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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not_interval_integrable_of_sub_inv_is_O_punctured {f : ℝ → F} {a b c : ℝ}
(hf : (λ x, (x - c)⁻¹) =O[𝓝[≠] c] f) (hne : a ≠ b) (hc : c ∈ [a, b]) :
¬interval_integrable f volume a b | begin
have A : ∀ᶠ x in 𝓝[≠] c, has_deriv_at (λ x, real.log (x - c)) (x - c)⁻¹ x,
{ filter_upwards [self_mem_nhds_within] with x hx,
simpa using ((has_deriv_at_id x).sub_const c).log (sub_ne_zero.2 hx) },
have B : tendsto (λ x, ‖real.log (x - c)‖) (𝓝[≠] c) at_top,
{ refine tendsto_abs_at_bot_at_top.comp (r... | lemma | not_interval_integrable_of_sub_inv_is_O_punctured | analysis.special_functions | src/analysis/special_functions/non_integrable.lean | [
"analysis.special_functions.log.deriv",
"measure_theory.integral.fund_thm_calculus"
] | [
"has_deriv_at",
"has_deriv_at_id",
"interval_integrable",
"not_interval_integrable_of_tendsto_norm_at_top_of_deriv_is_O_punctured",
"one_ne_zero",
"real.log",
"self_mem_nhds_within"
] | If `f` grows in the punctured neighborhood of `c : ℝ` at least as fast as `1 / (x - c)`,
then it is not interval integrable on any nontrivial interval `a..b`, `c ∈ [a, b]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
interval_integrable_sub_inv_iff {a b c : ℝ} :
interval_integrable (λ x, (x - c)⁻¹) volume a b ↔ a = b ∨ c ∉ [a, b] | begin
split,
{ refine λ h, or_iff_not_imp_left.2 (λ hne hc, _),
exact not_interval_integrable_of_sub_inv_is_O_punctured (is_O_refl _ _) hne hc h },
{ rintro (rfl|h₀),
{ exact interval_integrable.refl },
refine ((continuous_sub_right c).continuous_on.inv₀ _).interval_integrable,
exact λ x hx, sub_n... | lemma | interval_integrable_sub_inv_iff | analysis.special_functions | src/analysis/special_functions/non_integrable.lean | [
"analysis.special_functions.log.deriv",
"measure_theory.integral.fund_thm_calculus"
] | [
"continuous_on.inv₀",
"interval_integrable",
"interval_integrable.refl",
"ne_of_mem_of_not_mem",
"not_interval_integrable_of_sub_inv_is_O_punctured"
] | The function `λ x, (x - c)⁻¹` is integrable on `a..b` if and only if `a = b` or `c ∉ [a, b]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
interval_integrable_inv_iff {a b : ℝ} :
interval_integrable (λ x, x⁻¹) volume a b ↔ a = b ∨ (0 : ℝ) ∉ [a, b] | by simp only [← interval_integrable_sub_inv_iff, sub_zero] | lemma | interval_integrable_inv_iff | analysis.special_functions | src/analysis/special_functions/non_integrable.lean | [
"analysis.special_functions.log.deriv",
"measure_theory.integral.fund_thm_calculus"
] | [
"interval_integrable",
"interval_integrable_sub_inv_iff"
] | The function `λ x, x⁻¹` is integrable on `a..b` if and only if `a = b` or `0 ∉ [a, b]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
polar_coord : local_homeomorph (ℝ × ℝ) (ℝ × ℝ) | { to_fun := λ q, (real.sqrt (q.1^2 + q.2^2), complex.arg (complex.equiv_real_prod.symm q)),
inv_fun := λ p, (p.1 * cos p.2, p.1 * sin p.2),
source := {q | 0 < q.1} ∪ {q | q.2 ≠ 0},
target := Ioi (0 : ℝ) ×ˢ Ioo (-π) π,
map_target' :=
begin
rintros ⟨r, θ⟩ ⟨hr, hθ⟩,
dsimp at hr hθ,
rcases eq_or_ne θ ... | def | polar_coord | analysis.special_functions | src/analysis/special_functions/polar_coord.lean | [
"analysis.special_functions.trigonometric.deriv",
"measure_theory.function.jacobian"
] | [
"complex.I",
"complex.I_im",
"complex.I_re",
"complex.abs",
"complex.abs_def",
"complex.abs_mul_cos_add_sin_mul_I",
"complex.add_im",
"complex.add_re",
"complex.arg",
"complex.arg_lt_pi_iff",
"complex.arg_mul_cos_add_sin_mul_I",
"complex.continuous_at_arg",
"complex.equiv_real_prod_symm_appl... | The polar coordinates local homeomorphism in `ℝ^2`, mapping `(r cos θ, r sin θ)` to `(r, θ)`.
It is a homeomorphism between `ℝ^2 - (-∞, 0]` and `(0, +∞) × (-π, π)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_fderiv_at_polar_coord_symm (p : ℝ × ℝ) :
has_fderiv_at polar_coord.symm
(matrix.to_lin (basis.fin_two_prod ℝ) (basis.fin_two_prod ℝ)
(!![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2])).to_continuous_linear_map p | begin
rw matrix.to_lin_fin_two_prod_to_continuous_linear_map,
convert has_fderiv_at.prod
(has_fderiv_at_fst.mul ((has_deriv_at_cos p.2).comp_has_fderiv_at p has_fderiv_at_snd))
(has_fderiv_at_fst.mul ((has_deriv_at_sin p.2).comp_has_fderiv_at p has_fderiv_at_snd)) using 2;
simp only [smul_smul, add_comm, ... | lemma | has_fderiv_at_polar_coord_symm | analysis.special_functions | src/analysis/special_functions/polar_coord.lean | [
"analysis.special_functions.trigonometric.deriv",
"measure_theory.function.jacobian"
] | [
"basis.fin_two_prod",
"has_fderiv_at",
"has_fderiv_at.prod",
"has_fderiv_at_snd",
"matrix.to_lin",
"matrix.to_lin_fin_two_prod_to_continuous_linear_map",
"neg_mul",
"neg_smul",
"smul_neg",
"smul_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
polar_coord_source_ae_eq_univ :
polar_coord.source =ᵐ[volume] univ | begin
have A : polar_coord.sourceᶜ ⊆ (linear_map.snd ℝ ℝ ℝ).ker,
{ assume x hx,
simp only [polar_coord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_set_of_eq, not_lt,
not_not] at hx,
exact hx.2 },
have B : volume ((linear_map.snd ℝ ℝ ℝ).ker : set (ℝ × ℝ)) = 0,
{ apply measure.add_haar_su... | lemma | polar_coord_source_ae_eq_univ | analysis.special_functions | src/analysis/special_functions/polar_coord.lean | [
"analysis.special_functions.trigonometric.deriv",
"measure_theory.function.jacobian"
] | [
"bot_le",
"linear_map.ker_eq_top",
"linear_map.snd",
"not_not"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_comp_polar_coord_symm
{E : Type*} [normed_add_comm_group E] [normed_space ℝ E] [complete_space E] (f : ℝ × ℝ → E) :
∫ p in polar_coord.target, p.1 • f (polar_coord.symm p) = ∫ p, f p | begin
set B : (ℝ × ℝ) → ((ℝ × ℝ) →L[ℝ] (ℝ × ℝ)) := λ p,
(matrix.to_lin (basis.fin_two_prod ℝ) (basis.fin_two_prod ℝ)
!![cos p.2, -p.1 * sin p.2; sin p.2, p.1 * cos p.2]).to_continuous_linear_map with hB,
have A : ∀ p ∈ polar_coord.symm.source, has_fderiv_at polar_coord.symm (B p) p :=
λ p hp, has_fder... | theorem | integral_comp_polar_coord_symm | analysis.special_functions | src/analysis/special_functions/polar_coord.lean | [
"analysis.special_functions.trigonometric.deriv",
"measure_theory.function.jacobian"
] | [
"abs_of_pos",
"basis.fin_two_prod",
"complete_space",
"has_fderiv_at",
"has_fderiv_at_polar_coord_symm",
"linear_map.det_to_continuous_linear_map",
"linear_map.det_to_lin",
"matrix.det_fin_two_of",
"matrix.to_lin",
"neg_mul",
"normed_add_comm_group",
"normed_space",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eventually_no_roots (hP : P ≠ 0) : ∀ᶠ x in at_top, ¬ P.is_root x | at_top_le_cofinite $ (finite_set_of_is_root hP).compl_mem_cofinite | lemma | polynomial.eventually_no_roots | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent_at_top_lead :
(λ x, eval x P) ~[at_top] (λ x, P.leading_coeff * x ^ P.nat_degree) | begin
by_cases h : P = 0,
{ simp [h] },
{ simp only [polynomial.eval_eq_sum_range, sum_range_succ],
exact is_o.add_is_equivalent (is_o.sum $ λ i hi, is_o.const_mul_left
(is_o.const_mul_right (λ hz, h $ leading_coeff_eq_zero.mp hz) $
is_o_pow_pow_at_top_of_lt (mem_range.mp hi)) _) is_equivalent.r... | lemma | polynomial.is_equivalent_at_top_lead | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [
"polynomial.eval_eq_sum_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_top_of_leading_coeff_nonneg (hdeg : 0 < P.degree) (hnng : 0 ≤ P.leading_coeff) :
tendsto (λ x, eval x P) at_top at_top | P.is_equivalent_at_top_lead.symm.tendsto_at_top $
tendsto_const_mul_pow_at_top (nat_degree_pos_iff_degree_pos.2 hdeg).ne' $
hnng.lt_of_ne' $ leading_coeff_ne_zero.mpr $ ne_zero_of_degree_gt hdeg | lemma | polynomial.tendsto_at_top_of_leading_coeff_nonneg | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_top_iff_leading_coeff_nonneg :
tendsto (λ x, eval x P) at_top at_top ↔ 0 < P.degree ∧ 0 ≤ P.leading_coeff | begin
refine ⟨λ h, _, λ h, tendsto_at_top_of_leading_coeff_nonneg P h.1 h.2⟩,
have : tendsto (λ x, P.leading_coeff * x ^ P.nat_degree) at_top at_top :=
(is_equivalent_at_top_lead P).tendsto_at_top h,
rw [tendsto_const_mul_pow_at_top_iff, ← pos_iff_ne_zero, nat_degree_pos_iff_degree_pos] at this,
exact ⟨this... | lemma | polynomial.tendsto_at_top_iff_leading_coeff_nonneg | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_bot_iff_leading_coeff_nonpos :
tendsto (λ x, eval x P) at_top at_bot ↔ 0 < P.degree ∧ P.leading_coeff ≤ 0 | by simp only [← tendsto_neg_at_top_iff, ← eval_neg, tendsto_at_top_iff_leading_coeff_nonneg,
degree_neg, leading_coeff_neg, neg_nonneg] | lemma | polynomial.tendsto_at_bot_iff_leading_coeff_nonpos | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_at_bot_of_leading_coeff_nonpos (hdeg : 0 < P.degree) (hnps : P.leading_coeff ≤ 0) :
tendsto (λ x, eval x P) at_top at_bot | P.tendsto_at_bot_iff_leading_coeff_nonpos.2 ⟨hdeg, hnps⟩ | lemma | polynomial.tendsto_at_bot_of_leading_coeff_nonpos | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_tendsto_at_top (hdeg : 0 < P.degree) :
tendsto (λ x, abs $ eval x P) at_top at_top | begin
cases le_total 0 P.leading_coeff with hP hP,
{ exact tendsto_abs_at_top_at_top.comp (P.tendsto_at_top_of_leading_coeff_nonneg hdeg hP) },
{ exact tendsto_abs_at_bot_at_top.comp (P.tendsto_at_bot_of_leading_coeff_nonpos hdeg hP) }
end | lemma | polynomial.abs_tendsto_at_top | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_is_bounded_under_iff :
is_bounded_under (≤) at_top (λ x, |eval x P|) ↔ P.degree ≤ 0 | begin
refine ⟨λ h, _, λ h, ⟨|P.coeff 0|, eventually_map.mpr (eventually_of_forall
(forall_imp (λ _, le_of_eq) (λ x, congr_arg abs $ trans (congr_arg (eval x)
(eq_C_of_degree_le_zero h)) (eval_C))))⟩⟩,
contrapose! h,
exact not_is_bounded_under_of_tendsto_at_top (abs_tendsto_at_top P h)
end | lemma | polynomial.abs_is_bounded_under_iff | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [
"forall_imp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_tendsto_at_top_iff :
tendsto (λ x, abs $ eval x P) at_top at_top ↔ 0 < P.degree | ⟨λ h, not_le.mp (mt (abs_is_bounded_under_iff P).mpr (not_is_bounded_under_of_tendsto_at_top h)),
abs_tendsto_at_top P⟩ | lemma | polynomial.abs_tendsto_at_top_iff | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_nhds_iff {c : 𝕜} :
tendsto (λ x, eval x P) at_top (𝓝 c) ↔ P.leading_coeff = c ∧ P.degree ≤ 0 | begin
refine ⟨λ h, _, λ h, _⟩,
{ have := P.is_equivalent_at_top_lead.tendsto_nhds h,
by_cases hP : P.leading_coeff = 0,
{ simp only [hP, zero_mul, tendsto_const_nhds_iff] at this,
refine ⟨trans hP this, by simp [leading_coeff_eq_zero.1 hP]⟩ },
{ rw [tendsto_const_mul_pow_nhds_iff hP, nat_degree_eq... | lemma | polynomial.tendsto_nhds_iff | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [
"mul_one",
"pow_zero",
"tendsto_const_mul_pow_nhds_iff",
"tendsto_const_nhds",
"tendsto_const_nhds_iff",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_equivalent_at_top_div :
(λ x, (eval x P)/(eval x Q)) ~[at_top]
λ x, P.leading_coeff/Q.leading_coeff * x^(P.nat_degree - Q.nat_degree : ℤ) | begin
by_cases hP : P = 0,
{ simp [hP] },
by_cases hQ : Q = 0,
{ simp [hQ] },
refine (P.is_equivalent_at_top_lead.symm.div
Q.is_equivalent_at_top_lead.symm).symm.trans
(eventually_eq.is_equivalent ((eventually_gt_at_top 0).mono $ λ x hx, _)),
simp [← div_mul_div_comm, hP, hQ, zpow_sub₀ hx... | lemma | polynomial.is_equivalent_at_top_div | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [
"div_mul_div_comm",
"zpow_sub₀"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_tendsto_zero_of_degree_lt (hdeg : P.degree < Q.degree) :
tendsto (λ x, (eval x P)/(eval x Q)) at_top (𝓝 0) | begin
by_cases hP : P = 0,
{ simp [hP, tendsto_const_nhds] },
rw ← nat_degree_lt_nat_degree_iff hP at hdeg,
refine (is_equivalent_at_top_div P Q).symm.tendsto_nhds _,
rw ← mul_zero,
refine (tendsto_zpow_at_top_zero _).const_mul _,
linarith
end | lemma | polynomial.div_tendsto_zero_of_degree_lt | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [
"mul_zero",
"tendsto_const_nhds",
"tendsto_zpow_at_top_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_tendsto_zero_iff_degree_lt (hQ : Q ≠ 0) :
tendsto (λ x, (eval x P)/(eval x Q)) at_top (𝓝 0) ↔ P.degree < Q.degree | begin
refine ⟨λ h, _, div_tendsto_zero_of_degree_lt P Q⟩,
by_cases hPQ : P.leading_coeff / Q.leading_coeff = 0,
{ simp only [div_eq_mul_inv, inv_eq_zero, mul_eq_zero] at hPQ,
cases hPQ with hP0 hQ0,
{ rw [leading_coeff_eq_zero.1 hP0, degree_zero],
exact bot_lt_iff_ne_bot.2 (λ hQ', hQ (degree_eq_bot.... | lemma | polynomial.div_tendsto_zero_iff_degree_lt | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [
"div_eq_mul_inv",
"int.coe_nat_lt",
"inv_eq_zero",
"mul_eq_zero",
"tendsto_const_mul_zpow_at_top_nhds_iff",
"tendsto_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_tendsto_leading_coeff_div_of_degree_eq (hdeg : P.degree = Q.degree) :
tendsto (λ x, (eval x P)/(eval x Q)) at_top (𝓝 $ P.leading_coeff / Q.leading_coeff) | begin
refine (is_equivalent_at_top_div P Q).symm.tendsto_nhds _,
rw show (P.nat_degree : ℤ) = Q.nat_degree, by simp [hdeg, nat_degree],
simp [tendsto_const_nhds]
end | lemma | polynomial.div_tendsto_leading_coeff_div_of_degree_eq | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [
"tendsto_const_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_tendsto_at_top_of_degree_gt' (hdeg : Q.degree < P.degree)
(hpos : 0 < P.leading_coeff/Q.leading_coeff) :
tendsto (λ x, (eval x P)/(eval x Q)) at_top at_top | begin
have hQ : Q ≠ 0 := λ h, by {simp only [h, div_zero, leading_coeff_zero] at hpos, linarith},
rw ← nat_degree_lt_nat_degree_iff hQ at hdeg,
refine (is_equivalent_at_top_div P Q).symm.tendsto_at_top _,
apply tendsto.const_mul_at_top hpos,
apply tendsto_zpow_at_top_at_top,
linarith
end | lemma | polynomial.div_tendsto_at_top_of_degree_gt' | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [
"div_zero",
"tendsto_zpow_at_top_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_tendsto_at_top_of_degree_gt (hdeg : Q.degree < P.degree)
(hQ : Q ≠ 0) (hnng : 0 ≤ P.leading_coeff/Q.leading_coeff) :
tendsto (λ x, (eval x P)/(eval x Q)) at_top at_top | have ratio_pos : 0 < P.leading_coeff/Q.leading_coeff,
from lt_of_le_of_ne hnng
(div_ne_zero (λ h, ne_zero_of_degree_gt hdeg $ leading_coeff_eq_zero.mp h)
(λ h, hQ $ leading_coeff_eq_zero.mp h)).symm,
div_tendsto_at_top_of_degree_gt' P Q hdeg ratio_pos | lemma | polynomial.div_tendsto_at_top_of_degree_gt | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [
"div_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_tendsto_at_bot_of_degree_gt' (hdeg : Q.degree < P.degree)
(hneg : P.leading_coeff/Q.leading_coeff < 0) :
tendsto (λ x, (eval x P)/(eval x Q)) at_top at_bot | begin
have hQ : Q ≠ 0 := λ h, by {simp only [h, div_zero, leading_coeff_zero] at hneg, linarith},
rw ← nat_degree_lt_nat_degree_iff hQ at hdeg,
refine (is_equivalent_at_top_div P Q).symm.tendsto_at_bot _,
apply tendsto.neg_const_mul_at_top hneg,
apply tendsto_zpow_at_top_at_top,
linarith
end | lemma | polynomial.div_tendsto_at_bot_of_degree_gt' | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [
"div_zero",
"tendsto_zpow_at_top_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
div_tendsto_at_bot_of_degree_gt (hdeg : Q.degree < P.degree)
(hQ : Q ≠ 0) (hnps : P.leading_coeff/Q.leading_coeff ≤ 0) :
tendsto (λ x, (eval x P)/(eval x Q)) at_top at_bot | have ratio_neg : P.leading_coeff/Q.leading_coeff < 0,
from lt_of_le_of_ne hnps
(div_ne_zero (λ h, ne_zero_of_degree_gt hdeg $ leading_coeff_eq_zero.mp h)
(λ h, hQ $ leading_coeff_eq_zero.mp h)),
div_tendsto_at_bot_of_degree_gt' P Q hdeg ratio_neg | lemma | polynomial.div_tendsto_at_bot_of_degree_gt | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [
"div_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_div_tendsto_at_top_of_degree_gt (hdeg : Q.degree < P.degree)
(hQ : Q ≠ 0) :
tendsto (λ x, |(eval x P)/(eval x Q)|) at_top at_top | begin
by_cases h : 0 ≤ P.leading_coeff/Q.leading_coeff,
{ exact tendsto_abs_at_top_at_top.comp (P.div_tendsto_at_top_of_degree_gt Q hdeg hQ h) },
{ push_neg at h,
exact tendsto_abs_at_bot_at_top.comp (P.div_tendsto_at_bot_of_degree_gt Q hdeg hQ h.le) }
end | lemma | polynomial.abs_div_tendsto_at_top_of_degree_gt | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_of_degree_le (h : P.degree ≤ Q.degree) :
(λ x, eval x P) =O[at_top] (λ x, eval x Q) | begin
by_cases hp : P = 0,
{ simpa [hp] using is_O_zero (λ x, eval x Q) at_top },
{ have hq : Q ≠ 0 := ne_zero_of_degree_ge_degree h hp,
have hPQ : ∀ᶠ (x : 𝕜) in at_top, eval x Q = 0 → eval x P = 0 :=
filter.mem_of_superset (polynomial.eventually_no_roots Q hq) (λ x h h', absurd h' h),
cases le_iff... | theorem | polynomial.is_O_of_degree_le | analysis.special_functions | src/analysis/special_functions/polynomials.lean | [
"analysis.asymptotics.asymptotic_equivalent",
"analysis.asymptotics.specific_asymptotics",
"data.polynomial.ring_division"
] | [
"filter.mem_of_superset",
"polynomial.eventually_no_roots"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sq_local_homeomorph : local_homeomorph ℝ ℝ | { to_fun := λ x, x ^ 2,
inv_fun := sqrt,
source := Ioi 0,
target := Ioi 0,
map_source' := λ x hx, mem_Ioi.2 (pow_pos hx _),
map_target' := λ x hx, mem_Ioi.2 (sqrt_pos.2 hx),
left_inv' := λ x hx, sqrt_sq (le_of_lt hx),
right_inv' := λ x hx, sq_sqrt (le_of_lt hx),
open_source := is_open_Ioi,
open_target... | def | real.sq_local_homeomorph | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"continuous_on",
"continuous_pow",
"inv_fun",
"is_open_Ioi",
"local_homeomorph",
"pow_pos"
] | Local homeomorph between `(0, +∞)` and `(0, +∞)` with `to_fun = λ x, x ^ 2` and
`inv_fun = sqrt`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
deriv_sqrt_aux {x : ℝ} (hx : x ≠ 0) :
has_strict_deriv_at sqrt (1 / (2 * sqrt x)) x ∧ ∀ n, cont_diff_at ℝ n sqrt x | begin
cases hx.lt_or_lt with hx hx,
{ rw [sqrt_eq_zero_of_nonpos hx.le, mul_zero, div_zero],
have : sqrt =ᶠ[𝓝 x] (λ _, 0) := (gt_mem_nhds hx).mono (λ x hx, sqrt_eq_zero_of_nonpos hx.le),
exact ⟨(has_strict_deriv_at_const x (0 : ℝ)).congr_of_eventually_eq this.symm,
λ n, cont_diff_at_const.congr_of_ev... | lemma | real.deriv_sqrt_aux | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"cont_diff_at",
"div_zero",
"gt_mem_nhds",
"has_deriv_at_pow",
"has_strict_deriv_at",
"has_strict_deriv_at_const",
"has_strict_deriv_at_pow",
"mul_zero",
"two_ne_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at_sqrt {x : ℝ} (hx : x ≠ 0) :
has_strict_deriv_at sqrt (1 / (2 * sqrt x)) x | (deriv_sqrt_aux hx).1 | lemma | real.has_strict_deriv_at_sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"has_strict_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at_sqrt {x : ℝ} {n : ℕ∞} (hx : x ≠ 0) :
cont_diff_at ℝ n sqrt x | (deriv_sqrt_aux hx).2 n | lemma | real.cont_diff_at_sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at_sqrt {x : ℝ} (hx : x ≠ 0) : has_deriv_at sqrt (1 / (2 * sqrt x)) x | (has_strict_deriv_at_sqrt hx).has_deriv_at | lemma | real.has_deriv_at_sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"has_deriv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_within_at.sqrt (hf : has_deriv_within_at f f' s x) (hx : f x ≠ 0) :
has_deriv_within_at (λ y, sqrt (f y)) (f' / (2 * sqrt (f x))) s x | by simpa only [(∘), div_eq_inv_mul, mul_one]
using (has_deriv_at_sqrt hx).comp_has_deriv_within_at x hf | lemma | has_deriv_within_at.sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"div_eq_inv_mul",
"has_deriv_within_at",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_deriv_at.sqrt (hf : has_deriv_at f f' x) (hx : f x ≠ 0) :
has_deriv_at (λ y, sqrt (f y)) (f' / (2 * sqrt(f x))) x | by simpa only [(∘), div_eq_inv_mul, mul_one] using (has_deriv_at_sqrt hx).comp x hf | lemma | has_deriv_at.sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"div_eq_inv_mul",
"has_deriv_at",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_deriv_at.sqrt (hf : has_strict_deriv_at f f' x) (hx : f x ≠ 0) :
has_strict_deriv_at (λ t, sqrt (f t)) (f' / (2 * sqrt (f x))) x | by simpa only [(∘), div_eq_inv_mul, mul_one] using (has_strict_deriv_at_sqrt hx).comp x hf | lemma | has_strict_deriv_at.sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"div_eq_inv_mul",
"has_strict_deriv_at",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_within_sqrt (hf : differentiable_within_at ℝ f s x) (hx : f x ≠ 0)
(hxs : unique_diff_within_at ℝ s x) :
deriv_within (λx, sqrt (f x)) s x = (deriv_within f s x) / (2 * sqrt (f x)) | (hf.has_deriv_within_at.sqrt hx).deriv_within hxs | lemma | deriv_within_sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"deriv_within",
"differentiable_within_at",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
deriv_sqrt (hf : differentiable_at ℝ f x) (hx : f x ≠ 0) :
deriv (λx, sqrt (f x)) x = (deriv f x) / (2 * sqrt (f x)) | (hf.has_deriv_at.sqrt hx).deriv | lemma | deriv_sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"deriv",
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_at.sqrt (hf : has_fderiv_at f f' x) (hx : f x ≠ 0) :
has_fderiv_at (λ y, sqrt (f y)) ((1 / (2 * sqrt (f x))) • f') x | (has_deriv_at_sqrt hx).comp_has_fderiv_at x hf | lemma | has_fderiv_at.sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"has_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_strict_fderiv_at.sqrt (hf : has_strict_fderiv_at f f' x) (hx : f x ≠ 0) :
has_strict_fderiv_at (λ y, sqrt (f y)) ((1 / (2 * sqrt (f x))) • f') x | (has_strict_deriv_at_sqrt hx).comp_has_strict_fderiv_at x hf | lemma | has_strict_fderiv_at.sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"has_strict_fderiv_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_fderiv_within_at.sqrt (hf : has_fderiv_within_at f f' s x) (hx : f x ≠ 0) :
has_fderiv_within_at (λ y, sqrt (f y)) ((1 / (2 * sqrt (f x))) • f') s x | (has_deriv_at_sqrt hx).comp_has_fderiv_within_at x hf | lemma | has_fderiv_within_at.sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"has_fderiv_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_within_at.sqrt (hf : differentiable_within_at ℝ f s x) (hx : f x ≠ 0) :
differentiable_within_at ℝ (λ y, sqrt (f y)) s x | (hf.has_fderiv_within_at.sqrt hx).differentiable_within_at | lemma | differentiable_within_at.sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"differentiable_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_at.sqrt (hf : differentiable_at ℝ f x) (hx : f x ≠ 0) :
differentiable_at ℝ (λ y, sqrt (f y)) x | (hf.has_fderiv_at.sqrt hx).differentiable_at | lemma | differentiable_at.sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"differentiable_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable_on.sqrt (hf : differentiable_on ℝ f s) (hs : ∀ x ∈ s, f x ≠ 0) :
differentiable_on ℝ (λ y, sqrt (f y)) s | λ x hx, (hf x hx).sqrt (hs x hx) | lemma | differentiable_on.sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"differentiable_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
differentiable.sqrt (hf : differentiable ℝ f) (hs : ∀ x, f x ≠ 0) :
differentiable ℝ (λ y, sqrt (f y)) | λ x, (hf x).sqrt (hs x) | lemma | differentiable.sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"differentiable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_within_sqrt (hf : differentiable_within_at ℝ f s x) (hx : f x ≠ 0)
(hxs : unique_diff_within_at ℝ s x) :
fderiv_within ℝ (λx, sqrt (f x)) s x = (1 / (2 * sqrt (f x))) • fderiv_within ℝ f s x | (hf.has_fderiv_within_at.sqrt hx).fderiv_within hxs | lemma | fderiv_within_sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"differentiable_within_at",
"fderiv_within",
"unique_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
fderiv_sqrt (hf : differentiable_at ℝ f x) (hx : f x ≠ 0) :
fderiv ℝ (λx, sqrt (f x)) x = (1 / (2 * sqrt (f x))) • fderiv ℝ f x | (hf.has_fderiv_at.sqrt hx).fderiv | lemma | fderiv_sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"differentiable_at",
"fderiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_at.sqrt (hf : cont_diff_at ℝ n f x) (hx : f x ≠ 0) :
cont_diff_at ℝ n (λ y, sqrt (f y)) x | (cont_diff_at_sqrt hx).comp x hf | lemma | cont_diff_at.sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"cont_diff_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_within_at.sqrt (hf : cont_diff_within_at ℝ n f s x) (hx : f x ≠ 0) :
cont_diff_within_at ℝ n (λ y, sqrt (f y)) s x | (cont_diff_at_sqrt hx).comp_cont_diff_within_at x hf | lemma | cont_diff_within_at.sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"cont_diff_within_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff_on.sqrt (hf : cont_diff_on ℝ n f s) (hs : ∀ x ∈ s, f x ≠ 0) :
cont_diff_on ℝ n (λ y, sqrt (f y)) s | λ x hx, (hf x hx).sqrt (hs x hx) | lemma | cont_diff_on.sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"cont_diff_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cont_diff.sqrt (hf : cont_diff ℝ n f) (h : ∀ x, f x ≠ 0) :
cont_diff ℝ n (λ y, sqrt (f y)) | cont_diff_iff_cont_diff_at.2 $ λ x, (hf.cont_diff_at.sqrt (h x)) | lemma | cont_diff.sqrt | analysis.special_functions | src/analysis/special_functions/sqrt.lean | [
"analysis.calculus.cont_diff"
] | [
"cont_diff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stirling_seq (n : ℕ) : ℝ | n! / (sqrt (2 * n) * (n / exp 1) ^ n) | def | stirling.stirling_seq | analysis.special_functions | src/analysis/special_functions/stirling.lean | [
"analysis.p_series",
"data.real.pi.wallis"
] | [
"exp"
] | Define `stirling_seq n` as $\frac{n!}{\sqrt{2n}(\frac{n}{e})^n}$.
Stirling's formula states that this sequence has limit $\sqrt(π)$. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stirling_seq_zero : stirling_seq 0 = 0 | by rw [stirling_seq, cast_zero, mul_zero, real.sqrt_zero, zero_mul, div_zero] | lemma | stirling.stirling_seq_zero | analysis.special_functions | src/analysis/special_functions/stirling.lean | [
"analysis.p_series",
"data.real.pi.wallis"
] | [
"div_zero",
"mul_zero",
"real.sqrt_zero",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
stirling_seq_one : stirling_seq 1 = exp 1 / sqrt 2 | by rw [stirling_seq, pow_one, factorial_one, cast_one, mul_one, mul_one_div, one_div_div] | lemma | stirling.stirling_seq_one | analysis.special_functions | src/analysis/special_functions/stirling.lean | [
"analysis.p_series",
"data.real.pi.wallis"
] | [
"exp",
"mul_one",
"mul_one_div",
"one_div_div",
"pow_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
log_stirling_seq_formula (n : ℕ) : log (stirling_seq n.succ) =
log n.succ!- 1 / 2 * log (2 * n.succ) - n.succ * log (n.succ / exp 1) | by rw [stirling_seq, log_div, log_mul, sqrt_eq_rpow, log_rpow, real.log_pow, tsub_tsub];
try { apply ne_of_gt }; positivity | lemma | stirling.log_stirling_seq_formula | analysis.special_functions | src/analysis/special_functions/stirling.lean | [
"analysis.p_series",
"data.real.pi.wallis"
] | [
"exp",
"real.log_pow",
"tsub_tsub"
] | We have the expression
`log (stirling_seq (n + 1)) = log(n + 1)! - 1 / 2 * log(2 * n) - n * log ((n + 1) / e)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
log_stirling_seq_diff_has_sum (m : ℕ) :
has_sum (λ k : ℕ, (1 : ℝ) / (2 * k.succ + 1) * ((1 / (2 * m.succ + 1)) ^ 2) ^ k.succ)
(log (stirling_seq m.succ) - log (stirling_seq m.succ.succ)) | begin
change has_sum ((λ b : ℕ, 1 / (2 * (b : ℝ) + 1) * ((1 / (2 * m.succ + 1)) ^ 2) ^ b) ∘ succ) _,
refine (has_sum_nat_add_iff 1).mpr _,
convert (has_sum_log_one_add_inv $ cast_pos.mpr (succ_pos m)).mul_left ((m.succ : ℝ) + 1 / 2),
{ ext k,
rw [← pow_mul, pow_add],
push_cast,
have : 2 * (k : ℝ) + ... | lemma | stirling.log_stirling_seq_diff_has_sum | analysis.special_functions | src/analysis/special_functions/stirling.lean | [
"analysis.p_series",
"data.real.pi.wallis"
] | [
"div_self",
"has_sum",
"has_sum_nat_add_iff",
"inv_eq_one_div",
"mul_ne_zero",
"pow_add",
"pow_mul",
"ring"
] | The sequence `log (stirling_seq (m + 1)) - log (stirling_seq (m + 2))` has the series expansion
`∑ 1 / (2 * (k + 1) + 1) * (1 / 2 * (m + 1) + 1)^(2 * (k + 1))` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
log_stirling_seq'_antitone : antitone (real.log ∘ stirling_seq ∘ succ) | antitone_nat_of_succ_le $ λ n, sub_nonneg.mp $ (log_stirling_seq_diff_has_sum n).nonneg $ λ m,
by positivity | lemma | stirling.log_stirling_seq'_antitone | analysis.special_functions | src/analysis/special_functions/stirling.lean | [
"analysis.p_series",
"data.real.pi.wallis"
] | [
"antitone",
"antitone_nat_of_succ_le",
"real.log"
] | The sequence `log ∘ stirling_seq ∘ succ` is monotone decreasing | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
log_stirling_seq_diff_le_geo_sum (n : ℕ) :
log (stirling_seq n.succ) - log (stirling_seq n.succ.succ) ≤
(1 / (2 * n.succ + 1)) ^ 2 / (1 - (1 / (2 * n.succ + 1)) ^ 2) | begin
have h_nonneg : 0 ≤ ((1 / (2 * (n.succ : ℝ) + 1)) ^ 2) := sq_nonneg _,
have g : has_sum (λ k : ℕ, ((1 / (2 * (n.succ : ℝ) + 1)) ^ 2) ^ k.succ)
((1 / (2 * n.succ + 1)) ^ 2 / (1 - (1 / (2 * n.succ + 1)) ^ 2)),
{ have := (has_sum_geometric_of_lt_1 h_nonneg _).mul_left ((1 / (2 * (n.succ : ℝ) + 1)) ^ 2),
... | lemma | stirling.log_stirling_seq_diff_le_geo_sum | analysis.special_functions | src/analysis/special_functions/stirling.lean | [
"analysis.p_series",
"data.real.pi.wallis"
] | [
"has_sum",
"has_sum_geometric_of_lt_1",
"has_sum_le",
"inv_le_one",
"inv_lt_one",
"inv_pow",
"mul_le_of_le_one_left",
"one_div",
"one_lt_pow",
"pow_nonneg",
"sq_nonneg",
"two_ne_zero"
] | We have a bound for successive elements in the sequence `log (stirling_seq k)`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
log_stirling_seq_sub_log_stirling_seq_succ (n : ℕ) :
log (stirling_seq n.succ) - log (stirling_seq n.succ.succ) ≤ 1 / (4 * n.succ ^ 2) | begin
have h₁ : 0 < 4 * ((n : ℝ) + 1) ^ 2 := by positivity,
have h₃ : 0 < (2 * ((n : ℝ) + 1) + 1) ^ 2 := by positivity,
have h₂ : 0 < 1 - (1 / (2 * ((n : ℝ) + 1) + 1)) ^ 2,
{ rw ← mul_lt_mul_right h₃,
have H : 0 < (2 * ((n : ℝ) + 1) + 1) ^ 2 - 1 := by nlinarith [@cast_nonneg ℝ _ n],
convert H using 1; f... | lemma | stirling.log_stirling_seq_sub_log_stirling_seq_succ | analysis.special_functions | src/analysis/special_functions/stirling.lean | [
"analysis.p_series",
"data.real.pi.wallis"
] | [
"div_le_div_iff",
"div_le_div_right",
"mul_lt_mul_right"
] | We have the bound `log (stirling_seq n) - log (stirling_seq (n+1))` ≤ 1/(4 n^2) | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
log_stirling_seq_bounded_aux :
∃ (c : ℝ), ∀ (n : ℕ), log (stirling_seq 1) - log (stirling_seq n.succ) ≤ c | begin
let d := ∑' k : ℕ, (1 : ℝ) / k.succ ^ 2,
use (1 / 4 * d : ℝ),
let log_stirling_seq' : ℕ → ℝ := λ k, log (stirling_seq k.succ),
intro n,
have h₁ : ∀ k, log_stirling_seq' k - log_stirling_seq' (k + 1) ≤ 1 / 4 * (1 / k.succ ^ 2) :=
by { intro k, convert log_stirling_seq_sub_log_stirling_seq_succ k using ... | lemma | stirling.log_stirling_seq_bounded_aux | analysis.special_functions | src/analysis/special_functions/stirling.lean | [
"analysis.p_series",
"data.real.pi.wallis"
] | [
"mul_le_mul_of_nonneg_left",
"one_lt_two",
"sum_le_tsum",
"summable_nat_add_iff"
] | For any `n`, we have `log_stirling_seq 1 - log_stirling_seq n ≤ 1/4 * ∑' 1/k^2` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
log_stirling_seq_bounded_by_constant : ∃ c, ∀ (n : ℕ), c ≤ log (stirling_seq n.succ) | begin
obtain ⟨d, h⟩ := log_stirling_seq_bounded_aux,
exact ⟨log (stirling_seq 1) - d, λ n, sub_le_comm.mp (h n)⟩,
end | lemma | stirling.log_stirling_seq_bounded_by_constant | analysis.special_functions | src/analysis/special_functions/stirling.lean | [
"analysis.p_series",
"data.real.pi.wallis"
] | [] | The sequence `log_stirling_seq` is bounded below for `n ≥ 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stirling_seq'_pos (n : ℕ) : 0 < stirling_seq n.succ | by { unfold stirling_seq, positivity } | lemma | stirling.stirling_seq'_pos | analysis.special_functions | src/analysis/special_functions/stirling.lean | [
"analysis.p_series",
"data.real.pi.wallis"
] | [] | The sequence `stirling_seq` is positive for `n > 0` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stirling_seq'_bounded_by_pos_constant : ∃ a, 0 < a ∧ ∀ n : ℕ, a ≤ stirling_seq n.succ | begin
cases log_stirling_seq_bounded_by_constant with c h,
refine ⟨exp c, exp_pos _, λ n, _⟩,
rw ← le_log_iff_exp_le (stirling_seq'_pos n),
exact h n,
end | lemma | stirling.stirling_seq'_bounded_by_pos_constant | analysis.special_functions | src/analysis/special_functions/stirling.lean | [
"analysis.p_series",
"data.real.pi.wallis"
] | [] | The sequence `stirling_seq` has a positive lower bound. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stirling_seq'_antitone : antitone (stirling_seq ∘ succ) | λ n m h, (log_le_log (stirling_seq'_pos m) (stirling_seq'_pos n)).mp (log_stirling_seq'_antitone h) | lemma | stirling.stirling_seq'_antitone | analysis.special_functions | src/analysis/special_functions/stirling.lean | [
"analysis.p_series",
"data.real.pi.wallis"
] | [
"antitone"
] | The sequence `stirling_seq ∘ succ` is monotone decreasing | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stirling_seq_has_pos_limit_a :
∃ (a : ℝ), 0 < a ∧ tendsto stirling_seq at_top (𝓝 a) | begin
obtain ⟨x, x_pos, hx⟩ := stirling_seq'_bounded_by_pos_constant,
have hx' : x ∈ lower_bounds (set.range (stirling_seq ∘ succ)) := by simpa [lower_bounds] using hx,
refine ⟨_, lt_of_lt_of_le x_pos (le_cInf (set.range_nonempty _) hx'), _⟩,
rw ←filter.tendsto_add_at_top_iff_nat 1,
exact tendsto_at_top_cinfi... | lemma | stirling.stirling_seq_has_pos_limit_a | analysis.special_functions | src/analysis/special_functions/stirling.lean | [
"analysis.p_series",
"data.real.pi.wallis"
] | [
"le_cInf",
"lower_bounds",
"set.range",
"set.range_nonempty",
"tendsto_at_top_cinfi"
] | The limit `a` of the sequence `stirling_seq` satisfies `0 < a` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_self_div_two_mul_self_add_one :
tendsto (λ (n : ℕ), (n : ℝ) / (2 * n + 1)) at_top (𝓝 (1 / 2)) | begin
conv { congr, skip, skip, rw [one_div, ←add_zero (2 : ℝ)] },
refine (((tendsto_const_div_at_top_nhds_0_nat 1).const_add (2 : ℝ)).inv₀
((add_zero (2 : ℝ)).symm ▸ two_ne_zero)).congr' (eventually_at_top.mpr ⟨1, λ n hn, _⟩),
rw [add_div' (1 : ℝ) 2 n (cast_ne_zero.mpr (one_le_iff_ne_zero.mp hn)), inv_div],
... | lemma | stirling.tendsto_self_div_two_mul_self_add_one | analysis.special_functions | src/analysis/special_functions/stirling.lean | [
"analysis.p_series",
"data.real.pi.wallis"
] | [
"add_div'",
"inv_div",
"one_div",
"tendsto_const_div_at_top_nhds_0_nat",
"two_ne_zero"
] | The sequence `n / (2 * n + 1)` tends to `1/2` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
stirling_seq_pow_four_div_stirling_seq_pow_two_eq (n : ℕ) (hn : n ≠ 0) :
((stirling_seq n) ^ 4 / (stirling_seq (2 * n)) ^ 2) * (n / (2 * n + 1)) = wallis.W n | begin
rw [bit0_eq_two_mul, stirling_seq, pow_mul, stirling_seq, wallis.W_eq_factorial_ratio],
simp_rw [div_pow, mul_pow],
rw [sq_sqrt, sq_sqrt],
any_goals { positivity },
have : (n : ℝ) ≠ 0, from cast_ne_zero.mpr hn,
have : (exp 1) ≠ 0, from exp_ne_zero 1,
have : ((2 * n)!: ℝ) ≠ 0, from cast_ne_zero.mpr (... | lemma | stirling.stirling_seq_pow_four_div_stirling_seq_pow_two_eq | analysis.special_functions | src/analysis/special_functions/stirling.lean | [
"analysis.p_series",
"data.real.pi.wallis"
] | [
"bit0_eq_two_mul",
"div_pow",
"exp",
"mul_comm",
"mul_pow",
"pow_mul",
"ring"
] | For any `n ≠ 0`, we have the identity
`(stirling_seq n)^4 / (stirling_seq (2*n))^2 * (n / (2 * n + 1)) = W n`, where `W n` is the
`n`-th partial product of Wallis' formula for `π / 2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
second_wallis_limit (a : ℝ) (hane : a ≠ 0) (ha : tendsto stirling_seq at_top (𝓝 a)) :
tendsto wallis.W at_top (𝓝 (a ^ 2 / 2)) | begin
refine tendsto.congr' (eventually_at_top.mpr ⟨1, λ n hn,
stirling_seq_pow_four_div_stirling_seq_pow_two_eq n (one_le_iff_ne_zero.mp hn)⟩) _,
have h : a ^ 2 / 2 = (a ^ 4 / a ^ 2) * (1 / 2),
{ rw [mul_one_div, ←mul_one_div (a ^ 4) (a ^ 2), one_div, ←pow_sub_of_lt a],
norm_num },
rw h,
exact ((ha.p... | lemma | stirling.second_wallis_limit | analysis.special_functions | src/analysis/special_functions/stirling.lean | [
"analysis.p_series",
"data.real.pi.wallis"
] | [
"mul_one_div",
"one_div",
"pow_ne_zero"
] | Suppose the sequence `stirling_seq` (defined above) has the limit `a ≠ 0`.
Then the Wallis sequence `W n` has limit `a^2 / 2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_stirling_seq_sqrt_pi : tendsto (λ (n : ℕ), stirling_seq n) at_top (𝓝 (sqrt π)) | begin
obtain ⟨a, hapos, halimit⟩ := stirling_seq_has_pos_limit_a,
have hπ : π / 2 = a ^ 2 / 2 := tendsto_nhds_unique wallis.tendsto_W_nhds_pi_div_two
(second_wallis_limit a hapos.ne' halimit),
rwa [(div_left_inj' (two_ne_zero' ℝ)).mp hπ, sqrt_sq hapos.le],
end | theorem | stirling.tendsto_stirling_seq_sqrt_pi | analysis.special_functions | src/analysis/special_functions/stirling.lean | [
"analysis.p_series",
"data.real.pi.wallis"
] | [
"div_left_inj'",
"tendsto_nhds_unique",
"two_ne_zero'"
] | **Stirling's Formula** | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arg (x : ℂ) : ℝ | if 0 ≤ x.re
then real.arcsin (x.im / x.abs)
else if 0 ≤ x.im
then real.arcsin ((-x).im / x.abs) + π
else real.arcsin ((-x).im / x.abs) - π | def | complex.arg | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"real.arcsin"
] | `arg` returns values in the range (-π, π], such that for `x ≠ 0`,
`sin (arg x) = x.im / x.abs` and `cos (arg x) = x.re / x.abs`,
`arg 0` defaults to `0` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
sin_arg (x : ℂ) : real.sin (arg x) = x.im / x.abs | by unfold arg; split_ifs;
simp [sub_eq_add_neg, arg, real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1
(abs_le.1 (abs_im_div_abs_le_one x)).2, real.sin_add, neg_div, real.arcsin_neg,
real.sin_neg] | lemma | complex.sin_arg | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"neg_div",
"real.arcsin_neg",
"real.sin",
"real.sin_add",
"real.sin_arcsin",
"real.sin_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cos_arg {x : ℂ} (hx : x ≠ 0) : real.cos (arg x) = x.re / x.abs | begin
have habs : 0 < abs x := abs.pos hx,
have him : |im x / abs x| ≤ 1,
{ rw [_root_.abs_div, abs_abs],
exact div_le_one_of_le x.abs_im_le_abs (abs.nonneg x) },
rw abs_le at him,
rw arg, split_ifs with h₁ h₂ h₂,
{ rw [real.cos_arcsin], field_simp [real.sqrt_sq, habs.le, *] },
{ rw [real.cos_add_pi, ... | lemma | complex.cos_arg | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"abs_abs",
"abs_le",
"div_le_one_of_le",
"real.cos",
"real.cos_add_pi",
"real.cos_arcsin",
"real.cos_sub_pi",
"real.sqrt_div",
"real.sqrt_sq",
"real.sqrt_sq_eq_abs",
"sq_nonneg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_mul_exp_arg_mul_I (x : ℂ) : ↑(abs x) * exp (arg x * I) = x | begin
rcases eq_or_ne x 0 with (rfl|hx),
{ simp },
{ have : abs x ≠ 0 := abs.ne_zero hx,
ext; field_simp [sin_arg, cos_arg hx, this, mul_comm (abs x)] }
end | lemma | complex.abs_mul_exp_arg_mul_I | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"eq_or_ne",
"exp",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_mul_cos_add_sin_mul_I (x : ℂ) :
(abs x * (cos (arg x) + sin (arg x) * I) : ℂ) = x | by rw [← exp_mul_I, abs_mul_exp_arg_mul_I] | lemma | complex.abs_mul_cos_add_sin_mul_I | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z | begin
refine ⟨λ hz, ⟨arg z, _⟩, _⟩,
{ calc exp (arg z * I) = abs z * exp (arg z * I) : by rw [hz, of_real_one, one_mul]
... = z : abs_mul_exp_arg_mul_I z },
{ rintro ⟨θ, rfl⟩,
exact complex.abs_exp_of_real_mul_I θ },
end | lemma | complex.abs_eq_one_iff | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"complex.abs_exp_of_real_mul_I",
"exp",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_exp_mul_I : range (λ x : ℝ, exp (x * I)) = metric.sphere 0 1 | by { ext x, simp only [mem_sphere_zero_iff_norm, norm_eq_abs, abs_eq_one_iff, mem_range] } | lemma | complex.range_exp_mul_I | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"exp",
"metric.sphere"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ | begin
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one],
simp only [of_real_mul_re, of_real_mul_im, neg_im, ← of_real_cos, ← of_real_sin,
← mk_eq_add_mul_I, neg_div, mul_div_cancel_left _ hr.ne',
mul_nonneg_iff_right_nonneg_of_pos hr],
by_cases h₁ : θ ∈ Icc (-(π / 2)) (π / 2),
... | lemma | complex.arg_mul_cos_add_sin_mul_I | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"abs_of_nonneg",
"map_mul",
"mul_div_cancel_left",
"mul_nonneg_iff_right_nonneg_of_pos",
"mul_one",
"neg_div",
"not_and_distrib",
"real.arcsin_sin",
"real.arcsin_sin'",
"real.cos",
"real.cos_add_pi",
"real.cos_neg_of_pi_div_two_lt_of_lt",
"real.cos_nonneg_of_mem_Icc",
"real.cos_pos_of_mem_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_cos_add_sin_mul_I {θ : ℝ} (hθ : θ ∈ Ioc (-π) π) :
arg (cos θ + sin θ * I) = θ | by rw [← one_mul (_ + _), ← of_real_one, arg_mul_cos_add_sin_mul_I zero_lt_one hθ] | lemma | complex.arg_cos_add_sin_mul_I | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"one_mul",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_zero : arg 0 = 0 | by simp [arg, le_refl] | lemma | complex.arg_zero | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_abs_arg {x y : ℂ} (h₁ : x.abs = y.abs) (h₂ : x.arg = y.arg) : x = y | by rw [← abs_mul_exp_arg_mul_I x, ← abs_mul_exp_arg_mul_I y, h₁, h₂] | lemma | complex.ext_abs_arg | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext_abs_arg_iff {x y : ℂ} : x = y ↔ abs x = abs y ∧ arg x = arg y | ⟨λ h, h ▸ ⟨rfl, rfl⟩, and_imp.2 ext_abs_arg⟩ | lemma | complex.ext_abs_arg_iff | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_mem_Ioc (z : ℂ) : arg z ∈ Ioc (-π) π | begin
have hπ : 0 < π := real.pi_pos,
rcases eq_or_ne z 0 with (rfl|hz), simp [hπ, hπ.le],
rcases exists_unique_add_zsmul_mem_Ioc real.two_pi_pos (arg z) (-π) with ⟨N, hN, -⟩,
rw [two_mul, neg_add_cancel_left, ← two_mul, zsmul_eq_mul] at hN,
rw [← abs_mul_cos_add_sin_mul_I z, ← cos_add_int_mul_two_pi _ N,
... | lemma | complex.arg_mem_Ioc | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"eq_or_ne",
"exists_unique_add_zsmul_mem_Ioc",
"real.pi_pos",
"real.two_pi_pos",
"two_mul",
"zsmul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
range_arg : range arg = Ioc (-π) π | (range_subset_iff.2 arg_mem_Ioc).antisymm (λ x hx, ⟨_, arg_cos_add_sin_mul_I hx⟩) | lemma | complex.range_arg | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_le_pi (x : ℂ) : arg x ≤ π | (arg_mem_Ioc x).2 | lemma | complex.arg_le_pi | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
neg_pi_lt_arg (x : ℂ) : -π < arg x | (arg_mem_Ioc x).1 | lemma | complex.neg_pi_lt_arg | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
abs_arg_le_pi (z : ℂ) : |arg z| ≤ π | abs_le.2 ⟨(neg_pi_lt_arg z).le, arg_le_pi z⟩ | lemma | complex.abs_arg_le_pi | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im | begin
rcases eq_or_ne z 0 with (rfl|h₀), { simp },
calc 0 ≤ arg z ↔ 0 ≤ real.sin (arg z) :
⟨λ h, real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩,
by { contrapose!, intro h, exact real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _) }⟩
... ↔ _ : by rw [sin_arg, le_div_iff (abs.pos h₀), zero_mul]
end | lemma | complex.arg_nonneg_iff | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"eq_or_ne",
"le_div_iff",
"real.sin",
"real.sin_neg_of_neg_of_neg_pi_lt",
"real.sin_nonneg_of_mem_Icc",
"zero_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_neg_iff {z : ℂ} : arg z < 0 ↔ z.im < 0 | lt_iff_lt_of_le_iff_le arg_nonneg_iff | lemma | complex.arg_neg_iff | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"lt_iff_lt_of_le_iff_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_real_mul (x : ℂ) {r : ℝ} (hr : 0 < r) : arg (r * x) = arg x | begin
rcases eq_or_ne x 0 with (rfl|hx), { rw mul_zero },
conv_lhs { rw [← abs_mul_cos_add_sin_mul_I x, ← mul_assoc, ← of_real_mul,
arg_mul_cos_add_sin_mul_I (mul_pos hr (abs.pos hx)) x.arg_mem_Ioc] }
end | lemma | complex.arg_real_mul | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"eq_or_ne",
"mul_assoc",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_eq_arg_iff {x y : ℂ} (hx : x ≠ 0) (hy : y ≠ 0) :
arg x = arg y ↔ (abs y / abs x : ℂ) * x = y | begin
simp only [ext_abs_arg_iff, map_mul, map_div₀, abs_of_real, abs_abs,
div_mul_cancel _ (abs.ne_zero hx), eq_self_iff_true, true_and],
rw [← of_real_div, arg_real_mul],
exact div_pos (abs.pos hy) (abs.pos hx)
end | lemma | complex.arg_eq_arg_iff | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"abs_abs",
"div_mul_cancel",
"div_pos",
"map_div₀",
"map_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_one : arg 1 = 0 | by simp [arg, zero_le_one] | lemma | complex.arg_one | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_neg_one : arg (-1) = π | by simp [arg, le_refl, not_le.2 (zero_lt_one' ℝ)] | lemma | complex.arg_neg_one | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"zero_lt_one'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_I : arg I = π / 2 | by simp [arg, le_refl] | lemma | complex.arg_I | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_neg_I : arg (-I) = -(π / 2) | by simp [arg, le_refl] | lemma | complex.arg_neg_I | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tan_arg (x : ℂ) : real.tan (arg x) = x.im / x.re | begin
by_cases h : x = 0,
{ simp only [h, zero_div, complex.zero_im, complex.arg_zero, real.tan_zero, complex.zero_re] },
rw [real.tan_eq_sin_div_cos, sin_arg, cos_arg h,
div_div_div_cancel_right _ (abs.ne_zero h)]
end | lemma | complex.tan_arg | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"complex.arg_zero",
"complex.zero_im",
"complex.zero_re",
"div_div_div_cancel_right",
"real.tan",
"real.tan_eq_sin_div_cos",
"real.tan_zero",
"zero_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_of_real_of_nonneg {x : ℝ} (hx : 0 ≤ x) : arg x = 0 | by simp [arg, hx] | lemma | complex.arg_of_real_of_nonneg | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_eq_zero_iff {z : ℂ} : arg z = 0 ↔ 0 ≤ z.re ∧ z.im = 0 | begin
refine ⟨λ h, _, _⟩,
{ rw [←abs_mul_cos_add_sin_mul_I z, h],
simp [abs.nonneg] },
{ cases z with x y,
rintro ⟨h, rfl : y = 0⟩,
exact arg_of_real_of_nonneg h }
end | lemma | complex.arg_eq_zero_iff | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_eq_pi_iff {z : ℂ} : arg z = π ↔ z.re < 0 ∧ z.im = 0 | begin
by_cases h₀ : z = 0, { simp [h₀, lt_irrefl, real.pi_ne_zero.symm] },
split,
{ intro h, rw [← abs_mul_cos_add_sin_mul_I z, h], simp [h₀] },
{ cases z with x y, rintro ⟨h : x < 0, rfl : y = 0⟩,
rw [← arg_neg_one, ← arg_real_mul (-1) (neg_pos.2 h)], simp [← of_real_def] }
end | lemma | complex.arg_eq_pi_iff | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_lt_pi_iff {z : ℂ} : arg z < π ↔ 0 ≤ z.re ∨ z.im ≠ 0 | by rw [(arg_le_pi z).lt_iff_ne, not_iff_comm, not_or_distrib, not_le, not_not, arg_eq_pi_iff] | lemma | complex.arg_lt_pi_iff | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"not_iff_comm",
"not_not",
"not_or_distrib"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_of_real_of_neg {x : ℝ} (hx : x < 0) : arg x = π | arg_eq_pi_iff.2 ⟨hx, rfl⟩ | lemma | complex.arg_of_real_of_neg | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arg_eq_pi_div_two_iff {z : ℂ} : arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im | begin
by_cases h₀ : z = 0, { simp [h₀, lt_irrefl, real.pi_div_two_pos.ne] },
split,
{ intro h, rw [← abs_mul_cos_add_sin_mul_I z, h], simp [h₀] },
{ cases z with x y, rintro ⟨rfl : x = 0, hy : 0 < y⟩,
rw [← arg_I, ← arg_real_mul I hy, of_real_mul', I_re, I_im, mul_zero, mul_one] }
end | lemma | complex.arg_eq_pi_div_two_iff | analysis.special_functions.complex | src/analysis/special_functions/complex/arg.lean | [
"analysis.special_functions.trigonometric.angle",
"analysis.special_functions.trigonometric.inverse"
] | [
"mul_one",
"mul_zero"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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