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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is_o_exp_neg_mul_sq_cocompact {a : ℂ} (ha : 0 < a.re) (s : ℝ) :
(λ x : ℝ, complex.exp (-a * x ^ 2)) =o[cocompact ℝ] (λ x : ℝ, |x| ^ s) | begin
rw ←is_o_norm_left,
simp_rw norm_cexp_neg_mul_sq,
apply is_o_of_tendsto',
{ refine eventually.filter_mono cocompact_le_cofinite _,
refine (eventually_cofinite_ne 0).mp (eventually_of_forall (λ x hx h, _)),
exact ((rpow_pos_of_pos (abs_pos.mpr hx) _).ne' h).elim },
{ refine tendsto.congr' (eventu... | lemma | is_o_exp_neg_mul_sq_cocompact | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"abs_nonneg",
"complex.exp",
"div_eq_mul_inv",
"mul_comm",
"norm_cexp_neg_mul_sq",
"norm_mul",
"tendsto_rpow_abs_mul_exp_neg_mul_sq_cocompact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
complex.tsum_exp_neg_mul_int_sq {a : ℂ} (ha : 0 < a.re) :
∑' (n : ℤ), cexp (-π * a * n ^ 2) = 1 / a ^ (1 / 2 : ℂ) * ∑' (n : ℤ), cexp (-π / a * n ^ 2) | begin
let f := λ x : ℝ, cexp (-π * a * x ^ 2),
have h1 : 0 < (↑π * a).re,
{ rw [of_real_mul_re],
exact mul_pos pi_pos ha },
have h2 : 0 < (↑π / a).re,
{ rw [div_eq_mul_inv, of_real_mul_re, inv_re],
refine mul_pos pi_pos (div_pos ha $ norm_sq_pos.mpr _),
contrapose! ha,
rw [ha, zero_re] },
ha... | lemma | complex.tsum_exp_neg_mul_int_sq | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"continuous",
"div_eq_mul_inv",
"div_pos",
"fourier_transform_gaussian_pi",
"is_o_exp_neg_mul_sq_cocompact",
"one_lt_two",
"real.tsum_eq_tsum_fourier_integral_of_rpow_decay",
"ring",
"tsum_mul_left"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.tsum_exp_neg_mul_int_sq {a : ℝ} (ha : 0 < a) :
∑' (n : ℤ), exp (-π * a * n ^ 2) = 1 / a ^ (1 / 2 : ℝ) * ∑' (n : ℤ), exp (-π / a * n ^ 2) | by simpa only [←of_real_inj, of_real_mul, of_real_tsum, of_real_exp, of_real_div, of_real_pow,
of_real_int_cast, of_real_neg, of_real_cpow ha.le, of_real_bit0, of_real_one] using
complex.tsum_exp_neg_mul_int_sq (by rwa [of_real_re] : 0 < (a : ℂ).re) | lemma | real.tsum_exp_neg_mul_int_sq | analysis.special_functions | src/analysis/special_functions/gaussian.lean | [
"analysis.special_functions.gamma.basic",
"analysis.special_functions.polar_coord",
"analysis.convex.complex",
"analysis.complex.cauchy_integral",
"analysis.fourier.poisson_summation"
] | [
"complex.tsum_exp_neg_mul_int_sq",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_on_exp_Iic (c : ℝ) : integrable_on exp (Iic c) | begin
refine integrable_on_Iic_of_interval_integral_norm_bounded (exp c) c (λ y,
interval_integrable_exp.1) tendsto_id (eventually_of_mem (Iic_mem_at_bot 0) (λ y hy, _)),
simp_rw [(norm_of_nonneg (exp_pos _).le), integral_exp, sub_le_self_iff],
exact (exp_pos _).le,
end | lemma | integrable_on_exp_Iic | analysis.special_functions | src/analysis/special_functions/improper_integrals.lean | [
"analysis.special_functions.integrals",
"measure_theory.group.integration",
"measure_theory.integral.exp_decay",
"measure_theory.integral.integral_eq_improper",
"measure_theory.measure.lebesgue.integral"
] | [
"exp",
"integral_exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_exp_Iic (c : ℝ) : ∫ (x : ℝ) in Iic c, exp x = exp c | begin
refine tendsto_nhds_unique (interval_integral_tendsto_integral_Iic _ (integrable_on_exp_Iic _)
tendsto_id) _,
simp_rw [integral_exp, (show 𝓝 (exp c) = 𝓝 (exp c - 0), by rw sub_zero)],
exact tendsto_exp_at_bot.const_sub _,
end | lemma | integral_exp_Iic | analysis.special_functions | src/analysis/special_functions/improper_integrals.lean | [
"analysis.special_functions.integrals",
"measure_theory.group.integration",
"measure_theory.integral.exp_decay",
"measure_theory.integral.integral_eq_improper",
"measure_theory.measure.lebesgue.integral"
] | [
"exp",
"integrable_on_exp_Iic",
"integral_exp",
"tendsto_nhds_unique"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_exp_Iic_zero : ∫ (x : ℝ) in Iic 0, exp x = 1 | exp_zero ▸ integral_exp_Iic 0 | lemma | integral_exp_Iic_zero | analysis.special_functions | src/analysis/special_functions/improper_integrals.lean | [
"analysis.special_functions.integrals",
"measure_theory.group.integration",
"measure_theory.integral.exp_decay",
"measure_theory.integral.integral_eq_improper",
"measure_theory.measure.lebesgue.integral"
] | [
"exp",
"exp_zero",
"integral_exp_Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_exp_neg_Ioi (c : ℝ) : ∫ (x : ℝ) in Ioi c, exp (-x) = exp (-c) | by simpa only [integral_comp_neg_Ioi] using integral_exp_Iic (-c) | lemma | integral_exp_neg_Ioi | analysis.special_functions | src/analysis/special_functions/improper_integrals.lean | [
"analysis.special_functions.integrals",
"measure_theory.group.integration",
"measure_theory.integral.exp_decay",
"measure_theory.integral.integral_eq_improper",
"measure_theory.measure.lebesgue.integral"
] | [
"exp",
"integral_comp_neg_Ioi",
"integral_exp_Iic"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_exp_neg_Ioi_zero : ∫ (x : ℝ) in Ioi 0, exp (-x) = 1 | by simpa only [neg_zero, exp_zero] using integral_exp_neg_Ioi 0 | lemma | integral_exp_neg_Ioi_zero | analysis.special_functions | src/analysis/special_functions/improper_integrals.lean | [
"analysis.special_functions.integrals",
"measure_theory.group.integration",
"measure_theory.integral.exp_decay",
"measure_theory.integral.integral_eq_improper",
"measure_theory.measure.lebesgue.integral"
] | [
"exp",
"exp_zero",
"integral_exp_neg_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_on_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 < c) :
integrable_on (λ t : ℝ, t ^ a) (Ioi c) | begin
have hd : ∀ (x : ℝ) (hx : x ∈ Ici c), has_deriv_at (λ t, t ^ (a + 1) / (a + 1)) (x ^ a) x,
{ intros x hx,
convert (has_deriv_at_rpow_const (or.inl (hc.trans_le hx).ne')).div_const _,
field_simp [show a + 1 ≠ 0, from ne_of_lt (by linarith), mul_comm] },
have ht : tendsto (λ t, t ^ (a + 1) / (a + 1)) ... | lemma | integrable_on_Ioi_rpow_of_lt | analysis.special_functions | src/analysis/special_functions/improper_integrals.lean | [
"analysis.special_functions.integrals",
"measure_theory.group.integration",
"measure_theory.integral.exp_decay",
"measure_theory.integral.integral_eq_improper",
"measure_theory.measure.lebesgue.integral"
] | [
"has_deriv_at",
"mul_comm",
"tendsto_rpow_neg_at_top"
] | If `0 < c`, then `(λ t : ℝ, t ^ a)` is integrable on `(c, ∞)` for all `a < -1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_Ioi_rpow_of_lt {a : ℝ} (ha : a < -1) {c : ℝ} (hc : 0 < c) :
∫ (t : ℝ) in Ioi c, t ^ a = -c ^ (a + 1) / (a + 1) | begin
have hd : ∀ (x : ℝ) (hx : x ∈ Ici c), has_deriv_at (λ t, t ^ (a + 1) / (a + 1)) (x ^ a) x,
{ intros x hx,
convert (has_deriv_at_rpow_const (or.inl (hc.trans_le hx).ne')).div_const _,
field_simp [show a + 1 ≠ 0, from ne_of_lt (by linarith), mul_comm] },
have ht : tendsto (λ t, t ^ (a + 1) / (a + 1)) ... | lemma | integral_Ioi_rpow_of_lt | analysis.special_functions | src/analysis/special_functions/improper_integrals.lean | [
"analysis.special_functions.integrals",
"measure_theory.group.integration",
"measure_theory.integral.exp_decay",
"measure_theory.integral.integral_eq_improper",
"measure_theory.measure.lebesgue.integral"
] | [
"has_deriv_at",
"integrable_on_Ioi_rpow_of_lt",
"mul_comm",
"neg_div",
"tendsto_rpow_neg_at_top",
"zero_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_on_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0 < c) :
integrable_on (λ t : ℝ, (t : ℂ) ^ a) (Ioi c) | begin
rw [integrable_on, ←integrable_norm_iff, ←integrable_on],
refine (integrable_on_Ioi_rpow_of_lt ha hc).congr_fun (λ x hx, _) measurable_set_Ioi,
{ dsimp only,
rw [complex.norm_eq_abs, complex.abs_cpow_eq_rpow_re_of_pos (hc.trans hx)] },
{ refine continuous_on.ae_strongly_measurable (λ t ht, _) measurab... | lemma | integrable_on_Ioi_cpow_of_lt | analysis.special_functions | src/analysis/special_functions/improper_integrals.lean | [
"analysis.special_functions.integrals",
"measure_theory.group.integration",
"measure_theory.integral.exp_decay",
"measure_theory.integral.integral_eq_improper",
"measure_theory.measure.lebesgue.integral"
] | [
"complex.abs_cpow_eq_rpow_re_of_pos",
"complex.continuous_at_of_real_cpow_const",
"complex.norm_eq_abs",
"continuous_on.ae_strongly_measurable",
"continuous_within_at",
"integrable_on_Ioi_rpow_of_lt",
"measurable_set_Ioi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_Ioi_cpow_of_lt {a : ℂ} (ha : a.re < -1) {c : ℝ} (hc : 0 < c) :
∫ (t : ℝ) in Ioi c, (t : ℂ) ^ a = -(c : ℂ) ^ (a + 1) / (a + 1) | begin
refine tendsto_nhds_unique (interval_integral_tendsto_integral_Ioi c
(integrable_on_Ioi_cpow_of_lt ha hc) tendsto_id) _,
suffices : tendsto (λ (x : ℝ), ((x : ℂ) ^ (a + 1) - (c : ℂ) ^ (a + 1)) / (a + 1)) at_top
(𝓝 $ -c ^ (a + 1) / (a + 1)),
{ refine this.congr' ((eventually_gt_at_top 0).mp (eventual... | lemma | integral_Ioi_cpow_of_lt | analysis.special_functions | src/analysis/special_functions/improper_integrals.lean | [
"analysis.special_functions.integrals",
"measure_theory.group.integration",
"measure_theory.integral.exp_decay",
"measure_theory.integral.integral_eq_improper",
"measure_theory.measure.lebesgue.integral"
] | [
"complex.abs_cpow_eq_rpow_re_of_pos",
"complex.add_re",
"complex.neg_re",
"complex.norm_eq_abs",
"complex.one_re",
"integrable_on_Ioi_cpow_of_lt",
"integral_cpow",
"sub_div",
"tendsto_nhds_unique",
"tendsto_rpow_neg_at_top"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interval_integrable_pow : interval_integrable (λ x, x^n) μ a b | (continuous_pow n).interval_integrable a b | lemma | interval_integral.interval_integrable_pow | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"continuous_pow",
"interval_integrable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interval_integrable_zpow {n : ℤ} (h : 0 ≤ n ∨ (0 : ℝ) ∉ [a, b]) :
interval_integrable (λ x, x ^ n) μ a b | (continuous_on_id.zpow₀ n $ λ x hx, h.symm.imp (ne_of_mem_of_not_mem hx) id).interval_integrable | lemma | interval_integral.interval_integrable_zpow | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"interval_integrable",
"ne_of_mem_of_not_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interval_integrable_rpow {r : ℝ} (h : 0 ≤ r ∨ (0 : ℝ) ∉ [a, b]) :
interval_integrable (λ x, x ^ r) μ a b | (continuous_on_id.rpow_const $ λ x hx, h.symm.imp (ne_of_mem_of_not_mem hx) id).interval_integrable | lemma | interval_integral.interval_integrable_rpow | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"interval_integrable",
"ne_of_mem_of_not_mem"
] | See `interval_integrable_rpow'` for a version with a weaker hypothesis on `r`, but assuming the
measure is volume. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
interval_integrable_rpow' {r : ℝ} (h : -1 < r) :
interval_integrable (λ x, x ^ r) volume a b | begin
suffices : ∀ (c : ℝ), interval_integrable (λ x, x ^ r) volume 0 c,
{ exact interval_integrable.trans (this a).symm (this b) },
have : ∀ (c : ℝ), (0 ≤ c) → interval_integrable (λ x, x ^ r) volume 0 c,
{ intros c hc,
rw [interval_integrable_iff, uIoc_of_le hc],
have hderiv : ∀ x ∈ Ioo 0 c, has_deriv... | lemma | interval_integral.interval_integrable_rpow' | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"algebra.id.smul_eq_mul",
"has_deriv_at",
"interval_integrable",
"interval_integrable.iff_comp_neg",
"interval_integrable.trans",
"interval_integrable_iff",
"measurable_set_Ioc",
"mul_comm",
"pi.smul_apply",
"real.has_deriv_at_rpow_const",
"ring"
] | See `interval_integrable_rpow` for a version applying to any locally finite measure, but with a
stronger hypothesis on `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
interval_integrable_cpow {r : ℂ} (h : 0 ≤ r.re ∨ (0 : ℝ) ∉ [a, b]) :
interval_integrable (λ x : ℝ, (x : ℂ) ^ r) μ a b | begin
by_cases h2 : (0:ℝ) ∉ [a,b],
{ -- Easy case #1: 0 ∉ [a, b] -- use continuity.
refine (continuous_at.continuous_on (λ x hx, _)).interval_integrable,
exact complex.continuous_at_of_real_cpow_const _ _ (or.inr $ ne_of_mem_of_not_mem hx h2) },
rw [eq_false_intro h2, or_false] at h,
rcases lt_or_eq_of_... | lemma | interval_integral.interval_integrable_cpow | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"complex.I",
"complex.abs_cpow_eq_rpow_re_of_pos",
"complex.continuous_at_of_real_cpow_const",
"complex.continuous_of_real_cpow_const",
"complex.norm_eq_abs",
"complex.of_real_cpow_of_nonpos",
"continuous_at.continuous_on",
"interval_integrable",
"interval_integrable.interval_integrable_norm_iff",
... | See `interval_integrable_cpow'` for a version with a weaker hypothesis on `r`, but assuming the
measure is volume. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
interval_integrable_cpow' {r : ℂ} (h : -1 < r.re) :
interval_integrable (λ x:ℝ, (x:ℂ) ^ r) volume a b | begin
suffices : ∀ (c : ℝ), interval_integrable (λ x, (x : ℂ) ^ r) volume 0 c,
{ exact interval_integrable.trans (this a).symm (this b) },
have : ∀ (c : ℝ), (0 ≤ c) → interval_integrable (λ x, (x : ℂ) ^ r) volume 0 c,
{ intros c hc,
rw ←interval_integrable.interval_integrable_norm_iff,
{ rw interval_int... | lemma | interval_integral.interval_integrable_cpow' | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"complex.I",
"complex.abs_cpow_eq_rpow_re_of_pos",
"complex.exp",
"complex.norm_eq_abs",
"complex.of_real_cpow_of_nonpos",
"complex.of_real_re",
"continuous_at.continuous_on",
"continuous_at_cpow_const",
"continuous_on.ae_strongly_measurable",
"interval_integrable",
"interval_integrable.iff_comp... | See `interval_integrable_cpow` for a version applying to any locally finite measure, but with a
stronger hypothesis on `r`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
interval_integrable_id : interval_integrable (λ x, x) μ a b | continuous_id.interval_integrable a b | lemma | interval_integral.interval_integrable_id | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"interval_integrable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interval_integrable_const : interval_integrable (λ x, c) μ a b | continuous_const.interval_integrable a b | lemma | interval_integral.interval_integrable_const | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"interval_integrable",
"interval_integrable_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interval_integrable_one_div (h : ∀ x : ℝ, x ∈ [a, b] → f x ≠ 0)
(hf : continuous_on f [a, b]) :
interval_integrable (λ x, 1 / f x) μ a b | (continuous_on_const.div hf h).interval_integrable | lemma | interval_integral.interval_integrable_one_div | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"continuous_on",
"interval_integrable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interval_integrable_inv (h : ∀ x : ℝ, x ∈ [a, b] → f x ≠ 0)
(hf : continuous_on f [a, b]) :
interval_integrable (λ x, (f x)⁻¹) μ a b | by simpa only [one_div] using interval_integrable_one_div h hf | lemma | interval_integral.interval_integrable_inv | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"continuous_on",
"interval_integrable",
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interval_integrable_exp : interval_integrable exp μ a b | continuous_exp.interval_integrable a b | lemma | interval_integral.interval_integrable_exp | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"exp",
"interval_integrable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.interval_integrable.log
(hf : continuous_on f [a, b]) (h : ∀ x : ℝ, x ∈ [a, b] → f x ≠ 0) :
interval_integrable (λ x, log (f x)) μ a b | (continuous_on.log hf h).interval_integrable | lemma | interval_integrable.log | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"continuous_on",
"continuous_on.log",
"interval_integrable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interval_integrable_log (h : (0:ℝ) ∉ [a, b]) :
interval_integrable log μ a b | interval_integrable.log continuous_on_id $ λ x hx, ne_of_mem_of_not_mem hx h | lemma | interval_integral.interval_integrable_log | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"continuous_on_id",
"interval_integrable",
"interval_integrable.log",
"ne_of_mem_of_not_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interval_integrable_sin : interval_integrable sin μ a b | continuous_sin.interval_integrable a b | lemma | interval_integral.interval_integrable_sin | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"interval_integrable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interval_integrable_cos : interval_integrable cos μ a b | continuous_cos.interval_integrable a b | lemma | interval_integral.interval_integrable_cos | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"interval_integrable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interval_integrable_one_div_one_add_sq : interval_integrable (λ x : ℝ, 1 / (1 + x^2)) μ a b | begin
refine (continuous_const.div _ (λ x, _)).interval_integrable a b,
{ continuity },
{ nlinarith },
end | lemma | interval_integral.interval_integrable_one_div_one_add_sq | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"continuity",
"interval_integrable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interval_integrable_inv_one_add_sq : interval_integrable (λ x : ℝ, (1 + x^2)⁻¹) μ a b | by simpa only [one_div] using interval_integrable_one_div_one_add_sq | lemma | interval_integral.interval_integrable_inv_one_add_sq | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"interval_integrable",
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_integral_comp_mul_right : c * ∫ x in a..b, f (x * c) = ∫ x in a*c..b*c, f x | smul_integral_comp_mul_right f c | lemma | interval_integral.mul_integral_comp_mul_right | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_integral_comp_mul_left : c * ∫ x in a..b, f (c * x) = ∫ x in c*a..c*b, f x | smul_integral_comp_mul_left f c | lemma | interval_integral.mul_integral_comp_mul_left | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_integral_comp_div : c⁻¹ * ∫ x in a..b, f (x / c) = ∫ x in a/c..b/c, f x | inv_smul_integral_comp_div f c | lemma | interval_integral.inv_mul_integral_comp_div | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_integral_comp_mul_add : c * ∫ x in a..b, f (c * x + d) = ∫ x in c*a+d..c*b+d, f x | smul_integral_comp_mul_add f c d | lemma | interval_integral.mul_integral_comp_mul_add | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_integral_comp_add_mul : c * ∫ x in a..b, f (d + c * x) = ∫ x in d+c*a..d+c*b, f x | smul_integral_comp_add_mul f c d | lemma | interval_integral.mul_integral_comp_add_mul | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_integral_comp_div_add : c⁻¹ * ∫ x in a..b, f (x / c + d) = ∫ x in a/c+d..b/c+d, f x | inv_smul_integral_comp_div_add f c d | lemma | interval_integral.inv_mul_integral_comp_div_add | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_integral_comp_add_div : c⁻¹ * ∫ x in a..b, f (d + x / c) = ∫ x in d+a/c..d+b/c, f x | inv_smul_integral_comp_add_div f c d | lemma | interval_integral.inv_mul_integral_comp_add_div | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_integral_comp_mul_sub : c * ∫ x in a..b, f (c * x - d) = ∫ x in c*a-d..c*b-d, f x | smul_integral_comp_mul_sub f c d | lemma | interval_integral.mul_integral_comp_mul_sub | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mul_integral_comp_sub_mul : c * ∫ x in a..b, f (d - c * x) = ∫ x in d-c*b..d-c*a, f x | smul_integral_comp_sub_mul f c d | lemma | interval_integral.mul_integral_comp_sub_mul | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_integral_comp_div_sub : c⁻¹ * ∫ x in a..b, f (x / c - d) = ∫ x in a/c-d..b/c-d, f x | inv_smul_integral_comp_div_sub f c d | lemma | interval_integral.inv_mul_integral_comp_div_sub | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inv_mul_integral_comp_sub_div : c⁻¹ * ∫ x in a..b, f (d - x / c) = ∫ x in d-b/c..d-a/c, f x | inv_smul_integral_comp_sub_div f c d | lemma | interval_integral.inv_mul_integral_comp_sub_div | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_cpow {r : ℂ} (h : -1 < r.re ∨ (r ≠ -1 ∧ (0 : ℝ) ∉ [a, b])) :
∫ (x : ℝ) in a..b, (x : ℂ) ^ r = (b ^ (r + 1) - a ^ (r + 1)) / (r + 1) | begin
rw sub_div,
have hr : r + 1 ≠ 0,
{ cases h,
{ apply_fun complex.re,
rw [complex.add_re, complex.one_re, complex.zero_re, ne.def, add_eq_zero_iff_eq_neg],
exact h.ne' },
{ rw [ne.def, ←add_eq_zero_iff_eq_neg] at h, exact h.1 } },
by_cases hab : (0:ℝ) ∉ [a, b],
{ refine integral_eq_sub... | lemma | integral_cpow | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"complex.add_re",
"complex.continuous_of_real_cpow_const",
"complex.one_re",
"complex.zero_cpow",
"complex.zero_re",
"continuous_on",
"has_deriv_at_of_real_cpow",
"has_deriv_within_at",
"ne_of_mem_of_not_mem",
"ring",
"sub_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_rpow {r : ℝ} (h : -1 < r ∨ (r ≠ -1 ∧ (0 : ℝ) ∉ [a, b])) :
∫ x in a..b, x ^ r = (b ^ (r + 1) - a ^ (r + 1)) / (r + 1) | begin
have h' : -1 < (r:ℂ).re ∨ (r:ℂ) ≠ -1 ∧ (0:ℝ) ∉ [a, b],
{ cases h,
{ left, rwa complex.of_real_re },
{ right, rwa [←complex.of_real_one, ←complex.of_real_neg, ne.def, complex.of_real_inj] } },
have : ∫ x in a..b, (x:ℂ) ^ (r :ℂ) = ((b:ℂ) ^ (r + 1 : ℂ) - (a:ℂ) ^ (r + 1 : ℂ)) / (r + 1),
from integra... | lemma | integral_rpow | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"complex.of_real_inj",
"complex.of_real_mul_re",
"complex.of_real_re",
"complex.real_smul",
"complex.sub_re",
"div_eq_inv_mul",
"integral_cpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_zpow {n : ℤ} (h : 0 ≤ n ∨ n ≠ -1 ∧ (0 : ℝ) ∉ [a, b]) :
∫ x in a..b, x ^ n = (b ^ (n + 1) - a ^ (n + 1)) / (n + 1) | begin
replace h : -1 < (n : ℝ) ∨ (n : ℝ) ≠ -1 ∧ (0 : ℝ) ∉ [a, b], by exact_mod_cast h,
exact_mod_cast integral_rpow h,
end | lemma | integral_zpow | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_pow : ∫ x in a..b, x ^ n = (b ^ (n + 1) - a ^ (n + 1)) / (n + 1) | by simpa only [←int.coe_nat_succ, zpow_coe_nat] using integral_zpow (or.inl (int.coe_nat_nonneg n)) | lemma | integral_pow | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"int.coe_nat_nonneg",
"integral_zpow",
"zpow_coe_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_pow_abs_sub_uIoc :
∫ x in Ι a b, |x - a| ^ n = |b - a| ^ (n + 1) / (n + 1) | begin
cases le_or_lt a b with hab hab,
{ calc ∫ x in Ι a b, |x - a| ^ n = ∫ x in a..b, |x - a| ^ n :
by rw [uIoc_of_le hab, ← integral_of_le hab]
... = ∫ x in 0..(b - a), x ^ n :
begin
simp only [integral_comp_sub_right (λ x, |x| ^ n), sub_self],
refine integral_congr (λ x hx, congr_... | lemma | integral_pow_abs_sub_uIoc | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"abs_of_neg",
"abs_of_nonneg",
"abs_of_nonpos",
"congr_arg2"
] | Integral of `|x - a| ^ n` over `Ι a b`. This integral appears in the proof of the
Picard-Lindelöf/Cauchy-Lipschitz theorem. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_id : ∫ x in a..b, x = (b ^ 2 - a ^ 2) / 2 | by simpa using integral_pow 1 | lemma | integral_id | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_one : ∫ x in a..b, (1 : ℝ) = b - a | by simp only [mul_one, smul_eq_mul, integral_const] | lemma | integral_one | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"mul_one",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_const_on_unit_interval : ∫ x in a..(a + 1), b = b | by simp | lemma | integral_const_on_unit_interval | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_inv (h : (0:ℝ) ∉ [a, b]) : ∫ x in a..b, x⁻¹ = log (b / a) | begin
have h' := λ x hx, ne_of_mem_of_not_mem hx h,
rw [integral_deriv_eq_sub' _ deriv_log' (λ x hx, differentiable_at_log (h' x hx))
(continuous_on_inv₀.mono $ subset_compl_singleton_iff.mpr h),
log_div (h' b right_mem_uIcc) (h' a left_mem_uIcc)],
end | lemma | integral_inv | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"ne_of_mem_of_not_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_inv_of_pos (ha : 0 < a) (hb : 0 < b) : ∫ x in a..b, x⁻¹ = log (b / a) | integral_inv $ not_mem_uIcc_of_lt ha hb | lemma | integral_inv_of_pos | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_inv_of_neg (ha : a < 0) (hb : b < 0) : ∫ x in a..b, x⁻¹ = log (b / a) | integral_inv $ not_mem_uIcc_of_gt ha hb | lemma | integral_inv_of_neg | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_inv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_one_div (h : (0:ℝ) ∉ [a, b]) : ∫ x : ℝ in a..b, 1/x = log (b / a) | by simp only [one_div, integral_inv h] | lemma | integral_one_div | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_inv",
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_one_div_of_pos (ha : 0 < a) (hb : 0 < b) : ∫ x : ℝ in a..b, 1/x = log (b / a) | by simp only [one_div, integral_inv_of_pos ha hb] | lemma | integral_one_div_of_pos | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_inv_of_pos",
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_one_div_of_neg (ha : a < 0) (hb : b < 0) : ∫ x : ℝ in a..b, 1/x = log (b / a) | by simp only [one_div, integral_inv_of_neg ha hb] | lemma | integral_one_div_of_neg | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_inv_of_neg",
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_exp : ∫ x in a..b, exp x = exp b - exp a | by rw integral_deriv_eq_sub'; norm_num [continuous_on_exp] | lemma | integral_exp | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"continuous_on_exp",
"exp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_exp_mul_complex {c : ℂ} (hc : c ≠ 0) :
∫ x in a..b, complex.exp (c * x) = (complex.exp (c * b) - complex.exp (c * a)) / c | begin
have D : ∀ (x : ℝ), has_deriv_at (λ (y : ℝ), complex.exp (c * y) / c) (complex.exp (c * x)) x,
{ intro x,
conv { congr, skip, rw ←mul_div_cancel (complex.exp (c * x)) hc, },
convert ((complex.has_deriv_at_exp _).comp x _).div_const c using 1,
simpa only [mul_one] using ((has_deriv_at_id (x:ℂ)).con... | lemma | integral_exp_mul_complex | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"complex.exp",
"complex.has_deriv_at_exp",
"continuity",
"continuous.continuous_on",
"deriv",
"differentiable_at",
"has_deriv_at",
"has_deriv_at_id",
"mul_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_log (h : (0:ℝ) ∉ [a, b]) :
∫ x in a..b, log x = b * log b - a * log a - b + a | begin
obtain ⟨h', heq⟩ := ⟨λ x hx, ne_of_mem_of_not_mem hx h, λ x hx, mul_inv_cancel (h' x hx)⟩,
convert integral_mul_deriv_eq_deriv_mul (λ x hx, has_deriv_at_log (h' x hx))
(λ x hx, has_deriv_at_id x)
(continuous_on_inv₀.mono $ subset_compl_singleton_iff.mpr h).interval_integrable
continuous_on_c... | lemma | integral_log | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"has_deriv_at_id",
"interval_integrable",
"mul_comm",
"mul_inv_cancel",
"ne_of_mem_of_not_mem"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_log_of_pos (ha : 0 < a) (hb : 0 < b) :
∫ x in a..b, log x = b * log b - a * log a - b + a | integral_log $ not_mem_uIcc_of_lt ha hb | lemma | integral_log_of_pos | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_log"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_log_of_neg (ha : a < 0) (hb : b < 0) :
∫ x in a..b, log x = b * log b - a * log a - b + a | integral_log $ not_mem_uIcc_of_gt ha hb | lemma | integral_log_of_neg | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_log"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sin : ∫ x in a..b, sin x = cos a - cos b | by rw integral_deriv_eq_sub' (λ x, -cos x); norm_num [continuous_on_sin] | lemma | integral_sin | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_cos : ∫ x in a..b, cos x = sin b - sin a | by rw integral_deriv_eq_sub'; norm_num [continuous_on_cos] | lemma | integral_cos | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_cos_mul_complex {z : ℂ} (hz : z ≠ 0) (a b : ℝ) :
∫ x in a..b, complex.cos (z * x) = complex.sin (z * b) / z - complex.sin (z * a) / z | begin
apply integral_eq_sub_of_has_deriv_at,
swap,
{ apply continuous.interval_integrable,
exact complex.continuous_cos.comp (continuous_const.mul complex.continuous_of_real) },
intros x hx,
have a := complex.has_deriv_at_sin (↑x * z),
have b : has_deriv_at (λ y, y * z : ℂ → ℂ) z ↑x := has_deriv_at_mul_... | lemma | integral_cos_mul_complex | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"complex.continuous_of_real",
"complex.cos",
"complex.has_deriv_at_sin",
"complex.sin",
"continuous.interval_integrable",
"has_deriv_at",
"has_deriv_at.comp",
"has_deriv_at.comp_of_real",
"has_deriv_at_mul_const",
"mul_comm",
"mul_div_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_cos_sq_sub_sin_sq :
∫ x in a..b, cos x ^ 2 - sin x ^ 2 = sin b * cos b - sin a * cos a | by simpa only [sq, sub_eq_add_neg, neg_mul_eq_mul_neg] using integral_deriv_mul_eq_sub
(λ x hx, has_deriv_at_sin x) (λ x hx, has_deriv_at_cos x) continuous_on_cos.interval_integrable
continuous_on_sin.neg.interval_integrable | lemma | integral_cos_sq_sub_sin_sq | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"neg_mul_eq_mul_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_inv_one_add_sq : ∫ x : ℝ in a..b, (1 + x^2)⁻¹ = arctan b - arctan a | begin
simp only [← one_div],
refine integral_deriv_eq_sub' _ _ _ (continuous_const.div _ (λ x, _)).continuous_on,
{ norm_num },
{ norm_num },
{ continuity },
{ nlinarith },
end | lemma | integral_inv_one_add_sq | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"continuity",
"continuous_on",
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_one_div_one_add_sq : ∫ x : ℝ in a..b, 1 / (1 + x^2) = arctan b - arctan a | by simp only [one_div, integral_inv_one_add_sq] | lemma | integral_one_div_one_add_sq | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_inv_one_add_sq",
"one_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_mul_cpow_one_add_sq {t : ℂ} (ht : t ≠ -1) :
∫ x : ℝ in a..b, (x:ℂ) * (1 + x ^ 2) ^ t =
(1 + b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + a ^ 2) ^ (t + 1) / (2 * (t + 1)) | begin
have : t + 1 ≠ 0 := by { contrapose! ht, rwa add_eq_zero_iff_eq_neg at ht },
apply integral_eq_sub_of_has_deriv_at,
{ intros x hx,
have f : has_deriv_at (λ y:ℂ, 1 + y ^ 2) (2 * x) x,
{ convert (has_deriv_at_pow 2 (x:ℂ)).const_add 1, { norm_cast }, { simp } },
have g : ∀ {z : ℂ}, (0 < z.re) → has... | lemma | integral_mul_cpow_one_add_sq | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"continuous.cpow",
"continuous.interval_integrable",
"continuous_const",
"has_deriv_at",
"has_deriv_at.comp",
"has_deriv_at.cpow_const",
"has_deriv_at_id",
"has_deriv_at_pow",
"ring",
"sq_nonneg",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_mul_rpow_one_add_sq {t : ℝ} (ht : t ≠ -1) :
∫ x : ℝ in a..b, x * (1 + x ^ 2) ^ t =
(1 + b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + a ^ 2) ^ (t + 1) / (2 * (t + 1)) | begin
have : ∀ (x s : ℝ), (((1 + x ^ 2) ^ s : ℝ) : ℂ) = (1 + (x : ℂ) ^ 2) ^ ↑s,
{ intros x s,
rw [of_real_cpow, of_real_add, of_real_pow, of_real_one],
exact add_nonneg zero_le_one (sq_nonneg x), },
rw ←of_real_inj,
convert integral_mul_cpow_one_add_sq (_ : (t:ℂ) ≠ -1),
{ rw ←interval_integral.integra... | lemma | integral_mul_rpow_one_add_sq | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_mul_cpow_one_add_sq",
"sq_nonneg",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sin_pow_aux :
∫ x in a..b, sin x ^ (n + 2) = sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b
+ (n + 1) * (∫ x in a..b, sin x ^ n) - (n + 1) * ∫ x in a..b, sin x ^ (n + 2) | begin
let C := sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b,
have h : ∀ α β γ : ℝ, (β * α * γ) * α = β * (α * α * γ) := λ α β γ, by ring,
have hu : ∀ x ∈ _, has_deriv_at (λ y, sin y ^ (n + 1)) ((n + 1 : ℕ) * cos x * sin x ^ n) x :=
λ x hx, by simpa only [mul_right_comm] using (has_deriv_at_sin x).pow (n+... | lemma | integral_sin_pow_aux | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"continuity",
"continuous.interval_integrable",
"has_deriv_at",
"mul_right_comm",
"pow_add",
"pow_succ'",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sin_pow :
∫ x in a..b, sin x ^ (n + 2) = (sin a ^ (n + 1) * cos a - sin b ^ (n + 1) * cos b) / (n + 2)
+ (n + 1) / (n + 2) * ∫ x in a..b, sin x ^ n | begin
have : (n : ℝ) + 2 ≠ 0 := by exact_mod_cast succ_ne_zero n.succ,
field_simp,
convert eq_sub_iff_add_eq.mp (integral_sin_pow_aux n),
ring,
end | lemma | integral_sin_pow | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_sin_pow_aux",
"ring"
] | The reduction formula for the integral of `sin x ^ n` for any natural `n ≥ 2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_sin_sq : ∫ x in a..b, sin x ^ 2 = (sin a * cos a - sin b * cos b + b - a) / 2 | by field_simp [integral_sin_pow, add_sub_assoc] | lemma | integral_sin_sq | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_sin_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sin_pow_odd :
∫ x in 0..π, sin x ^ (2 * n + 1) = 2 * ∏ i in range n, (2 * i + 2) / (2 * i + 3) | begin
induction n with k ih, { norm_num },
rw [prod_range_succ_comm, mul_left_comm, ← ih, mul_succ, integral_sin_pow],
norm_cast,
simp [-cast_add] with field_simps,
end | theorem | integral_sin_pow_odd | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"ih",
"integral_sin_pow",
"mul_left_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sin_pow_even :
∫ x in 0..π, sin x ^ (2 * n) = π * ∏ i in range n, (2 * i + 1) / (2 * i + 2) | begin
induction n with k ih, { simp },
rw [prod_range_succ_comm, mul_left_comm, ← ih, mul_succ, integral_sin_pow],
norm_cast,
simp [-cast_add] with field_simps,
end | theorem | integral_sin_pow_even | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"ih",
"integral_sin_pow",
"mul_left_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sin_pow_pos : 0 < ∫ x in 0..π, sin x ^ n | begin
rcases even_or_odd' n with ⟨k, (rfl | rfl)⟩;
simp only [integral_sin_pow_even, integral_sin_pow_odd];
refine mul_pos (by norm_num [pi_pos]) (prod_pos (λ n hn, div_pos _ _));
norm_cast;
linarith,
end | lemma | integral_sin_pow_pos | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"div_pos",
"integral_sin_pow_even",
"integral_sin_pow_odd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sin_pow_succ_le : ∫ x in 0..π, sin x ^ (n + 1) ≤ ∫ x in 0..π, sin x ^ n | let H := λ x h, pow_le_pow_of_le_one (sin_nonneg_of_mem_Icc h) (sin_le_one x) (n.le_add_right 1) in
by refine integral_mono_on pi_pos.le _ _ H; exact (continuous_sin.pow _).interval_integrable 0 π | lemma | integral_sin_pow_succ_le | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"interval_integrable",
"pow_le_pow_of_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sin_pow_antitone : antitone (λ n : ℕ, ∫ x in 0..π, sin x ^ n) | antitone_nat_of_succ_le integral_sin_pow_succ_le | lemma | integral_sin_pow_antitone | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"antitone",
"antitone_nat_of_succ_le",
"integral_sin_pow_succ_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_cos_pow_aux :
∫ x in a..b, cos x ^ (n + 2) = cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a
+ (n + 1) * (∫ x in a..b, cos x ^ n) - (n + 1) * ∫ x in a..b, cos x ^ (n + 2) | begin
let C := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a,
have h : ∀ α β γ : ℝ, (β * α * γ) * α = β * (α * α * γ) := λ α β γ, by ring,
have hu : ∀ x ∈ _, has_deriv_at (λ y, cos y ^ (n + 1)) (-(n + 1 : ℕ) * sin x * cos x ^ n) x :=
λ x hx, by simpa only [mul_right_comm, neg_mul, mul_neg]
using (ha... | lemma | integral_cos_pow_aux | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"continuity",
"continuous.interval_integrable",
"has_deriv_at",
"mul_neg",
"mul_right_comm",
"neg_mul",
"pow_add",
"pow_succ'",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_cos_pow :
∫ x in a..b, cos x ^ (n + 2) = (cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a) / (n + 2)
+ (n + 1) / (n + 2) * ∫ x in a..b, cos x ^ n | begin
have : (n : ℝ) + 2 ≠ 0 := by exact_mod_cast succ_ne_zero n.succ,
field_simp,
convert eq_sub_iff_add_eq.mp (integral_cos_pow_aux n),
ring,
end | lemma | integral_cos_pow | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_cos_pow_aux",
"ring"
] | The reduction formula for the integral of `cos x ^ n` for any natural `n ≥ 2`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_cos_sq : ∫ x in a..b, cos x ^ 2 = (cos b * sin b - cos a * sin a + b - a) / 2 | by field_simp [integral_cos_pow, add_sub_assoc] | lemma | integral_cos_sq | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_cos_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sin_pow_mul_cos_pow_odd (m n : ℕ) :
∫ x in a..b, sin x ^ m * cos x ^ (2 * n + 1) = ∫ u in sin a..sin b, u ^ m * (1 - u ^ 2) ^ n | have hc : continuous (λ u : ℝ, u ^ m * (1 - u ^ 2) ^ n), by continuity,
calc ∫ x in a..b, sin x ^ m * cos x ^ (2 * n + 1)
= ∫ x in a..b, sin x ^ m * (1 - sin x ^ 2) ^ n * cos x : by simp only [pow_succ', ← mul_assoc,
pow_mul, cos_sq']
... =... | lemma | integral_sin_pow_mul_cos_pow_odd | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"continuity",
"continuous",
"mul_assoc",
"pow_mul",
"pow_succ'"
] | Simplification of the integral of `sin x ^ m * cos x ^ n`, case `n` is odd. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_sin_mul_cos₁ :
∫ x in a..b, sin x * cos x = (sin b ^ 2 - sin a ^ 2) / 2 | by simpa using integral_sin_pow_mul_cos_pow_odd 1 0 | lemma | integral_sin_mul_cos₁ | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_sin_pow_mul_cos_pow_odd"
] | The integral of `sin x * cos x`, given in terms of sin².
See `integral_sin_mul_cos₂` below for the integral given in terms of cos². | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_sin_sq_mul_cos :
∫ x in a..b, sin x ^ 2 * cos x = (sin b ^ 3 - sin a ^ 3) / 3 | by simpa using integral_sin_pow_mul_cos_pow_odd 2 0 | lemma | integral_sin_sq_mul_cos | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_sin_pow_mul_cos_pow_odd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_cos_pow_three :
∫ x in a..b, cos x ^ 3 = sin b - sin a - (sin b ^ 3 - sin a ^ 3) / 3 | by simpa using integral_sin_pow_mul_cos_pow_odd 0 1 | lemma | integral_cos_pow_three | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_sin_pow_mul_cos_pow_odd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sin_pow_odd_mul_cos_pow (m n : ℕ) :
∫ x in a..b, sin x ^ (2 * m + 1) * cos x ^ n = ∫ u in cos b..cos a, u ^ n * (1 - u ^ 2) ^ m | have hc : continuous (λ u : ℝ, u ^ n * (1 - u ^ 2) ^ m), by continuity,
calc ∫ x in a..b, sin x ^ (2 * m + 1) * cos x ^ n
= -∫ x in b..a, sin x ^ (2 * m + 1) * cos x ^ n : by rw integral_symm
... = ∫ x in b..a, (1 - cos x ^ 2) ^ m * -sin x * cos x ^ n : by simp [pow_succ', pow_mul, sin_sq]
... = ∫ x in... | lemma | integral_sin_pow_odd_mul_cos_pow | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"continuity",
"continuous",
"pow_mul",
"pow_succ'",
"ring"
] | Simplification of the integral of `sin x ^ m * cos x ^ n`, case `m` is odd. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_sin_mul_cos₂ :
∫ x in a..b, sin x * cos x = (cos a ^ 2 - cos b ^ 2) / 2 | by simpa using integral_sin_pow_odd_mul_cos_pow 0 1 | lemma | integral_sin_mul_cos₂ | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_sin_pow_odd_mul_cos_pow"
] | The integral of `sin x * cos x`, given in terms of cos².
See `integral_sin_mul_cos₁` above for the integral given in terms of sin². | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_sin_mul_cos_sq :
∫ x in a..b, sin x * cos x ^ 2 = (cos a ^ 3 - cos b ^ 3) / 3 | by simpa using integral_sin_pow_odd_mul_cos_pow 0 2 | lemma | integral_sin_mul_cos_sq | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_sin_pow_odd_mul_cos_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sin_pow_three :
∫ x in a..b, sin x ^ 3 = cos a - cos b - (cos a ^ 3 - cos b ^ 3) / 3 | by simpa using integral_sin_pow_odd_mul_cos_pow 1 0 | lemma | integral_sin_pow_three | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"integral_sin_pow_odd_mul_cos_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integral_sin_pow_even_mul_cos_pow_even (m n : ℕ) :
∫ x in a..b, sin x ^ (2 * m) * cos x ^ (2 * n)
= ∫ x in a..b, ((1 - cos (2 * x)) / 2) ^ m * ((1 + cos (2 * x)) / 2) ^ n | by field_simp [pow_mul, sin_sq, cos_sq, ← sub_sub, (by ring : (2:ℝ) - 1 = 1)] | lemma | integral_sin_pow_even_mul_cos_pow_even | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"pow_mul",
"ring"
] | Simplification of the integral of `sin x ^ m * cos x ^ n`, case `m` and `n` are both even. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
integral_sin_sq_mul_cos_sq :
∫ x in a..b, sin x ^ 2 * cos x ^ 2 = (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 | begin
convert integral_sin_pow_even_mul_cos_pow_even 1 1 using 1,
have h1 : ∀ c : ℝ, (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4 := λ c, by ring,
have h2 : continuous (λ x, cos (2 * x) ^ 2) := by continuity,
have h3 : ∀ x, cos x * sin x = sin (2 * x) / 2, { intro, rw sin_two_mul, ring },
have h4 : ∀ d : ℝ, ... | lemma | integral_sin_sq_mul_cos_sq | analysis.special_functions | src/analysis/special_functions/integrals.lean | [
"measure_theory.integral.fund_thm_calculus",
"analysis.special_functions.trigonometric.arctan_deriv"
] | [
"continuity",
"continuous",
"integral_sin_pow_even_mul_cos_pow_even",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sqrt_one_add_norm_sq_le (x : E) : real.sqrt (1 + ‖x‖^2) ≤ 1 + ‖x‖ | begin
refine le_of_pow_le_pow 2 (by positivity) two_pos _,
simp [sq_sqrt (zero_lt_one_add_norm_sq x).le, add_pow_two],
end | lemma | sqrt_one_add_norm_sq_le | analysis.special_functions | src/analysis/special_functions/japanese_bracket.lean | [
"measure_theory.measure.lebesgue.eq_haar",
"measure_theory.integral.layercake"
] | [
"le_of_pow_le_pow",
"real.sqrt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
one_add_norm_le_sqrt_two_mul_sqrt (x : E) : 1 + ‖x‖ ≤ (real.sqrt 2) * sqrt (1 + ‖x‖^2) | begin
suffices : (sqrt 2 * sqrt (1 + ‖x‖ ^ 2)) ^ 2 - (1 + ‖x‖) ^ 2 = (1 - ‖x‖) ^2,
{ refine le_of_pow_le_pow 2 (by positivity) (by norm_num) _,
rw [←sub_nonneg, this],
positivity, },
rw [mul_pow, sq_sqrt (zero_lt_one_add_norm_sq x).le, add_pow_two, sub_pow_two],
norm_num,
ring,
end | lemma | one_add_norm_le_sqrt_two_mul_sqrt | analysis.special_functions | src/analysis/special_functions/japanese_bracket.lean | [
"measure_theory.measure.lebesgue.eq_haar",
"measure_theory.integral.layercake"
] | [
"le_of_pow_le_pow",
"mul_pow",
"real.sqrt",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_neg_one_add_norm_sq_le {r : ℝ} (x : E) (hr : 0 < r) :
(1 + ‖x‖^2)^(-r/2) ≤ 2^(r/2) * (1 + ‖x‖)^(-r) | begin
have h1 : 0 ≤ (2 : ℝ) := by positivity,
have h3 : 0 < sqrt 2 := by positivity,
have h4 : 0 < 1 + ‖x‖ := by positivity,
have h5 : 0 < sqrt (1 + ‖x‖ ^ 2) := by positivity,
have h6 : 0 < sqrt 2 * sqrt (1 + ‖x‖^2) := mul_pos h3 h5,
rw [rpow_div_two_eq_sqrt _ h1, rpow_div_two_eq_sqrt _ (zero_lt_one_add_nor... | lemma | rpow_neg_one_add_norm_sq_le | analysis.special_functions | src/analysis/special_functions/japanese_bracket.lean | [
"measure_theory.measure.lebesgue.eq_haar",
"measure_theory.integral.layercake"
] | [
"inv_le_inv",
"one_add_norm_le_sqrt_two_mul_sqrt"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_rpow_one_add_norm_iff_norm_le {r t : ℝ} (hr : 0 < r) (ht : 0 < t) (x : E) :
t ≤ (1 + ‖x‖) ^ -r ↔ ‖x‖ ≤ t ^ -r⁻¹ - 1 | begin
rw [le_sub_iff_add_le', neg_inv],
exact (real.le_rpow_inv_iff_of_neg (by positivity) ht (neg_lt_zero.mpr hr)).symm,
end | lemma | le_rpow_one_add_norm_iff_norm_le | analysis.special_functions | src/analysis/special_functions/japanese_bracket.lean | [
"measure_theory.measure.lebesgue.eq_haar",
"measure_theory.integral.layercake"
] | [
"neg_inv",
"real.le_rpow_inv_iff_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closed_ball_rpow_sub_one_eq_empty_aux {r t : ℝ} (hr : 0 < r) (ht : 1 < t) :
metric.closed_ball (0 : E) (t^(-r⁻¹) - 1) = ∅ | begin
rw [metric.closed_ball_eq_empty, sub_neg],
exact real.rpow_lt_one_of_one_lt_of_neg ht (by simp only [hr, right.neg_neg_iff, inv_pos]),
end | lemma | closed_ball_rpow_sub_one_eq_empty_aux | analysis.special_functions | src/analysis/special_functions/japanese_bracket.lean | [
"measure_theory.measure.lebesgue.eq_haar",
"measure_theory.integral.layercake"
] | [
"inv_pos",
"metric.closed_ball",
"metric.closed_ball_eq_empty",
"real.rpow_lt_one_of_one_lt_of_neg"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_integral_rpow_sub_one_pow_aux {r : ℝ} (n : ℕ) (hnr : (n : ℝ) < r) :
∫⁻ (x : ℝ) in Ioc 0 1, ennreal.of_real ((x ^ -r⁻¹ - 1) ^ n) < ∞ | begin
have hr : 0 < r := lt_of_le_of_lt n.cast_nonneg hnr,
have h_int : ∀ (x : ℝ) (hx : x ∈ Ioc (0 : ℝ) 1),
ennreal.of_real ((x ^ -r⁻¹ - 1) ^ n) ≤ ennreal.of_real (x ^ -(r⁻¹ * n)) :=
begin
intros x hx,
have hxr : 0 ≤ x^ -r⁻¹ := rpow_nonneg_of_nonneg hx.1.le _,
apply ennreal.of_real_le_of_real,
... | lemma | finite_integral_rpow_sub_one_pow_aux | analysis.special_functions | src/analysis/special_functions/japanese_bracket.lean | [
"measure_theory.measure.lebesgue.eq_haar",
"measure_theory.integral.layercake"
] | [
"ennreal.of_real",
"ennreal.of_real_le_of_real",
"interval_integral.interval_integrable_rpow'",
"inv_mul_lt_iff'",
"inv_nonneg",
"measurability",
"one_mul",
"pow_le_pow_of_le_left",
"real.one_le_rpow_of_pos_of_le_one_of_nonpos",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finite_integral_one_add_norm [measure_space E] [borel_space E]
[(@volume E _).is_add_haar_measure] {r : ℝ} (hnr : (finrank ℝ E : ℝ) < r) :
∫⁻ (x : E), ennreal.of_real ((1 + ‖x‖) ^ -r) < ∞ | begin
have hr : 0 < r := lt_of_le_of_lt (finrank ℝ E).cast_nonneg hnr,
-- We start by applying the layer cake formula
have h_meas : measurable (λ (ω : E), (1 + ‖ω‖) ^ -r) := by measurability,
have h_pos : ∀ x : E, 0 ≤ (1 + ‖x‖) ^ -r :=
by { intros x, positivity },
rw lintegral_eq_lintegral_meas_le volume h... | lemma | finite_integral_one_add_norm | analysis.special_functions | src/analysis/special_functions/japanese_bracket.lean | [
"measure_theory.measure.lebesgue.eq_haar",
"measure_theory.integral.layercake"
] | [
"borel_space",
"closed_ball_rpow_sub_one_eq_empty_aux",
"disjoint",
"disjoint_iff",
"ennreal",
"ennreal.add_lt_top",
"ennreal.mul_lt_top_iff",
"ennreal.of_real",
"finite_dimensional.finrank",
"le_rpow_one_add_norm_iff_norm_le",
"measurability",
"measurable",
"measurable_set_Ioc",
"measurab... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_one_add_norm [measure_space E] [borel_space E] [(@volume E _).is_add_haar_measure]
{r : ℝ} (hnr : (finrank ℝ E : ℝ) < r) :
integrable (λ (x : E), (1 + ‖x‖) ^ -r) | begin
refine ⟨by measurability, _⟩,
-- Lower Lebesgue integral
have : ∫⁻ (a : E), ‖(1 + ‖a‖) ^ -r‖₊ = ∫⁻ (a : E), ennreal.of_real ((1 + ‖a‖) ^ -r) :=
lintegral_nnnorm_eq_of_nonneg (λ _, rpow_nonneg_of_nonneg (by positivity) _),
rw [has_finite_integral, this],
exact finite_integral_one_add_norm hnr,
end | lemma | integrable_one_add_norm | analysis.special_functions | src/analysis/special_functions/japanese_bracket.lean | [
"measure_theory.measure.lebesgue.eq_haar",
"measure_theory.integral.layercake"
] | [
"borel_space",
"ennreal.of_real",
"finite_integral_one_add_norm",
"measurability"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
integrable_rpow_neg_one_add_norm_sq [measure_space E] [borel_space E]
[(@volume E _).is_add_haar_measure] {r : ℝ} (hnr : (finrank ℝ E : ℝ) < r) :
integrable (λ (x : E), (1 + ‖x‖^2) ^ (-r/2)) | begin
have hr : 0 < r := lt_of_le_of_lt (finrank ℝ E).cast_nonneg hnr,
refine ((integrable_one_add_norm hnr).const_mul $ 2 ^ (r / 2)).mono (by measurability)
(eventually_of_forall $ λ x, _),
have h1 : 0 ≤ (1 + ‖x‖ ^ 2) ^ (-r/2) := by positivity,
have h2 : 0 ≤ (1 + ‖x‖) ^ -r := by positivity,
have h3 : 0 ≤... | lemma | integrable_rpow_neg_one_add_norm_sq | analysis.special_functions | src/analysis/special_functions/japanese_bracket.lean | [
"measure_theory.measure.lebesgue.eq_haar",
"measure_theory.integral.layercake"
] | [
"abs_of_nonneg",
"borel_space",
"integrable_one_add_norm",
"measurability",
"norm_mul",
"rpow_neg_one_add_norm_sq_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_interval_integrable_of_tendsto_norm_at_top_of_deriv_is_O_filter {f : ℝ → E} {g : ℝ → F}
{a b : ℝ} (l : filter ℝ) [ne_bot l] [tendsto_Ixx_class Icc l l] (hl : [a, b] ∈ l)
(hd : ∀ᶠ x in l, differentiable_at ℝ f x) (hf : tendsto (λ x, ‖f x‖) l at_top)
(hfg : deriv f =O[l] g) :
¬interval_integrable g volume a b | begin
intro hgi,
obtain ⟨C, hC₀, s, hsl, hsub, hfd, hg⟩ : ∃ (C : ℝ) (hC₀ : 0 ≤ C) (s ∈ l),
(∀ (x ∈ s) (y ∈ s), [x, y] ⊆ [a, b]) ∧
(∀ (x ∈ s) (y ∈ s) (z ∈ [x, y]), differentiable_at ℝ f z) ∧
(∀ (x ∈ s) (y ∈ s) (z ∈ [x, y]), ‖deriv f z‖ ≤ C * ‖g z‖),
{ rcases hfg.exists_nonneg with ⟨C, C₀, hC⟩,
have... | lemma | not_interval_integrable_of_tendsto_norm_at_top_of_deriv_is_O_filter | analysis.special_functions | src/analysis/special_functions/non_integrable.lean | [
"analysis.special_functions.log.deriv",
"measure_theory.integral.fund_thm_calculus"
] | [
"ae_strongly_measurable_deriv",
"deriv",
"differentiable_at",
"filter",
"filter.nonempty_of_mem",
"interval_integrable",
"measurable_set_uIoc"
] | If `f` is eventually differentiable along a nontrivial filter `l : filter ℝ` that is generated
by convex sets, the norm of `f` tends to infinity along `l`, and `f' = O(g)` along `l`, where `f'`
is the derivative of `f`, then `g` is not integrable on any interval `a..b` such that
`[a, b] ∈ l`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_interval_integrable_of_tendsto_norm_at_top_of_deriv_is_O_within_diff_singleton
{f : ℝ → E} {g : ℝ → F} {a b c : ℝ} (hne : a ≠ b) (hc : c ∈ [a, b])
(h_deriv : ∀ᶠ x in 𝓝[[a, b] \ {c}] c, differentiable_at ℝ f x)
(h_infty : tendsto (λ x, ‖f x‖) (𝓝[[a, b] \ {c}] c) at_top)
(hg : deriv f =O[𝓝[[a, b] \ {c}] c]... | begin
obtain ⟨l, hl, hl', hle, hmem⟩ : ∃ l : filter ℝ, tendsto_Ixx_class Icc l l ∧ l.ne_bot ∧
l ≤ 𝓝 c ∧ [a, b] \ {c} ∈ l,
{ cases (min_lt_max.2 hne).lt_or_lt c with hlt hlt,
{ refine ⟨𝓝[<] c, infer_instance, infer_instance, inf_le_left, _⟩,
rw ← Iic_diff_right,
exact diff_mem_nhds_within_diff ... | lemma | not_interval_integrable_of_tendsto_norm_at_top_of_deriv_is_O_within_diff_singleton | analysis.special_functions | src/analysis/special_functions/non_integrable.lean | [
"analysis.special_functions.log.deriv",
"measure_theory.integral.fund_thm_calculus"
] | [
"Icc_mem_nhds_within_Ici",
"Icc_mem_nhds_within_Iic",
"deriv",
"diff_mem_nhds_within_diff",
"differentiable_at",
"filter",
"inf_le_left",
"interval_integrable",
"le_inf",
"not_interval_integrable_of_tendsto_norm_at_top_of_deriv_is_O_filter"
] | If `a ≠ b`, `c ∈ [a, b]`, `f` is differentiable in the neighborhood of `c` within
`[a, b] \ {c}`, `‖f x‖ → ∞` as `x → c` within `[a, b] \ {c}`, and `f' = O(g)` along
`𝓝[[a, b] \ {c}] c`, where `f'` is the derivative of `f`, then `g` is not interval integrable on
`a..b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
not_interval_integrable_of_tendsto_norm_at_top_of_deriv_is_O_punctured {f : ℝ → E} {g : ℝ → F}
{a b c : ℝ} (h_deriv : ∀ᶠ x in 𝓝[≠] c, differentiable_at ℝ f x)
(h_infty : tendsto (λ x, ‖f x‖) (𝓝[≠] c) at_top) (hg : deriv f =O[𝓝[≠] c] g)
(hne : a ≠ b) (hc : c ∈ [a, b]) :
¬interval_integrable g volume a b | have 𝓝[[a, b] \ {c}] c ≤ 𝓝[≠] c, from nhds_within_mono _ (inter_subset_right _ _),
not_interval_integrable_of_tendsto_norm_at_top_of_deriv_is_O_within_diff_singleton hne hc
(h_deriv.filter_mono this) (h_infty.mono_left this) (hg.mono this) | lemma | not_interval_integrable_of_tendsto_norm_at_top_of_deriv_is_O_punctured | analysis.special_functions | src/analysis/special_functions/non_integrable.lean | [
"analysis.special_functions.log.deriv",
"measure_theory.integral.fund_thm_calculus"
] | [
"deriv",
"differentiable_at",
"interval_integrable",
"nhds_within_mono",
"not_interval_integrable_of_tendsto_norm_at_top_of_deriv_is_O_within_diff_singleton"
] | If `f` is differentiable in a punctured neighborhood of `c`, `‖f x‖ → ∞` as `x → c` (more
formally, along the filter `𝓝[≠] c`), and `f' = O(g)` along `𝓝[≠] c`, where `f'` is the derivative
of `f`, then `g` is not interval integrable on any nontrivial interval `a..b` such that
`c ∈ [a, b]`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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