | real.angle.zsmul_eq_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"quotient_add_group.zmultiples_zsmul_eq_zsmul_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nsmul_eq_iff {ψ θ : angle} {n : ℕ} (hz : n ≠ 0) :
n • ψ = n • θ ↔ (∃ k : fin n, ψ = θ + (k : ℕ) • (2 * π / n : ℝ)) | quotient_add_group.zmultiples_nsmul_eq_nsmul_iff hz | lemma | real.angle.nsmul_eq_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"quotient_add_group.zmultiples_nsmul_eq_nsmul_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_zsmul_eq_iff {ψ θ : angle} : (2 : ℤ) • ψ = (2 : ℤ) • θ ↔ (ψ = θ ∨ ψ = θ + π) | by rw [zsmul_eq_iff two_ne_zero, int.nat_abs_bit0, int.nat_abs_one,
fin.exists_fin_two, fin.coe_zero, fin.coe_one, zero_smul, add_zero, one_smul,
int.cast_two, mul_div_cancel_left (_ : ℝ) two_ne_zero] | lemma | real.angle.two_zsmul_eq_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"fin.coe_one",
"fin.coe_zero",
"fin.exists_fin_two",
"int.cast_two",
"mul_div_cancel_left",
"one_smul",
"two_ne_zero",
"zero_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_nsmul_eq_iff {ψ θ : angle} : (2 : ℕ) • ψ = (2 : ℕ) • θ ↔ (ψ = θ ∨ ψ = θ + π) | by simp_rw [←coe_nat_zsmul, int.coe_nat_bit0, int.coe_nat_one, two_zsmul_eq_iff] | lemma | real.angle.two_nsmul_eq_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"int.coe_nat_bit0"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_nsmul_eq_zero_iff {θ : angle} : (2 : ℕ) • θ = 0 ↔ (θ = 0 ∨ θ = π) | by convert two_nsmul_eq_iff; simp | lemma | real.angle.two_nsmul_eq_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_nsmul_ne_zero_iff {θ : angle} : (2 : ℕ) • θ ≠ 0 ↔ θ ≠ 0 ∧ θ ≠ π | by rw [←not_or_distrib, ←two_nsmul_eq_zero_iff] | lemma | real.angle.two_nsmul_ne_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_zsmul_eq_zero_iff {θ : angle} : (2 : ℤ) • θ = 0 ↔ (θ = 0 ∨ θ = π) | by simp_rw [two_zsmul, ←two_nsmul, two_nsmul_eq_zero_iff] | lemma | real.angle.two_zsmul_eq_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_zsmul_ne_zero_iff {θ : angle} : (2 : ℤ) • θ ≠ 0 ↔ θ ≠ 0 ∧ θ ≠ π | by rw [←not_or_distrib, ←two_zsmul_eq_zero_iff] | lemma | real.angle.two_zsmul_ne_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_neg_self_iff {θ : angle} : θ = -θ ↔ θ = 0 ∨ θ = π | by rw [←add_eq_zero_iff_eq_neg, ←two_nsmul, two_nsmul_eq_zero_iff] | lemma | real.angle.eq_neg_self_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"eq_neg_self_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_neg_self_iff {θ : angle} : θ ≠ -θ ↔ θ ≠ 0 ∧ θ ≠ π | by rw [←not_or_distrib, ←eq_neg_self_iff.not] | lemma | real.angle.ne_neg_self_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_eq_self_iff {θ : angle} : -θ = θ ↔ θ = 0 ∨ θ = π | by rw [eq_comm, eq_neg_self_iff] | lemma | real.angle.neg_eq_self_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"eq_neg_self_iff",
"neg_eq_self_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_ne_self_iff {θ : angle} : -θ ≠ θ ↔ θ ≠ 0 ∧ θ ≠ π | by rw [←not_or_distrib, ←neg_eq_self_iff.not] | lemma | real.angle.neg_ne_self_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_nsmul_eq_pi_iff {θ : angle} : (2 : ℕ) • θ = π ↔ (θ = (π / 2 : ℝ) ∨ θ = (-π / 2 : ℝ)) | begin
have h : (π : angle) = (2 : ℕ) • (π / 2 : ℝ), { rw [two_nsmul, ←coe_add, add_halves] },
nth_rewrite 0 h,
rw [two_nsmul_eq_iff],
congr',
rw [add_comm, ←coe_add, ←sub_eq_zero, ←coe_sub, add_sub_assoc, neg_div, sub_neg_eq_add,
add_halves, ←two_mul, coe_two_pi]
end | lemma | real.angle.two_nsmul_eq_pi_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"add_halves",
"neg_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
two_zsmul_eq_pi_iff {θ : angle} : (2 : ℤ) • θ = π ↔ (θ = (π / 2 : ℝ) ∨ θ = (-π / 2 : ℝ)) | by rw [two_zsmul, ←two_nsmul, two_nsmul_eq_pi_iff] | lemma | real.angle.two_zsmul_eq_pi_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_eq_iff_coe_eq_or_eq_neg {θ ψ : ℝ} :
cos θ = cos ψ ↔ (θ : angle) = ψ ∨ (θ : angle) = -ψ | begin
split,
{ intro Hcos,
rw [← sub_eq_zero, cos_sub_cos, mul_eq_zero, mul_eq_zero, neg_eq_zero,
eq_false_intro (two_ne_zero' ℝ), false_or, sin_eq_zero_iff, sin_eq_zero_iff] at Hcos,
rcases Hcos with ⟨n, hn⟩ | ⟨n, hn⟩,
{ right,
rw [eq_div_iff_mul_eq (two_ne_zero' ℝ), ← sub_eq_iff_eq_add] ... | theorem | real.angle.cos_eq_iff_coe_eq_or_eq_neg | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"eq_div_iff_mul_eq",
"mul_assoc",
"mul_comm",
"mul_div_cancel_left",
"mul_eq_zero",
"mul_zero",
"two_ne_zero'",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_eq_iff_coe_eq_or_add_eq_pi {θ ψ : ℝ} :
sin θ = sin ψ ↔ (θ : angle) = ψ ∨ (θ : angle) + ψ = π | begin
split,
{ intro Hsin, rw [← cos_pi_div_two_sub, ← cos_pi_div_two_sub] at Hsin,
cases cos_eq_iff_coe_eq_or_eq_neg.mp Hsin with h h,
{ left, rw [coe_sub, coe_sub] at h, exact sub_right_inj.1 h },
right, rw [coe_sub, coe_sub, eq_neg_iff_add_eq_zero, add_sub,
sub_add_eq_add_sub, ← coe_add, add_... | theorem | real.angle.sin_eq_iff_coe_eq_or_add_eq_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"add_div",
"add_halves",
"mul_assoc",
"mul_comm",
"mul_div_cancel_left",
"mul_zero",
"two_ne_zero'",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_sin_inj {θ ψ : ℝ} (Hcos : cos θ = cos ψ) (Hsin : sin θ = sin ψ) : (θ : angle) = ψ | begin
cases cos_eq_iff_coe_eq_or_eq_neg.mp Hcos with hc hc, { exact hc },
cases sin_eq_iff_coe_eq_or_add_eq_pi.mp Hsin with hs hs, { exact hs },
rw [eq_neg_iff_add_eq_zero, hs] at hc,
obtain ⟨n, hn⟩ : ∃ n, n • _ = _ := quotient_add_group.left_rel_apply.mp (quotient.exact' hc),
rw [← neg_one_mul, add_zero, ← s... | theorem | real.angle.cos_sin_inj | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"int.add_mul_mod_self",
"int.cast_add",
"int.cast_bit0",
"int.cast_inj",
"int.cast_mul",
"int.cast_one",
"int.cast_zero",
"mul_assoc",
"mul_eq_zero",
"neg_one_mul",
"one_ne_zero",
"quotient.exact'",
"zsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin (θ : angle) : ℝ | sin_periodic.lift θ | def | real.angle.sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | The sine of a `real.angle`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sin_coe (x : ℝ) : sin (x : angle) = real.sin x | rfl | lemma | real.angle.sin_coe | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_sin : continuous sin | real.continuous_sin.quotient_lift_on' _ | lemma | real.angle.continuous_sin | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos (θ : angle) : ℝ | cos_periodic.lift θ | def | real.angle.cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | The cosine of a `real.angle`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cos_coe (x : ℝ) : cos (x : angle) = real.cos x | rfl | lemma | real.angle.cos_coe | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_cos : continuous cos | real.continuous_cos.quotient_lift_on' _ | lemma | real.angle.continuous_cos | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_eq_real_cos_iff_eq_or_eq_neg {θ : angle} {ψ : ℝ} : cos θ = real.cos ψ ↔ θ = ψ ∨ θ = -ψ | begin
induction θ using real.angle.induction_on,
exact cos_eq_iff_coe_eq_or_eq_neg
end | lemma | real.angle.cos_eq_real_cos_iff_eq_or_eq_neg | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle.induction_on",
"real.cos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_eq_iff_eq_or_eq_neg {θ ψ : angle} : cos θ = cos ψ ↔ θ = ψ ∨ θ = -ψ | begin
induction ψ using real.angle.induction_on,
exact cos_eq_real_cos_iff_eq_or_eq_neg
end | lemma | real.angle.cos_eq_iff_eq_or_eq_neg | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_eq_real_sin_iff_eq_or_add_eq_pi {θ : angle} {ψ : ℝ} :
sin θ = real.sin ψ ↔ θ = ψ ∨ θ + ψ = π | begin
induction θ using real.angle.induction_on,
exact sin_eq_iff_coe_eq_or_add_eq_pi
end | lemma | real.angle.sin_eq_real_sin_iff_eq_or_add_eq_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle.induction_on",
"real.sin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_eq_iff_eq_or_add_eq_pi {θ ψ : angle} : sin θ = sin ψ ↔ θ = ψ ∨ θ + ψ = π | begin
induction ψ using real.angle.induction_on,
exact sin_eq_real_sin_iff_eq_or_add_eq_pi
end | lemma | real.angle.sin_eq_iff_eq_or_add_eq_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_zero : sin (0 : angle) = 0 | by rw [←coe_zero, sin_coe, real.sin_zero] | lemma | real.angle.sin_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.sin_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_coe_pi : sin (π : angle) = 0 | by rw [sin_coe, real.sin_pi] | lemma | real.angle.sin_coe_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.sin_pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_eq_zero_iff {θ : angle} : sin θ = 0 ↔ θ = 0 ∨ θ = π | begin
nth_rewrite 0 ←sin_zero,
rw sin_eq_iff_eq_or_add_eq_pi,
simp
end | lemma | real.angle.sin_eq_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_ne_zero_iff {θ : angle} : sin θ ≠ 0 ↔ θ ≠ 0 ∧ θ ≠ π | by rw [←not_or_distrib, ←sin_eq_zero_iff] | lemma | real.angle.sin_ne_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_neg (θ : angle) : sin (-θ) = -sin θ | begin
induction θ using real.angle.induction_on,
exact real.sin_neg _
end | lemma | real.angle.sin_neg | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle.induction_on",
"real.sin_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_antiperiodic : function.antiperiodic sin (π : angle) | begin
intro θ,
induction θ using real.angle.induction_on,
exact real.sin_antiperiodic θ
end | lemma | real.angle.sin_antiperiodic | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"function.antiperiodic",
"real.angle.induction_on",
"real.sin_antiperiodic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_add_pi (θ : angle) : sin (θ + π) = -sin θ | sin_antiperiodic θ | lemma | real.angle.sin_add_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_sub_pi (θ : angle) : sin (θ - π) = -sin θ | sin_antiperiodic.sub_eq θ | lemma | real.angle.sin_sub_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_zero : cos (0 : angle) = 1 | by rw [←coe_zero, cos_coe, real.cos_zero] | lemma | real.angle.cos_zero | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.cos_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_coe_pi : cos (π : angle) = -1 | by rw [cos_coe, real.cos_pi] | lemma | real.angle.cos_coe_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.cos_pi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_neg (θ : angle) : cos (-θ) = cos θ | begin
induction θ using real.angle.induction_on,
exact real.cos_neg _
end | lemma | real.angle.cos_neg | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle.induction_on",
"real.cos_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_antiperiodic : function.antiperiodic cos (π : angle) | begin
intro θ,
induction θ using real.angle.induction_on,
exact real.cos_antiperiodic θ
end | lemma | real.angle.cos_antiperiodic | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"function.antiperiodic",
"real.angle.induction_on",
"real.cos_antiperiodic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_add_pi (θ : angle) : cos (θ + π) = -cos θ | cos_antiperiodic θ | lemma | real.angle.cos_add_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_sub_pi (θ : angle) : cos (θ - π) = -cos θ | cos_antiperiodic.sub_eq θ | lemma | real.angle.cos_sub_pi | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_eq_zero_iff {θ : angle} : cos θ = 0 ↔ (θ = (π / 2 : ℝ) ∨ θ = (-π / 2 : ℝ)) | by rw [← cos_pi_div_two, ← cos_coe, cos_eq_iff_eq_or_eq_neg, ← coe_neg, ← neg_div] | lemma | real.angle.cos_eq_zero_iff | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"neg_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_add (θ₁ θ₂ : real.angle) : sin (θ₁ + θ₂) = sin θ₁ * cos θ₂ + cos θ₁ * sin θ₂ | begin
induction θ₁ using real.angle.induction_on,
induction θ₂ using real.angle.induction_on,
exact real.sin_add θ₁ θ₂
end | lemma | real.angle.sin_add | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle",
"real.angle.induction_on",
"real.sin_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_add (θ₁ θ₂ : real.angle) : cos (θ₁ + θ₂) = cos θ₁ * cos θ₂ - sin θ₁ * sin θ₂ | begin
induction θ₂ using real.angle.induction_on,
induction θ₁ using real.angle.induction_on,
exact real.cos_add θ₁ θ₂,
end | lemma | real.angle.cos_add | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle",
"real.angle.induction_on",
"real.cos_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_sq_add_sin_sq (θ : real.angle) : cos θ ^ 2 + sin θ ^ 2 = 1 | begin
induction θ using real.angle.induction_on,
exact real.cos_sq_add_sin_sq θ,
end | lemma | real.angle.cos_sq_add_sin_sq | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle",
"real.angle.induction_on",
"real.cos_sq_add_sin_sq"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sin_add_pi_div_two (θ : angle) : sin (θ + ↑(π / 2)) = cos θ | begin
induction θ using real.angle.induction_on,
exact sin_add_pi_div_two _
end | lemma | real.angle.sin_add_pi_div_two | analysis.special_functions.trigonometric | src/analysis/special_functions/trigonometric/angle.lean | [
"analysis.special_functions.trigonometric.basic",
"analysis.normed.group.add_circle",
"algebra.char_zero.quotient",
"topology.instances.sign"
] | [
"real.angle.induction_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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