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
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
Lp_add_le_tsum' {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : summable (λ i, (f i) ^ p))
(hg : summable (λ i, (g i) ^ p)) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, (f i) ^ p) ^ (1 / p) + (∑' i, (g i) ^ p) ^ (1 / p) | (Lp_add_le_tsum hp hf hg).2 | theorem | nnreal.Lp_add_le_tsum' | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Lp_add_le_has_sum {f g : ι → ℝ≥0} {A B : ℝ≥0} {p : ℝ} (hp : 1 ≤ p)
(hf : has_sum (λ i, (f i) ^ p) (A ^ p)) (hg : has_sum (λ i, (g i) ^ p) (B ^ p)) :
∃ C, C ≤ A + B ∧ has_sum (λ i, (f i + g i) ^ p) (C ^ p) | begin
have hp' : p ≠ 0 := (lt_of_lt_of_le zero_lt_one hp).ne',
obtain ⟨H₁, H₂⟩ := Lp_add_le_tsum hp hf.summable hg.summable,
have hA : A = (∑' (i : ι), f i ^ p) ^ (1 / p) := by rw [hf.tsum_eq, rpow_inv_rpow_self hp'],
have hB : B = (∑' (i : ι), g i ^ p) ^ (1 / p) := by rw [hg.tsum_eq, rpow_inv_rpow_self hp'],
... | theorem | nnreal.Lp_add_le_has_sum | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"has_sum",
"zero_lt_one"
] | Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `nnreal`-valued functions. For an alternative version, convenient if the
infinite sums are not already expressed ... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_le_Lp_mul_Lq (hpq : is_conjugate_exponent p q) :
∑ i in s, f i * g i ≤ (∑ i in s, |f i| ^ p) ^ (1 / p) * (∑ i in s, |g i| ^ q) ^ (1 / q) | begin
have := nnreal.coe_le_coe.2 (nnreal.inner_le_Lp_mul_Lq s (λ i, ⟨_, abs_nonneg (f i)⟩)
(λ i, ⟨_, abs_nonneg (g i)⟩) hpq),
push_cast at this,
refine le_trans (sum_le_sum $ λ i hi, _) this,
simp only [← abs_mul, le_abs_self]
end | theorem | real.inner_le_Lp_mul_Lq | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"abs_mul",
"abs_nonneg",
"le_abs_self",
"nnreal.inner_le_Lp_mul_Lq"
] | Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, |f i|) ^ p ≤ (card s) ^ (p - 1) * ∑ i in s, |f i| ^ p | begin
have := nnreal.coe_le_coe.2
(nnreal.rpow_sum_le_const_mul_sum_rpow s (λ i, ⟨_, abs_nonneg (f i)⟩) hp),
push_cast at this,
exact this, -- for some reason `exact_mod_cast` can't replace this argument
end | theorem | real.rpow_sum_le_const_mul_sum_rpow | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"abs_nonneg",
"nnreal.rpow_sum_le_const_mul_sum_rpow"
] | For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ`-valued functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, |f i + g i| ^ p) ^ (1 / p) ≤
(∑ i in s, |f i| ^ p) ^ (1 / p) + (∑ i in s, |g i| ^ p) ^ (1 / p) | begin
have := nnreal.coe_le_coe.2 (nnreal.Lp_add_le s (λ i, ⟨_, abs_nonneg (f i)⟩)
(λ i, ⟨_, abs_nonneg (g i)⟩) hp),
push_cast at this,
refine le_trans (rpow_le_rpow _ (sum_le_sum $ λ i hi, _) _) this;
simp [sum_nonneg, rpow_nonneg_of_nonneg, abs_nonneg, le_trans zero_le_one hp, abs_add,
rpow_le_rpo... | theorem | real.Lp_add_le | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"abs_add",
"abs_nonneg",
"nnreal.Lp_add_le",
"zero_le_one"
] | Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `real`-valued functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_le_Lp_mul_Lq_of_nonneg (hpq : is_conjugate_exponent p q)
(hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
∑ i in s, f i * g i ≤ (∑ i in s, (f i)^p) ^ (1 / p) * (∑ i in s, (g i)^q) ^ (1 / q) | by convert inner_le_Lp_mul_Lq s f g hpq using 3; apply sum_congr rfl; intros i hi;
simp only [abs_of_nonneg, hf i hi, hg i hi] | theorem | real.inner_le_Lp_mul_Lq_of_nonneg | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"abs_of_nonneg"
] | Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with real-valued nonnegative functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_le_Lp_mul_Lq_tsum_of_nonneg (hpq : p.is_conjugate_exponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : summable (λ i, (f i) ^ p)) (hg_sum : summable (λ i, (g i) ^ q)) :
summable (λ i, f i * g i) ∧
∑' i, f i * g i ≤ (∑' i, (f i) ^ p) ^ (1 / p) * (∑' i, (g i) ^ q) ^ (1 / q) | begin
lift f to (ι → ℝ≥0) using hf,
lift g to (ι → ℝ≥0) using hg,
norm_cast at *,
exact nnreal.inner_le_Lp_mul_Lq_tsum hpq hf_sum hg_sum,
end | theorem | real.inner_le_Lp_mul_Lq_tsum_of_nonneg | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"lift",
"nnreal.inner_le_Lp_mul_Lq_tsum",
"summable"
] | Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `ℝ`-valued functions.
For an alternative version, convenient if the infinite sums are already expressed as `p`-th powers,
see `inner_le_Lp_mul_Lq_has_su... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_mul_of_Lp_Lq_of_nonneg (hpq : p.is_conjugate_exponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : summable (λ i, (f i) ^ p)) (hg_sum : summable (λ i, (g i) ^ q)) :
summable (λ i, f i * g i) | (inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).1 | theorem | real.summable_mul_of_Lp_Lq_of_nonneg | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_le_Lp_mul_Lq_tsum_of_nonneg' (hpq : p.is_conjugate_exponent q) (hf : ∀ i, 0 ≤ f i)
(hg : ∀ i, 0 ≤ g i) (hf_sum : summable (λ i, (f i) ^ p)) (hg_sum : summable (λ i, (g i) ^ q)) :
∑' i, f i * g i ≤ (∑' i, (f i) ^ p) ^ (1 / p) * (∑' i, (g i) ^ q) ^ (1 / q) | (inner_le_Lp_mul_Lq_tsum_of_nonneg hpq hf hg hf_sum hg_sum).2 | theorem | real.inner_le_Lp_mul_Lq_tsum_of_nonneg' | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_le_Lp_mul_Lq_has_sum_of_nonneg (hpq : p.is_conjugate_exponent q) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : has_sum (λ i, (f i) ^ p) (A ^ p)) (hg_sum : has_sum (λ i, (g i) ^ q) (B ^ q)) :
∃ C : ℝ, 0 ≤ C ∧ C ≤ A * B ∧ has_sum (λ i, f i * g i) C | begin
lift f to (ι → ℝ≥0) using hf,
lift g to (ι → ℝ≥0) using hg,
lift A to ℝ≥0 using hA,
lift B to ℝ≥0 using hB,
norm_cast at hf_sum hg_sum,
obtain ⟨C, hC, H⟩ := nnreal.inner_le_Lp_mul_Lq_has_sum hpq hf_sum hg_sum,
refine ⟨C, C.prop, hC, _⟩,
norm_cast,
exact H
end | theorem | real.inner_le_Lp_mul_Lq_has_sum_of_nonneg | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"has_sum",
"lift",
"nnreal.inner_le_Lp_mul_Lq_has_sum"
] | Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. A version for `nnreal`-valued
functions. For an alternative version, convenient if the infinite sums are not already expressed as
`p`-th powers, see `inner_le_Lp_mul_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rpow_sum_le_const_mul_sum_rpow_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) :
(∑ i in s, f i) ^ p ≤ (card s) ^ (p - 1) * ∑ i in s, f i ^ p | by convert rpow_sum_le_const_mul_sum_rpow s f hp using 2; apply sum_congr rfl; intros i hi;
simp only [abs_of_nonneg, hf i hi] | theorem | real.rpow_sum_le_const_mul_sum_rpow_of_nonneg | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"abs_of_nonneg"
] | For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with nonnegative `ℝ`-valued
functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Lp_add_le_of_nonneg (hp : 1 ≤ p) (hf : ∀ i ∈ s, 0 ≤ f i) (hg : ∀ i ∈ s, 0 ≤ g i) :
(∑ i in s, (f i + g i) ^ p) ^ (1 / p) ≤
(∑ i in s, (f i) ^ p) ^ (1 / p) + (∑ i in s, (g i) ^ p) ^ (1 / p) | by convert Lp_add_le s f g hp using 2 ; [skip, congr' 1, congr' 1];
apply sum_congr rfl; intros i hi; simp only [abs_of_nonneg, hf i hi, hg i hi, add_nonneg] | theorem | real.Lp_add_le_of_nonneg | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"abs_of_nonneg"
] | Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ`-valued nonnegative
functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Lp_add_le_tsum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : summable (λ i, (f i) ^ p)) (hg_sum : summable (λ i, (g i) ^ p)) :
summable (λ i, (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, (f i) ^ p) ^ (1 / p) + (∑' i, (g i) ^ p) ^ (1 / p) | begin
lift f to (ι → ℝ≥0) using hf,
lift g to (ι → ℝ≥0) using hg,
norm_cast at *,
exact nnreal.Lp_add_le_tsum hp hf_sum hg_sum,
end | theorem | real.Lp_add_le_tsum_of_nonneg | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"lift",
"nnreal.Lp_add_le_tsum",
"summable"
] | Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are already expressed as `p`-th... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_Lp_add_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : summable (λ i, (f i) ^ p)) (hg_sum : summable (λ i, (g i) ^ p)) :
summable (λ i, (f i + g i) ^ p) | (Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).1 | theorem | real.summable_Lp_add_of_nonneg | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Lp_add_le_tsum_of_nonneg' (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i)
(hf_sum : summable (λ i, (f i) ^ p)) (hg_sum : summable (λ i, (g i) ^ p)) :
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤ (∑' i, (f i) ^ p) ^ (1 / p) + (∑' i, (g i) ^ p) ^ (1 / p) | (Lp_add_le_tsum_of_nonneg hp hf hg hf_sum hg_sum).2 | theorem | real.Lp_add_le_tsum_of_nonneg' | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Lp_add_le_has_sum_of_nonneg (hp : 1 ≤ p) (hf : ∀ i, 0 ≤ f i) (hg : ∀ i, 0 ≤ g i) {A B : ℝ}
(hA : 0 ≤ A) (hB : 0 ≤ B) (hfA : has_sum (λ i, (f i) ^ p) (A ^ p))
(hgB : has_sum (λ i, (g i) ^ p) (B ^ p)) :
∃ C, 0 ≤ C ∧ C ≤ A + B ∧ has_sum (λ i, (f i + g i) ^ p) (C ^ p) | begin
lift f to (ι → ℝ≥0) using hf,
lift g to (ι → ℝ≥0) using hg,
lift A to ℝ≥0 using hA,
lift B to ℝ≥0 using hB,
norm_cast at hfA hgB,
obtain ⟨C, hC₁, hC₂⟩ := nnreal.Lp_add_le_has_sum hp hfA hgB,
use C,
norm_cast,
exact ⟨zero_le _, hC₁, hC₂⟩,
end | theorem | real.Lp_add_le_has_sum_of_nonneg | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"has_sum",
"lift",
"nnreal.Lp_add_le_has_sum"
] | Minkowski inequality: the `L_p` seminorm of the infinite sum of two vectors is less than or
equal to the infinite sum of the `L_p`-seminorms of the summands, if these infinite sums both
exist. A version for `ℝ`-valued functions. For an alternative version, convenient if the infinite
sums are not already expressed as `p... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
inner_le_Lp_mul_Lq (hpq : p.is_conjugate_exponent q) :
(∑ i in s, f i * g i) ≤ (∑ i in s, (f i)^p) ^ (1/p) * (∑ i in s, (g i)^q) ^ (1/q) | begin
by_cases H : (∑ i in s, (f i)^p) ^ (1/p) = 0 ∨ (∑ i in s, (g i)^q) ^ (1/q) = 0,
{ replace H : (∀ i ∈ s, f i = 0) ∨ (∀ i ∈ s, g i = 0),
by simpa [ennreal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H,
have : ∀ i ∈ s, f i... | theorem | ennreal.inner_le_Lp_mul_Lq | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"ennreal.coe_rpow_of_nonneg",
"ennreal.rpow_eq_top_iff",
"ennreal.rpow_eq_zero_iff",
"ennreal.sum_eq_top_iff",
"ennreal.to_nnreal",
"nnreal.inner_le_Lp_mul_Lq",
"not_or_distrib",
"one_div",
"with_zero.coe_mul"
] | Hölder inequality: the scalar product of two functions is bounded by the product of their
`L^p` and `L^q` norms when `p` and `q` are conjugate exponents. Version for sums over finite sets,
with `ℝ≥0∞`-valued functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rpow_sum_le_const_mul_sum_rpow (hp : 1 ≤ p) :
(∑ i in s, f i) ^ p ≤ (card s) ^ (p - 1) * ∑ i in s, (f i) ^ p | begin
cases eq_or_lt_of_le hp with hp hp,
{ simp [← hp] },
let q : ℝ := p / (p - 1),
have hpq : p.is_conjugate_exponent q,
{ rw real.is_conjugate_exponent_iff hp },
have hp₁ : 1 / p * p = 1 := one_div_mul_cancel hpq.ne_zero,
have hq : 1 / q * p = (p - 1),
{ rw [← hpq.div_conj_eq_sub_one],
ring },
... | theorem | ennreal.rpow_sum_le_const_mul_sum_rpow | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"ennreal.mul_rpow_of_nonneg",
"ennreal.rpow_le_rpow",
"ennreal.rpow_mul",
"eq_or_lt_of_le",
"nat.smul_one_eq_coe",
"one_div_mul_cancel",
"one_mul",
"pi.one_apply",
"real.is_conjugate_exponent_iff",
"ring"
] | For `1 ≤ p`, the `p`-th power of the sum of `f i` is bounded above by a constant times the
sum of the `p`-th powers of `f i`. Version for sums over finite sets, with `ℝ≥0∞`-valued functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Lp_add_le (hp : 1 ≤ p) :
(∑ i in s, (f i + g i) ^ p)^(1/p) ≤ (∑ i in s, (f i)^p) ^ (1/p) + (∑ i in s, (g i)^p) ^ (1/p) | begin
by_cases H' : (∑ i in s, (f i)^p) ^ (1/p) = ⊤ ∨ (∑ i in s, (g i)^p) ^ (1/p) = ⊤,
{ cases H'; simp [H', -one_div] },
have pos : 0 < p := lt_of_lt_of_le zero_lt_one hp,
replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ (∀ i ∈ s, g i ≠ ⊤),
by simpa [ennreal.rpow_eq_top_iff, asymm pos, pos, ennreal.sum_eq_top_iff,
... | theorem | ennreal.Lp_add_le | analysis | src/analysis/mean_inequalities.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"data.real.conjugate_exponents"
] | [
"ennreal.coe_rpow_of_nonneg",
"ennreal.rpow_eq_top_iff",
"ennreal.sum_eq_top_iff",
"ennreal.to_nnreal",
"nnreal.Lp_add_le",
"not_or_distrib",
"one_div",
"zero_lt_one"
] | Minkowski inequality: the `L_p` seminorm of the sum of two vectors is less than or equal
to the sum of the `L_p`-seminorms of the summands. A version for `ℝ≥0∞` valued nonnegative
functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_arith_mean_le_arith_mean_pow (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) (n : ℕ) :
(∑ i in s, w i * z i) ^ n ≤ ∑ i in s, (w i * z i ^ n) | (convex_on_pow n).map_sum_le hw hw' hz | theorem | real.pow_arith_mean_le_arith_mean_pow | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"convex_on_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_arith_mean_le_arith_mean_pow_of_even (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) {n : ℕ} (hn : even n) :
(∑ i in s, w i * z i) ^ n ≤ ∑ i in s, (w i * z i ^ n) | hn.convex_on_pow.map_sum_le hw hw' (λ _ _, trivial) | theorem | real.pow_arith_mean_le_arith_mean_pow_of_even | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_sum_div_card_le_sum_pow {f : ι → ℝ} (n : ℕ) (hf : ∀ a ∈ s, 0 ≤ f a) :
(∑ x in s, f x) ^ (n + 1) / s.card ^ n ≤ ∑ x in s, (f x) ^ (n + 1) | begin
rcases s.eq_empty_or_nonempty with rfl | hs,
{ simp_rw [finset.sum_empty, zero_pow' _ (nat.succ_ne_zero n), zero_div] },
{ have hs0 : 0 < (s.card : ℝ) := nat.cast_pos.2 hs.card_pos,
suffices : (∑ x in s, f x / s.card) ^ (n + 1) ≤ ∑ x in s, (f x ^ (n + 1) / s.card),
{ rwa [← finset.sum_div, ← finset.... | lemma | real.pow_sum_div_card_le_sum_pow | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"algebra.id.smul_eq_mul",
"convex_on.map_sum_le",
"convex_on_pow",
"div_div",
"div_le_iff",
"div_mul",
"div_one",
"div_pow",
"div_self",
"finset.sum_div",
"inv_mul_eq_div",
"inv_nonneg",
"mul_inv_cancel",
"nat.cast_nonneg",
"nsmul_eq_mul",
"one_div",
"pow_succ'",
"set.Ici",
"zero... | Specific case of Jensen's inequality for sums of powers | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
zpow_arith_mean_le_arith_mean_zpow (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 < z i) (m : ℤ) :
(∑ i in s, w i * z i) ^ m ≤ ∑ i in s, (w i * z i ^ m) | (convex_on_zpow m).map_sum_le hw hw' hz | theorem | real.zpow_arith_mean_le_arith_mean_zpow | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"convex_on_zpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) {p : ℝ} (hp : 1 ≤ p) :
(∑ i in s, w i * z i) ^ p ≤ ∑ i in s, (w i * z i ^ p) | (convex_on_rpow hp).map_sum_le hw hw' hz | theorem | real.rpow_arith_mean_le_arith_mean_rpow | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"convex_on_rpow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
arith_mean_le_rpow_mean (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i)
(hw' : ∑ i in s, w i = 1) (hz : ∀ i ∈ s, 0 ≤ z i) {p : ℝ} (hp : 1 ≤ p) :
∑ i in s, w i * z i ≤ (∑ i in s, (w i * z i ^ p)) ^ (1 / p) | begin
have : 0 < p := by positivity,
rw [← rpow_le_rpow_iff _ _ this, ← rpow_mul, one_div_mul_cancel (ne_of_gt this), rpow_one],
exact rpow_arith_mean_le_arith_mean_rpow s w z hw hw' hz hp,
all_goals { apply_rules [sum_nonneg, rpow_nonneg_of_nonneg],
intros i hi,
apply_rules [mul_nonneg, rpow_nonneg_of_... | theorem | real.arith_mean_le_rpow_mean | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"one_div_mul_cancel"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pow_arith_mean_le_arith_mean_pow (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) (n : ℕ) :
(∑ i in s, w i * z i) ^ n ≤ ∑ i in s, (w i * z i ^ n) | by exact_mod_cast real.pow_arith_mean_le_arith_mean_pow s _ _ (λ i _, (w i).coe_nonneg)
(by exact_mod_cast hw') (λ i _, (z i).coe_nonneg) n | theorem | nnreal.pow_arith_mean_le_arith_mean_pow | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"real.pow_arith_mean_le_arith_mean_pow"
] | Weighted generalized mean inequality, version sums over finite sets, with `ℝ≥0`-valued
functions and natural exponent. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pow_sum_div_card_le_sum_pow (f : ι → ℝ≥0) (n : ℕ) :
(∑ x in s, f x) ^ (n + 1) / s.card ^ n ≤ ∑ x in s, (f x) ^ (n + 1) | by simpa only [← nnreal.coe_le_coe, nnreal.coe_sum, nonneg.coe_div, nnreal.coe_pow] using
@real.pow_sum_div_card_le_sum_pow ι s (coe ∘ f) n (λ _ _, nnreal.coe_nonneg _) | lemma | nnreal.pow_sum_div_card_le_sum_pow | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"nnreal.coe_le_coe",
"nnreal.coe_nonneg",
"nnreal.coe_pow",
"nnreal.coe_sum",
"nonneg.coe_div",
"real.pow_sum_div_card_le_sum_pow"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) {p : ℝ}
(hp : 1 ≤ p) :
(∑ i in s, w i * z i) ^ p ≤ ∑ i in s, (w i * z i ^ p) | by exact_mod_cast real.rpow_arith_mean_le_arith_mean_rpow s _ _ (λ i _, (w i).coe_nonneg)
(by exact_mod_cast hw') (λ i _, (z i).coe_nonneg) hp | theorem | nnreal.rpow_arith_mean_le_arith_mean_rpow | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"real.rpow_arith_mean_le_arith_mean_rpow"
] | Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0`-valued
functions and real exponents. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rpow_arith_mean_le_arith_mean2_rpow (w₁ w₂ z₁ z₂ : ℝ≥0) (hw' : w₁ + w₂ = 1) {p : ℝ}
(hp : 1 ≤ p) :
(w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p | begin
have h := rpow_arith_mean_le_arith_mean_rpow univ ![w₁, w₂] ![z₁, z₂] _ hp,
{ simpa [fin.sum_univ_succ] using h, },
{ simp [hw', fin.sum_univ_succ], },
end | theorem | nnreal.rpow_arith_mean_le_arith_mean2_rpow | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [] | Weighted generalized mean inequality, version for two elements of `ℝ≥0` and real exponents. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rpow_add_le_mul_rpow_add_rpow (z₁ z₂ : ℝ≥0) {p : ℝ} (hp : 1 ≤ p) :
(z₁ + z₂) ^ p ≤ 2^(p-1) * (z₁ ^ p + z₂ ^ p) | begin
rcases eq_or_lt_of_le hp with rfl|h'p,
{ simp only [rpow_one, sub_self, rpow_zero, one_mul] },
convert rpow_arith_mean_le_arith_mean2_rpow (1/2) (1/2) (2 * z₁) (2 * z₂) (add_halves 1) hp,
{ simp only [one_div, inv_mul_cancel_left₀, ne.def, bit0_eq_zero, one_ne_zero, not_false_iff] },
{ simp only [one_di... | theorem | nnreal.rpow_add_le_mul_rpow_add_rpow | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"add_halves",
"bit0_eq_zero",
"div_eq_inv_mul",
"eq_or_lt_of_le",
"inv_mul_cancel_left₀",
"mul_one",
"one_div",
"one_mul",
"one_ne_zero",
"ring"
] | Unweighted mean inequality, version for two elements of `ℝ≥0` and real exponents. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
arith_mean_le_rpow_mean (w z : ι → ℝ≥0) (hw' : ∑ i in s, w i = 1) {p : ℝ}
(hp : 1 ≤ p) :
∑ i in s, w i * z i ≤ (∑ i in s, (w i * z i ^ p)) ^ (1 / p) | by exact_mod_cast real.arith_mean_le_rpow_mean s _ _ (λ i _, (w i).coe_nonneg)
(by exact_mod_cast hw') (λ i _, (z i).coe_nonneg) hp | theorem | nnreal.arith_mean_le_rpow_mean | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"real.arith_mean_le_rpow_mean"
] | Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0`-valued
functions and real exponents. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_rpow_le_one_of_add_le_one {p : ℝ} (a b : ℝ≥0) (hab : a + b ≤ 1)
(hp1 : 1 ≤ p) :
a ^ p + b ^ p ≤ 1 | begin
have h_le_one : ∀ x : ℝ≥0, x ≤ 1 → x ^ p ≤ x, from λ x hx, rpow_le_self_of_le_one hx hp1,
have ha : a ≤ 1, from (self_le_add_right a b).trans hab,
have hb : b ≤ 1, from (self_le_add_left b a).trans hab,
exact (add_le_add (h_le_one a ha) (h_le_one b hb)).trans hab,
end | lemma | nnreal.add_rpow_le_one_of_add_le_one | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
add_rpow_le_rpow_add {p : ℝ} (a b : ℝ≥0) (hp1 : 1 ≤ p) :
a ^ p + b ^ p ≤ (a + b) ^ p | begin
have hp_pos : 0 < p := by positivity,
by_cases h_zero : a + b = 0,
{ simp [add_eq_zero_iff.mp h_zero, hp_pos.ne'] },
have h_nonzero : ¬(a = 0 ∧ b = 0), by rwa add_eq_zero_iff at h_zero,
have h_add : a/(a+b) + b/(a+b) = 1, by rw [div_add_div_same, div_self h_zero],
have h := add_rpow_le_one_of_add_le_o... | lemma | nnreal.add_rpow_le_rpow_add | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"div_add_div_same",
"div_eq_mul_inv",
"div_self",
"mul_comm",
"mul_inv_cancel",
"mul_le_mul_left",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_add_rpow_le_add {p : ℝ} (a b : ℝ≥0) (hp1 : 1 ≤ p) :
(a ^ p + b ^ p) ^ (1/p) ≤ a + b | begin
rw ←@nnreal.le_rpow_one_div_iff _ _ (1/p) (by simp [lt_of_lt_of_le zero_lt_one hp1]),
rw one_div_one_div,
exact add_rpow_le_rpow_add _ _ hp1,
end | lemma | nnreal.rpow_add_rpow_le_add | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"nnreal.le_rpow_one_div_iff",
"one_div_one_div",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_add_rpow_le {p q : ℝ} (a b : ℝ≥0) (hp_pos : 0 < p) (hpq : p ≤ q) :
(a ^ q + b ^ q) ^ (1/q) ≤ (a ^ p + b ^ p) ^ (1/p) | begin
have h_rpow : ∀ a : ℝ≥0, a^q = (a^p)^(q/p),
from λ a, by rw [←nnreal.rpow_mul, div_eq_inv_mul, ←mul_assoc,
_root_.mul_inv_cancel hp_pos.ne.symm, one_mul],
have h_rpow_add_rpow_le_add : ((a^p)^(q/p) + (b^p)^(q/p)) ^ (1/(q/p)) ≤ a^p + b^p,
{ refine rpow_add_rpow_le_add (a^p) (b^p) _,
rwa one_le_... | theorem | nnreal.rpow_add_rpow_le | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"div_eq_inv_mul",
"mul_comm",
"mul_one_div",
"nnreal.le_rpow_one_div_iff",
"one_div_div",
"one_le_div",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0) (hp : 0 ≤ p) (hp1 : p ≤ 1) :
(a + b) ^ p ≤ a ^ p + b ^ p | begin
rcases hp.eq_or_lt with rfl|hp_pos,
{ simp },
have h := rpow_add_rpow_le a b hp_pos hp1,
rw one_div_one at h,
repeat { rw nnreal.rpow_one at h },
exact (nnreal.le_rpow_one_div_iff hp_pos).mp h
end | lemma | nnreal.rpow_add_le_add_rpow | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"nnreal.le_rpow_one_div_iff",
"nnreal.rpow_one",
"one_div_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑ i in s, w i = 1) {p : ℝ}
(hp : 1 ≤ p) :
(∑ i in s, w i * z i) ^ p ≤ ∑ i in s, (w i * z i ^ p) | begin
have hp_pos : 0 < p, positivity,
have hp_nonneg : 0 ≤ p, positivity,
have hp_not_nonpos : ¬ p ≤ 0, by simp [hp_pos],
have hp_not_neg : ¬ p < 0, by simp [hp_nonneg],
have h_top_iff_rpow_top : ∀ (i : ι) (hi : i ∈ s), w i * z i = ⊤ ↔ w i * (z i) ^ p = ⊤,
by simp [ennreal.mul_eq_top, hp_pos, hp_nonneg, ... | theorem | ennreal.rpow_arith_mean_le_arith_mean_rpow | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"ennreal.mul_eq_top",
"ennreal.one_ne_top",
"nnreal.rpow_arith_mean_le_arith_mean_rpow"
] | Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0∞`-valued
functions and real exponents. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rpow_arith_mean_le_arith_mean2_rpow (w₁ w₂ z₁ z₂ : ℝ≥0∞) (hw' : w₁ + w₂ = 1) {p : ℝ}
(hp : 1 ≤ p) :
(w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p | begin
have h := rpow_arith_mean_le_arith_mean_rpow univ ![w₁, w₂] ![z₁, z₂] _ hp,
{ simpa [fin.sum_univ_succ] using h, },
{ simp [hw', fin.sum_univ_succ], },
end | theorem | ennreal.rpow_arith_mean_le_arith_mean2_rpow | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [] | Weighted generalized mean inequality, version for two elements of `ℝ≥0∞` and real
exponents. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
rpow_add_le_mul_rpow_add_rpow (z₁ z₂ : ℝ≥0∞) {p : ℝ} (hp : 1 ≤ p) :
(z₁ + z₂) ^ p ≤ 2^(p-1) * (z₁ ^ p + z₂ ^ p) | begin
rcases eq_or_lt_of_le hp with rfl|h'p,
{ simp only [rpow_one, sub_self, rpow_zero, one_mul, le_refl], },
convert rpow_arith_mean_le_arith_mean2_rpow
(1/2) (1/2) (2 * z₁) (2 * z₂) (ennreal.add_halves 1) hp,
{ simp [← mul_assoc, ennreal.inv_mul_cancel two_ne_zero two_ne_top] },
{ simp [← mul_assoc, en... | theorem | ennreal.rpow_add_le_mul_rpow_add_rpow | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"ennreal.add_halves",
"ennreal.div_eq_inv_mul",
"ennreal.inv_mul_cancel",
"eq_or_lt_of_le",
"mul_assoc",
"mul_one",
"one_mul",
"ring",
"two_ne_zero"
] | Unweighted mean inequality, version for two elements of `ℝ≥0∞` and real exponents. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
add_rpow_le_rpow_add {p : ℝ} (a b : ℝ≥0∞) (hp1 : 1 ≤ p) :
a ^ p + b ^ p ≤ (a + b) ^ p | begin
have hp_pos : 0 < p := by positivity,
by_cases h_top : a + b = ⊤,
{ rw ←@ennreal.rpow_eq_top_iff_of_pos (a + b) p hp_pos at h_top,
rw h_top,
exact le_top, },
obtain ⟨ha_top, hb_top⟩ := add_ne_top.mp h_top,
lift a to ℝ≥0 using ha_top,
lift b to ℝ≥0 using hb_top,
simpa [← ennreal.coe_rpow_of_n... | lemma | ennreal.add_rpow_le_rpow_add | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"ennreal.coe_rpow_of_nonneg",
"ennreal.rpow_eq_top_iff_of_pos",
"le_top",
"lift",
"nnreal.add_rpow_le_rpow_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_add_rpow_le_add {p : ℝ} (a b : ℝ≥0∞) (hp1 : 1 ≤ p) :
(a ^ p + b ^ p) ^ (1/p) ≤ a + b | begin
rw ←@ennreal.le_rpow_one_div_iff _ _ (1/p) (by simp [lt_of_lt_of_le zero_lt_one hp1]),
rw one_div_one_div,
exact add_rpow_le_rpow_add _ _ hp1,
end | lemma | ennreal.rpow_add_rpow_le_add | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"ennreal.le_rpow_one_div_iff",
"one_div_one_div",
"zero_lt_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_add_rpow_le {p q : ℝ} (a b : ℝ≥0∞) (hp_pos : 0 < p) (hpq : p ≤ q) :
(a ^ q + b ^ q) ^ (1/q) ≤ (a ^ p + b ^ p) ^ (1/p) | begin
have h_rpow : ∀ a : ℝ≥0∞, a^q = (a^p)^(q/p),
from λ a, by rw [← ennreal.rpow_mul, _root_.mul_div_cancel' _ hp_pos.ne'],
have h_rpow_add_rpow_le_add : ((a^p)^(q/p) + (b^p)^(q/p)) ^ (1/(q/p)) ≤ a^p + b^p,
{ refine rpow_add_rpow_le_add (a^p) (b^p) _,
rwa one_le_div hp_pos, },
rw [h_rpow a, h_rpow b, ... | theorem | ennreal.rpow_add_rpow_le | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"ennreal.le_rpow_one_div_iff",
"ennreal.rpow_mul",
"mul_comm",
"mul_one_div",
"one_div_div",
"one_le_div"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0∞) (hp : 0 ≤ p) (hp1 : p ≤ 1) :
(a + b) ^ p ≤ a ^ p + b ^ p | begin
rcases hp.eq_or_lt with rfl|hp_pos,
{ suffices : (1 : ℝ≥0∞) ≤ 1 + 1,
{ simpa using this },
norm_cast,
norm_num },
have h := rpow_add_rpow_le a b hp_pos hp1,
rw one_div_one at h,
repeat { rw ennreal.rpow_one at h },
exact (ennreal.le_rpow_one_div_iff hp_pos).mp h,
end | lemma | ennreal.rpow_add_le_add_rpow | analysis | src/analysis/mean_inequalities_pow.lean | [
"analysis.convex.jensen",
"analysis.convex.specific_functions.basic",
"analysis.special_functions.pow.nnreal",
"tactic.positivity"
] | [
"ennreal.le_rpow_one_div_iff",
"ennreal.rpow_one",
"one_div_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cpow_mul_of_real_nonneg {x : ℝ} (hx : 0 ≤ x) (y : ℝ) (z : ℂ) :
(x : ℂ) ^ (↑y * z) = (↑(x ^ y) : ℂ) ^ z | begin
rw [cpow_mul, of_real_cpow hx],
{ rw [←of_real_log hx, ←of_real_mul, of_real_im, neg_lt_zero], exact real.pi_pos },
{ rw [←of_real_log hx, ←of_real_mul, of_real_im], exact real.pi_pos.le },
end | lemma | complex.cpow_mul_of_real_nonneg | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"real.pi_pos"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mellin_convergent (f : ℝ → E) (s : ℂ) : Prop | integrable_on (λ t : ℝ, (t : ℂ) ^ (s - 1) • f t) (Ioi 0) | def | mellin_convergent | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [] | Predicate on `f` and `s` asserting that the Mellin integral is well-defined. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mellin_convergent.const_smul {f : ℝ → E} {s : ℂ} (hf : mellin_convergent f s)
{𝕜 : Type*} [nontrivially_normed_field 𝕜] [normed_space 𝕜 E] [smul_comm_class ℂ 𝕜 E] (c : 𝕜) :
mellin_convergent (λ t, c • f t) s | by simpa only [mellin_convergent, smul_comm] using hf.smul c | lemma | mellin_convergent.const_smul | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"mellin_convergent",
"nontrivially_normed_field",
"normed_space",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mellin_convergent.cpow_smul {f : ℝ → E} {s a : ℂ} :
mellin_convergent (λ t, (t : ℂ) ^ a • f t) s ↔ mellin_convergent f (s + a) | begin
refine integrable_on_congr_fun (λ t ht, _) measurable_set_Ioi,
simp_rw [←sub_add_eq_add_sub, cpow_add _ _ (of_real_ne_zero.2 $ ne_of_gt ht), mul_smul],
end | lemma | mellin_convergent.cpow_smul | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"measurable_set_Ioi",
"mellin_convergent"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mellin_convergent.div_const {f : ℝ → ℂ} {s : ℂ} (hf : mellin_convergent f s) (a : ℂ) :
mellin_convergent (λ t, f t / a) s | by simpa only [mellin_convergent, smul_eq_mul, ←mul_div_assoc] using hf.div_const a | lemma | mellin_convergent.div_const | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"mellin_convergent",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mellin_convergent.comp_mul_left {f : ℝ → E} {s : ℂ} {a : ℝ} (ha : 0 < a) :
mellin_convergent (λ t, f (a * t)) s ↔ mellin_convergent f s | begin
have := integrable_on_Ioi_comp_mul_left_iff (λ t : ℝ, (t : ℂ) ^ (s - 1) • f t) 0 ha,
rw mul_zero at this,
have h1 : eq_on (λ t : ℝ, (↑(a * t) : ℂ) ^ (s - 1) • f (a * t))
((a : ℂ) ^ (s - 1) • (λ t : ℝ, (t : ℂ) ^ (s - 1) • f (a * t))) (Ioi 0),
{ intros t ht,
simp only [of_real_mul, mul_cpow_of_real_... | lemma | mellin_convergent.comp_mul_left | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"measurable_set_Ioi",
"mellin_convergent",
"mul_zero",
"not_and_distrib",
"pi.smul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mellin_convergent.comp_rpow {f : ℝ → E} {s : ℂ} {a : ℝ} (ha : a ≠ 0) :
mellin_convergent (λ t, f (t ^ a)) s ↔ mellin_convergent f (s / a) | begin
simp_rw mellin_convergent,
letI u : normed_space ℝ E := normed_space.complex_to_real, -- why isn't this automatic?
conv_rhs { rw ←@integrable_on_Ioi_comp_rpow_iff' _ _ u _ a ha },
refine integrable_on_congr_fun (λ t ht, _) measurable_set_Ioi,
dsimp only [pi.smul_apply],
rw [←complex.coe_smul (t ^ (a -... | lemma | mellin_convergent.comp_rpow | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"measurable_set_Ioi",
"mellin_convergent",
"mul_div_cancel'",
"mul_one",
"normed_space",
"normed_space.complex_to_real",
"pi.smul_apply"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mellin (f : ℝ → E) (s : ℂ) : E | ∫ t : ℝ in Ioi 0, (t : ℂ) ^ (s - 1) • f t | def | mellin | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [] | The Mellin transform of a function `f` (for a complex exponent `s`), defined as the integral of
`t ^ (s - 1) • f` over `Ioi 0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mellin_cpow_smul (f : ℝ → E) (s a : ℂ) :
mellin (λ t, (t : ℂ) ^ a • f t) s = mellin f (s + a) | begin
refine set_integral_congr measurable_set_Ioi (λ t ht, _),
simp_rw [←sub_add_eq_add_sub, cpow_add _ _ (of_real_ne_zero.2 $ ne_of_gt ht), mul_smul],
end | lemma | mellin_cpow_smul | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"measurable_set_Ioi",
"mellin"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mellin_const_smul (f : ℝ → E) (s : ℂ)
{𝕜 : Type*} [nontrivially_normed_field 𝕜] [normed_space 𝕜 E] [smul_comm_class ℂ 𝕜 E] (c : 𝕜) :
mellin (λ t, c • f t) s = c • mellin f s | by simp only [mellin, smul_comm, integral_smul] | lemma | mellin_const_smul | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"mellin",
"nontrivially_normed_field",
"normed_space",
"smul_comm_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mellin_div_const (f : ℝ → ℂ) (s a : ℂ) :
mellin (λ t, f t / a) s = mellin f s / a | by simp_rw [mellin, smul_eq_mul, ←mul_div_assoc, integral_div] | lemma | mellin_div_const | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"mellin",
"smul_eq_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mellin_comp_rpow (f : ℝ → E) (s : ℂ) {a : ℝ} (ha : a ≠ 0) :
mellin (λ t, f (t ^ a)) s = |a|⁻¹ • mellin f (s / a) | begin
-- note: this is also true for a = 0 (both sides are zero), but this is mathematically
-- uninteresting and rather time-consuming to check
simp_rw mellin,
conv_rhs { rw [←integral_comp_rpow_Ioi _ ha, ←integral_smul] },
refine set_integral_congr measurable_set_Ioi (λ t ht, _),
dsimp only,
rw [←mul_sm... | lemma | mellin_comp_rpow | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"inv_mul_cancel",
"measurable_set_Ioi",
"mellin",
"mul_div_cancel'",
"mul_one",
"one_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mellin_comp_mul_left (f : ℝ → E) (s : ℂ) {a : ℝ} (ha : 0 < a) :
mellin (λ t, f (a * t)) s = (a : ℂ) ^ (-s) • mellin f s | begin
simp_rw mellin,
have : eq_on (λ t : ℝ, (t : ℂ) ^ (s - 1) • f (a * t))
(λ t : ℝ, (a : ℂ) ^ (1 - s) • (λ u : ℝ, (u : ℂ) ^ (s - 1) • f u) (a * t)) (Ioi 0),
{ intros t ht,
dsimp only,
rw [of_real_mul, mul_cpow_of_real_nonneg ha.le (le_of_lt ht), ←mul_smul,
(by ring : 1 - s = -(s - 1)), cpow_ne... | lemma | mellin_comp_mul_left | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"abs_of_pos",
"inv_mul_cancel_left₀",
"inv_mul_cancel_right₀",
"measurable_set_Ioi",
"mellin",
"mul_assoc",
"mul_comm",
"mul_zero",
"not_and_distrib",
"ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mellin_comp_mul_right (f : ℝ → E) (s : ℂ) {a : ℝ} (ha : 0 < a) :
mellin (λ t, f (t * a)) s = (a : ℂ) ^ (-s) • mellin f s | by simpa only [mul_comm] using mellin_comp_mul_left f s ha | lemma | mellin_comp_mul_right | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"mellin",
"mellin_comp_mul_left",
"mul_comm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mellin_comp_inv (f : ℝ → E) (s : ℂ) : mellin (λ t, f (t⁻¹)) s = mellin f (-s) | by simp_rw [←rpow_neg_one, mellin_comp_rpow _ _ (neg_ne_zero.mpr one_ne_zero), abs_neg, abs_one,
inv_one, one_smul, of_real_neg, of_real_one, div_neg, div_one] | lemma | mellin_comp_inv | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"abs_neg",
"abs_one",
"div_neg",
"div_one",
"inv_one",
"mellin",
"mellin_comp_rpow",
"one_ne_zero",
"one_smul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_mellin (f : ℝ → E) (s : ℂ) (m : E) : Prop | mellin_convergent f s ∧ mellin f s = m | def | has_mellin | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"mellin",
"mellin_convergent"
] | Predicate standing for "the Mellin transform of `f` is defined at `s` and equal to `m`". This
shortens some arguments. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_mellin_add {f g : ℝ → E} {s : ℂ}
(hf : mellin_convergent f s) (hg : mellin_convergent g s) :
has_mellin (λ t, f t + g t) s (mellin f s + mellin g s) | ⟨by simpa only [mellin_convergent, smul_add] using hf.add hg,
by simpa only [mellin, smul_add] using integral_add hf hg⟩ | lemma | has_mellin_add | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"has_mellin",
"mellin",
"mellin_convergent",
"smul_add"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_mellin_sub {f g : ℝ → E} {s : ℂ}
(hf : mellin_convergent f s) (hg : mellin_convergent g s) :
has_mellin (λ t, f t - g t) s (mellin f s - mellin g s) | ⟨by simpa only [mellin_convergent, smul_sub] using hf.sub hg,
by simpa only [mellin, smul_sub] using integral_sub hf hg⟩ | lemma | has_mellin_sub | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"has_mellin",
"mellin",
"mellin_convergent",
"smul_sub"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mellin_convergent_iff_norm [normed_space ℂ E] {f : ℝ → E}
{T : set ℝ} (hT : T ⊆ Ioi 0) (hT' : measurable_set T)
(hfc : ae_strongly_measurable f $ volume.restrict $ Ioi 0) {s : ℂ} :
integrable_on (λ t : ℝ, (t : ℂ) ^ (s - 1) • f t) T
↔ integrable_on (λ t : ℝ, t ^ (s.re - 1) * ‖f t‖) T | begin
have : ae_strongly_measurable (λ t : ℝ, (t : ℂ) ^ (s - 1) • f t) (volume.restrict T),
{ refine ((continuous_at.continuous_on _).ae_strongly_measurable hT').smul (hfc.mono_set hT),
exact λ t ht, continuous_at_of_real_cpow_const _ _ (or.inr $ ne_of_gt (hT ht)) },
rw [integrable_on, ←integrable_norm_iff th... | lemma | mellin_convergent_iff_norm | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"complex.norm_eq_abs",
"continuous_at.continuous_on",
"measurable_set",
"norm_smul",
"normed_space"
] | Auxiliary lemma to reduce convergence statements from vector-valued functions to real
scalar-valued functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mellin_convergent_top_of_is_O
{f : ℝ → ℝ} (hfc : ae_strongly_measurable f $ volume.restrict (Ioi 0))
{a s : ℝ} (hf : is_O at_top f (λ t, t ^ (-a))) (hs : s < a) :
∃ (c : ℝ), 0 < c ∧ integrable_on (λ t : ℝ, t ^ (s - 1) * f t) (Ioi c) | begin
obtain ⟨d, hd, hd'⟩ := hf.exists_pos,
simp_rw [is_O_with, eventually_at_top] at hd',
obtain ⟨e, he⟩ := hd',
have he' : 0 < max e 1, from zero_lt_one.trans_le (le_max_right _ _),
refine ⟨max e 1, he', _, _⟩,
{ refine ae_strongly_measurable.mul _ (hfc.mono_set (Ioi_subset_Ioi he'.le)),
refine (conti... | lemma | mellin_convergent_top_of_is_O | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"continuous_at.continuous_on",
"integrable_on_Ioi_rpow_of_lt",
"measurable_set_Ioi",
"mul_assoc",
"mul_comm",
"mul_le_mul_of_nonneg_left",
"norm_mul"
] | If `f` is a locally integrable real-valued function which is `O(x ^ (-a))` at `∞`, then for any
`s < a`, its Mellin transform converges on some neighbourhood of `+∞`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mellin_convergent_zero_of_is_O
{b : ℝ} {f : ℝ → ℝ} (hfc : ae_strongly_measurable f $ volume.restrict (Ioi 0))
(hf : is_O (𝓝[>] 0) f (λ t, t ^ (-b))) {s : ℝ} (hs : b < s) :
∃ (c : ℝ), 0 < c ∧ integrable_on (λ t : ℝ, t ^ (s - 1) * f t) (Ioc 0 c) | begin
obtain ⟨d, hd, hd'⟩ := hf.exists_pos,
simp_rw [is_O_with, eventually_nhds_within_iff, metric.eventually_nhds_iff, gt_iff_lt] at hd',
obtain ⟨ε, hε, hε'⟩ := hd',
refine ⟨ε, hε, integrable_on_Ioc_iff_integrable_on_Ioo.mpr ⟨_, _⟩⟩,
{ refine ae_strongly_measurable.mul _ (hfc.mono_set Ioo_subset_Ioi_self),
... | lemma | mellin_convergent_zero_of_is_O | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"continuous_at.continuous_on",
"eventually_nhds_within_iff",
"gt_iff_lt",
"interval_integral.interval_integrable_rpow'",
"measurable_set_Ioo",
"metric.eventually_nhds_iff",
"mul_assoc",
"mul_comm",
"mul_le_mul_of_nonneg_left",
"mul_le_mul_of_nonneg_right",
"norm_mul"
] | If `f` is a locally integrable real-valued function which is `O(x ^ (-b))` at `0`, then for any
`b < s`, its Mellin transform converges on some right neighbourhood of `0`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mellin_convergent_of_is_O_scalar
{a b : ℝ} {f : ℝ → ℝ} {s : ℝ}
(hfc : locally_integrable_on f $ Ioi 0)
(hf_top : is_O at_top f (λ t, t ^ (-a))) (hs_top : s < a)
(hf_bot : is_O (𝓝[>] 0) f (λ t, t ^ (-b))) (hs_bot : b < s) :
integrable_on (λ t : ℝ, t ^ (s - 1) * f t) (Ioi 0) | begin
obtain ⟨c1, hc1, hc1'⟩ := mellin_convergent_top_of_is_O hfc.ae_strongly_measurable hf_top hs_top,
obtain ⟨c2, hc2, hc2'⟩ := mellin_convergent_zero_of_is_O hfc.ae_strongly_measurable hf_bot hs_bot,
have : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1,
{ rw [union_assoc, Ioc_union_Ioi (le_max_right _ _), Ioc_union_... | lemma | mellin_convergent_of_is_O_scalar | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"continuous_at.continuous_on",
"is_open_Ioi",
"mellin_convergent_top_of_is_O",
"mellin_convergent_zero_of_is_O"
] | If `f` is a locally integrable real-valued function on `Ioi 0` which is `O(x ^ (-a))` at `∞`
and `O(x ^ (-b))` at `0`, then its Mellin transform integral converges for `b < s < a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mellin_convergent_of_is_O_rpow [normed_space ℂ E]
{a b : ℝ} {f : ℝ → E} {s : ℂ}
(hfc : locally_integrable_on f $ Ioi 0)
(hf_top : is_O at_top f (λ t, t ^ (-a))) (hs_top : s.re < a)
(hf_bot : is_O (𝓝[>] 0) f (λ t, t ^ (-b))) (hs_bot : b < s.re) :
mellin_convergent f s | begin
rw [mellin_convergent, mellin_convergent_iff_norm (subset_refl _) measurable_set_Ioi
hfc.ae_strongly_measurable],
exact mellin_convergent_of_is_O_scalar
hfc.norm hf_top.norm_left hs_top hf_bot.norm_left hs_bot,
end | lemma | mellin_convergent_of_is_O_rpow | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"measurable_set_Ioi",
"mellin_convergent",
"mellin_convergent_iff_norm",
"mellin_convergent_of_is_O_scalar",
"normed_space",
"subset_refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_O_rpow_top_log_smul [normed_space ℝ E] {a b : ℝ} {f : ℝ → E}
(hab : b < a) (hf : is_O at_top f (λ t, t ^ (-a))) :
is_O at_top (λ t : ℝ, log t • f t) (λ t, t ^ (-b)) | begin
refine ((is_o_log_rpow_at_top (sub_pos.mpr hab)).is_O.smul hf).congr'
(eventually_of_forall (λ t, by refl))
((eventually_gt_at_top 0).mp (eventually_of_forall (λ t ht, _))),
rw [smul_eq_mul, ←rpow_add ht, ←sub_eq_add_neg, sub_eq_add_neg a, add_sub_cancel'],
end | lemma | is_O_rpow_top_log_smul | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"is_o_log_rpow_at_top",
"normed_space",
"smul_eq_mul"
] | If `f` is `O(x ^ (-a))` as `x → +∞`, then `log • f` is `O(x ^ (-b))` for every `b < a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_O_rpow_zero_log_smul [normed_space ℝ E] {a b : ℝ} {f : ℝ → E}
(hab : a < b) (hf : is_O (𝓝[>] 0) f (λ t, t ^ (-a))) :
is_O (𝓝[>] 0) (λ t : ℝ, log t • f t) (λ t, t ^ (-b)) | begin
have : is_o (𝓝[>] 0) log (λ t : ℝ, t ^ (a - b)),
{ refine ((is_o_log_rpow_at_top (sub_pos.mpr hab)).neg_left.comp_tendsto
tendsto_inv_zero_at_top).congr'
(eventually_nhds_within_iff.mpr $ eventually_of_forall (λ t ht, _))
(eventually_nhds_within_iff.mpr $ eventually_of_forall (λ t ht, _... | lemma | is_O_rpow_zero_log_smul | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"is_o_log_rpow_at_top",
"normed_space",
"one_ne_zero",
"real.log_one",
"smul_eq_mul",
"tendsto_inv_zero_at_top"
] | If `f` is `O(x ^ (-a))` as `x → 0`, then `log • f` is `O(x ^ (-b))` for every `a < b`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mellin_has_deriv_of_is_O_rpow [complete_space E] [normed_space ℂ E]
{a b : ℝ} {f : ℝ → E} {s : ℂ}
(hfc : locally_integrable_on f $ Ioi 0)
(hf_top : is_O at_top f (λ t, t ^ (-a))) (hs_top : s.re < a)
(hf_bot : is_O (𝓝[>] 0) f (λ t, t ^ (-b))) (hs_bot : b < s.re) :
mellin_convergent (λ t, log t • f t) s ∧
ha... | begin
let F : ℂ → ℝ → E := λ z t, (t : ℂ) ^ (z - 1) • f t,
let F' : ℂ → ℝ → E := λ z t, ((t : ℂ) ^ (z - 1) * log t) • f t,
have hab : b < a := hs_bot.trans hs_top,
-- A convenient radius of ball within which we can uniformly bound the derivative.
obtain ⟨v, hv0, hv1, hv2⟩ : ∃ (v : ℝ), (0 < v) ∧ (v < s.re - b)... | theorem | mellin_has_deriv_of_is_O_rpow | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"abs_nonneg",
"bound",
"complete_space",
"complex.abs_of_real",
"complex.norm_eq_abs",
"continuous.comp_continuous_on",
"continuous_abs",
"continuous_at.continuous_on",
"continuous_on.ae_strongly_measurable",
"exists_between",
"has_deriv_at",
"has_deriv_at_id'",
"has_deriv_at_integral_of_dom... | Suppose `f` is locally integrable on `(0, ∞)`, is `O(x ^ (-a))` as `x → ∞`, and is
`O(x ^ (-b))` as `x → 0`. Then its Mellin transform is differentiable on the domain `b < re s < a`,
with derivative equal to the Mellin transform of `log • f`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mellin_differentiable_at_of_is_O_rpow [complete_space E] [normed_space ℂ E]
{a b : ℝ} {f : ℝ → E} {s : ℂ}
(hfc : locally_integrable_on f $ Ioi 0)
(hf_top : is_O at_top f (λ t, t ^ (-a))) (hs_top : s.re < a)
(hf_bot : is_O (𝓝[>] 0) f (λ t, t ^ (-b))) (hs_bot : b < s.re) :
differentiable_at ℂ (mellin f) s | (mellin_has_deriv_of_is_O_rpow hfc hf_top hs_top hf_bot hs_bot).2.differentiable_at | lemma | mellin_differentiable_at_of_is_O_rpow | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"complete_space",
"differentiable_at",
"mellin",
"mellin_has_deriv_of_is_O_rpow",
"normed_space"
] | Suppose `f` is locally integrable on `(0, ∞)`, is `O(x ^ (-a))` as `x → ∞`, and is
`O(x ^ (-b))` as `x → 0`. Then its Mellin transform is differentiable on the domain `b < re s < a`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mellin_convergent_of_is_O_rpow_exp [normed_space ℂ E]
{a b : ℝ} (ha : 0 < a) {f : ℝ → E} {s : ℂ}
(hfc : locally_integrable_on f $ Ioi 0)
(hf_top : is_O at_top f (λ t, exp (-a * t)))
(hf_bot : is_O (𝓝[>] 0) f (λ t, t ^ (-b))) (hs_bot : b < s.re) :
mellin_convergent f s | mellin_convergent_of_is_O_rpow hfc (hf_top.trans (is_o_exp_neg_mul_rpow_at_top ha _).is_O)
(lt_add_one _) hf_bot hs_bot | lemma | mellin_convergent_of_is_O_rpow_exp | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"exp",
"is_o_exp_neg_mul_rpow_at_top",
"lt_add_one",
"mellin_convergent",
"mellin_convergent_of_is_O_rpow",
"normed_space"
] | If `f` is locally integrable, decays exponentially at infinity, and is `O(x ^ (-b))` at 0, then
its Mellin transform converges for `b < s.re`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mellin_differentiable_at_of_is_O_rpow_exp [complete_space E] [normed_space ℂ E]
{a b : ℝ} (ha : 0 < a) {f : ℝ → E} {s : ℂ}
(hfc : locally_integrable_on f $ Ioi 0)
(hf_top : is_O at_top f (λ t, exp (-a * t)))
(hf_bot : is_O (𝓝[>] 0) f (λ t, t ^ (-b))) (hs_bot : b < s.re) :
differentiable_at ℂ (mellin f) s | mellin_differentiable_at_of_is_O_rpow hfc (hf_top.trans (is_o_exp_neg_mul_rpow_at_top ha _).is_O)
(lt_add_one _) hf_bot hs_bot | lemma | mellin_differentiable_at_of_is_O_rpow_exp | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"complete_space",
"differentiable_at",
"exp",
"is_o_exp_neg_mul_rpow_at_top",
"lt_add_one",
"mellin",
"mellin_differentiable_at_of_is_O_rpow",
"normed_space"
] | If `f` is locally integrable, decays exponentially at infinity, and is `O(x ^ (-b))` at 0, then
its Mellin transform is holomorphic on `b < s.re`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_mellin_one_Ioc {s : ℂ} (hs : 0 < re s) :
has_mellin (indicator (Ioc 0 1) (λ t, 1 : ℝ → ℂ)) s (1 / s) | begin
have aux1 : -1 < (s - 1).re, by simpa only [sub_re, one_re, sub_eq_add_neg]
using lt_add_of_pos_left _ hs,
have aux2 : s ≠ 0, by { contrapose! hs, rw [hs, zero_re] },
have aux3 : measurable_set (Ioc (0 : ℝ) 1), from measurable_set_Ioc,
simp_rw [has_mellin, mellin, mellin_convergent, ←indicator_smul, i... | lemma | has_mellin_one_Ioc | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"has_mellin",
"integral_cpow",
"measurable_set",
"measurable_set_Ioc",
"mellin",
"mellin_convergent",
"mul_one",
"smul_eq_mul",
"zero_le_one"
] | The Mellin transform of the indicator function of `Ioc 0 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
has_mellin_cpow_Ioc (a : ℂ) {s : ℂ} (hs : 0 < re s + re a) :
has_mellin (indicator (Ioc 0 1) (λ t, ↑t ^ a : ℝ → ℂ)) s (1 / (s + a)) | begin
have := has_mellin_one_Ioc (by rwa add_re : 0 < (s + a).re),
simp_rw [has_mellin, ←mellin_convergent.cpow_smul, ←mellin_cpow_smul, ←indicator_smul,
smul_eq_mul, mul_one] at this,
exact this
end | lemma | has_mellin_cpow_Ioc | analysis | src/analysis/mellin_transform.lean | [
"analysis.special_functions.improper_integrals",
"analysis.calculus.parametric_integral",
"measure_theory.measure.haar.normed_space"
] | [
"has_mellin",
"has_mellin_one_Ioc",
"mul_one",
"smul_eq_mul"
] | The Mellin transform of a power function restricted to `Ioc 0 1`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
le_sum_condensed' (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
(∑ k in Ico 1 (2 ^ n), f k) ≤ ∑ k in range n, (2 ^ k) • f (2 ^ k) | begin
induction n with n ihn, { simp },
suffices : (∑ k in Ico (2 ^ n) (2 ^ (n + 1)), f k) ≤ (2 ^ n) • f (2 ^ n),
{ rw [sum_range_succ, ← sum_Ico_consecutive],
exact add_le_add ihn this,
exacts [n.one_le_two_pow, nat.pow_le_pow_of_le_right zero_lt_two n.le_succ] },
have : ∀ k ∈ Ico (2 ^ n) (2 ^ (n + 1))... | lemma | finset.le_sum_condensed' | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"nat.pow_le_pow_of_le_right",
"pow_pos",
"pow_succ",
"two_mul",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_sum_condensed (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
(∑ k in range (2 ^ n), f k) ≤ f 0 + ∑ k in range n, (2 ^ k) • f (2 ^ k) | begin
convert add_le_add_left (le_sum_condensed' hf n) (f 0),
rw [← sum_range_add_sum_Ico _ n.one_le_two_pow, sum_range_succ, sum_range_zero, zero_add]
end | lemma | finset.le_sum_condensed | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_condensed_le' (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
(∑ k in range n, (2 ^ k) • f (2 ^ (k + 1))) ≤ ∑ k in Ico 2 (2 ^ n + 1), f k | begin
induction n with n ihn, { simp },
suffices : (2 ^ n) • f (2 ^ (n + 1)) ≤ ∑ k in Ico (2 ^ n + 1) (2 ^ (n + 1) + 1), f k,
{ rw [sum_range_succ, ← sum_Ico_consecutive],
exact add_le_add ihn this,
exacts [add_le_add_right n.one_le_two_pow _,
add_le_add_right (nat.pow_le_pow_of_le_right zero_lt_two... | lemma | finset.sum_condensed_le' | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"le_rfl",
"nat.pow_le_pow_of_le_right",
"pow_succ",
"two_mul",
"zero_lt_two"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_condensed_le (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
(∑ k in range (n + 1), (2 ^ k) • f (2 ^ k)) ≤ f 1 + 2 • ∑ k in Ico 2 (2 ^ n + 1), f k | begin
convert add_le_add_left (nsmul_le_nsmul_of_le_right (sum_condensed_le' hf n) 2) (f 1),
simp [sum_range_succ', add_comm, pow_succ, mul_nsmul, sum_nsmul]
end | lemma | finset.sum_condensed_le | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"pow_succ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_tsum_condensed (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) :
∑' k, f k ≤ f 0 + ∑' k : ℕ, (2 ^ k) * f (2 ^ k) | begin
rw [ennreal.tsum_eq_supr_nat' (nat.tendsto_pow_at_top_at_top_of_one_lt _root_.one_lt_two)],
refine supr_le (λ n, (finset.le_sum_condensed hf n).trans (add_le_add_left _ _)),
simp only [nsmul_eq_mul, nat.cast_pow, nat.cast_two],
apply ennreal.sum_le_tsum
end | lemma | ennreal.le_tsum_condensed | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"ennreal.sum_le_tsum",
"ennreal.tsum_eq_supr_nat'",
"finset.le_sum_condensed",
"nat.cast_pow",
"nat.cast_two",
"nat.tendsto_pow_at_top_at_top_of_one_lt",
"nsmul_eq_mul",
"supr_le"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tsum_condensed_le (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) :
∑' k : ℕ, (2 ^ k * f (2 ^ k)) ≤ f 1 + 2 * ∑' k, f k | begin
rw [ennreal.tsum_eq_supr_nat' (tendsto_at_top_mono nat.le_succ tendsto_id), two_mul, ← two_nsmul],
refine supr_le (λ n, le_trans _ (add_le_add_left (nsmul_le_nsmul_of_le_right
(ennreal.sum_le_tsum $ finset.Ico 2 (2^n + 1)) _) _)),
simpa using finset.sum_condensed_le hf n
end | lemma | ennreal.tsum_condensed_le | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"ennreal.sum_le_tsum",
"ennreal.tsum_eq_supr_nat'",
"finset.Ico",
"finset.sum_condensed_le",
"supr_le",
"two_mul"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
summable_condensed_iff {f : ℕ → ℝ≥0} (hf : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) :
summable (λ k : ℕ, (2 ^ k) * f (2 ^ k)) ↔ summable f | begin
simp only [← ennreal.tsum_coe_ne_top_iff_summable, ne.def, not_iff_not, ennreal.coe_mul,
ennreal.coe_pow, ennreal.coe_two],
split; intro h,
{ replace hf : ∀ m n, 1 < m → m ≤ n → (f n : ℝ≥0∞) ≤ f m :=
λ m n hm hmn, ennreal.coe_le_coe.2 (hf (zero_lt_one.trans hm) hmn),
simpa [h, ennreal.add_eq_t... | lemma | nnreal.summable_condensed_iff | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"ennreal.add_eq_top",
"ennreal.coe_mul",
"ennreal.coe_pow",
"ennreal.coe_two",
"ennreal.le_tsum_condensed",
"ennreal.mul_eq_top",
"ennreal.tsum_coe_ne_top_iff_summable",
"ennreal.tsum_condensed_le",
"not_iff_not",
"summable"
] | Cauchy condensation test for a series of `nnreal` version. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
summable_condensed_iff_of_nonneg {f : ℕ → ℝ} (h_nonneg : ∀ n, 0 ≤ f n)
(h_mono : ∀ ⦃m n⦄, 0 < m → m ≤ n → f n ≤ f m) :
summable (λ k : ℕ, (2 ^ k) * f (2 ^ k)) ↔ summable f | begin
lift f to ℕ → ℝ≥0 using h_nonneg,
simp only [nnreal.coe_le_coe] at *,
exact_mod_cast nnreal.summable_condensed_iff h_mono
end | lemma | summable_condensed_iff_of_nonneg | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"lift",
"nnreal.coe_le_coe",
"nnreal.summable_condensed_iff",
"summable"
] | Cauchy condensation test for series of nonnegative real numbers. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.summable_nat_rpow_inv {p : ℝ} : summable (λ n, (n ^ p)⁻¹ : ℕ → ℝ) ↔ 1 < p | begin
cases le_or_lt 0 p with hp hp,
/- Cauchy condensation test applies only to antitone sequences, so we consider the
cases `0 ≤ p` and `p < 0` separately. -/
{ rw ← summable_condensed_iff_of_nonneg,
{ simp_rw [nat.cast_pow, nat.cast_two, ← rpow_nat_cast, ← rpow_mul zero_lt_two.le, mul_comm _ p,
r... | lemma | real.summable_nat_rpow_inv | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"abs_of_pos",
"gt_mem_nhds",
"inv_le_inv_of_le",
"inv_lt_one_iff_of_pos",
"inv_pow",
"mul_comm",
"mul_pow",
"nat.cast_pos",
"nat.cast_pow",
"nat.cast_two",
"one_div",
"summable",
"summable_condensed_iff_of_nonneg",
"summable_geometric_iff_norm_lt_1",
"zero_lt_one",
"zero_lt_two"
] | Test for convergence of the `p`-series: the real-valued series `∑' n : ℕ, (n ^ p)⁻¹` converges
if and only if `1 < p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.summable_nat_rpow {p : ℝ} : summable (λ n, n ^ p : ℕ → ℝ) ↔ p < -1 | by { rcases neg_surjective p with ⟨p, rfl⟩, simp [rpow_neg] } | lemma | real.summable_nat_rpow | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.summable_one_div_nat_rpow {p : ℝ} : summable (λ n, 1 / n ^ p : ℕ → ℝ) ↔ 1 < p | by simp | lemma | real.summable_one_div_nat_rpow | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"summable"
] | Test for convergence of the `p`-series: the real-valued series `∑' n : ℕ, 1 / n ^ p` converges
if and only if `1 < p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.summable_nat_pow_inv {p : ℕ} : summable (λ n, (n ^ p)⁻¹ : ℕ → ℝ) ↔ 1 < p | by simp only [← rpow_nat_cast, real.summable_nat_rpow_inv, nat.one_lt_cast] | lemma | real.summable_nat_pow_inv | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"nat.one_lt_cast",
"real.summable_nat_rpow_inv",
"summable"
] | Test for convergence of the `p`-series: the real-valued series `∑' n : ℕ, (n ^ p)⁻¹` converges
if and only if `1 < p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.summable_one_div_nat_pow {p : ℕ} : summable (λ n, 1 / n ^ p : ℕ → ℝ) ↔ 1 < p | by simp | lemma | real.summable_one_div_nat_pow | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"summable"
] | Test for convergence of the `p`-series: the real-valued series `∑' n : ℕ, 1 / n ^ p` converges
if and only if `1 < p`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.summable_one_div_int_pow {p : ℕ} : summable (λ n:ℤ, 1 / (n : ℝ) ^ p) ↔ 1 < p | begin
refine ⟨λ h, real.summable_one_div_nat_pow.mp (h.comp_injective nat.cast_injective),
λ h, summable_int_of_summable_nat (real.summable_one_div_nat_pow.mpr h)
(((real.summable_one_div_nat_pow.mpr h).mul_left $ 1 / (-1) ^ p).congr $ λ n, _)⟩,
conv_rhs { rw [int.cast_neg, neg_eq_neg_one_mul, mul_pow, ←div... | lemma | real.summable_one_div_int_pow | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"int.cast_neg",
"mul_div",
"mul_one",
"mul_pow",
"nat.cast_injective",
"neg_eq_neg_one_mul",
"summable",
"summable_int_of_summable_nat"
] | Summability of the `p`-series over `ℤ`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.summable_abs_int_rpow {b : ℝ} (hb : 1 < b) : summable (λ n : ℤ, |(n : ℝ)| ^ (-b)) | begin
refine summable_int_of_summable_nat (_ : summable (λ n : ℕ, |(n : ℝ)| ^ _))
(_ : summable (λ n : ℕ, |((-n : ℤ) : ℝ)| ^ _)),
work_on_goal 2 { simp_rw [int.cast_neg, int.cast_coe_nat, abs_neg] },
all_goals { simp_rw (λ n : ℕ, abs_of_nonneg (n.cast_nonneg : 0 ≤ (n : ℝ))),
rwa [real.summable_nat_rpow, n... | lemma | real.summable_abs_int_rpow | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"abs_neg",
"abs_of_nonneg",
"int.cast_coe_nat",
"int.cast_neg",
"real.summable_nat_rpow",
"summable",
"summable_int_of_summable_nat"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
real.not_summable_nat_cast_inv : ¬summable (λ n, n⁻¹ : ℕ → ℝ) | have ¬summable (λ n, (n^1)⁻¹ : ℕ → ℝ), from mt real.summable_nat_pow_inv.1 (lt_irrefl 1),
by simpa | lemma | real.not_summable_nat_cast_inv | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"summable"
] | Harmonic series is not unconditionally summable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.not_summable_one_div_nat_cast : ¬summable (λ n, 1 / n : ℕ → ℝ) | by simpa only [inv_eq_one_div] using real.not_summable_nat_cast_inv | lemma | real.not_summable_one_div_nat_cast | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"inv_eq_one_div",
"real.not_summable_nat_cast_inv",
"summable"
] | Harmonic series is not unconditionally summable. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
real.tendsto_sum_range_one_div_nat_succ_at_top :
tendsto (λ n, ∑ i in finset.range n, (1 / (i + 1) : ℝ)) at_top at_top | begin
rw ← not_summable_iff_tendsto_nat_at_top_of_nonneg,
{ exact_mod_cast mt (summable_nat_add_iff 1).1 real.not_summable_one_div_nat_cast },
{ exact λ i, div_nonneg zero_le_one i.cast_add_one_pos.le }
end | lemma | real.tendsto_sum_range_one_div_nat_succ_at_top | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"div_nonneg",
"finset.range",
"not_summable_iff_tendsto_nat_at_top_of_nonneg",
"real.not_summable_one_div_nat_cast",
"summable_nat_add_iff",
"zero_le_one"
] | **Divergence of the Harmonic Series** | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
nnreal.summable_rpow_inv {p : ℝ} : summable (λ n, (n ^ p)⁻¹ : ℕ → ℝ≥0) ↔ 1 < p | by simp [← nnreal.summable_coe] | lemma | nnreal.summable_rpow_inv | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"nnreal.summable_coe",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnreal.summable_rpow {p : ℝ} : summable (λ n, n ^ p : ℕ → ℝ≥0) ↔ p < -1 | by simp [← nnreal.summable_coe] | lemma | nnreal.summable_rpow | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"nnreal.summable_coe",
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nnreal.summable_one_div_rpow {p : ℝ} : summable (λ n, 1 / n ^ p : ℕ → ℝ≥0) ↔ 1 < p | by simp | lemma | nnreal.summable_one_div_rpow | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"summable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_Ioc_inv_sq_le_sub {k n : ℕ} (hk : k ≠ 0) (h : k ≤ n) :
∑ i in Ioc k n, ((i ^ 2) ⁻¹ : α) ≤ k ⁻¹ - n ⁻¹ | begin
refine nat.le_induction _ _ n h,
{ simp only [Ioc_self, sum_empty, sub_self] },
assume n hn IH,
rw [sum_Ioc_succ_top hn],
apply (add_le_add IH le_rfl).trans,
simp only [sub_eq_add_neg, add_assoc, nat.cast_add, nat.cast_one, le_add_neg_iff_add_le,
add_le_iff_nonpos_right, neg_add_le_iff_le_add, add... | lemma | sum_Ioc_inv_sq_le_sub | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"div_le_div_iff",
"le_rfl",
"nat.cast_add",
"nat.cast_one",
"nat.le_induction"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum_Ioo_inv_sq_le (k n : ℕ) :
∑ i in Ioo k n, ((i ^ 2) ⁻¹ : α) ≤ 2 / (k + 1) | calc
∑ i in Ioo k n, ((i ^ 2) ⁻¹ : α) ≤ ∑ i in Ioc k (max (k+1) n), (i ^ 2) ⁻¹ :
begin
apply sum_le_sum_of_subset_of_nonneg,
{ assume x hx,
simp only [mem_Ioo] at hx,
simp only [hx, hx.2.le, mem_Ioc, le_max_iff, or_true, and_self] },
{ assume i hi hident,
exact inv_nonneg.2 (sq_nonneg _), }
end
... ≤ ... | lemma | sum_Ioo_inv_sq_le | analysis | src/analysis/p_series.lean | [
"analysis.special_functions.pow.nnreal"
] | [
"div_le_div",
"inv_nonneg",
"le_max_iff",
"le_rfl",
"nat.Icc_succ_left",
"nat.Ico_succ_right",
"nat.cast_nonneg",
"nat.cast_succ",
"one_div",
"one_le_two",
"pow_le_pow",
"ring",
"sq_nonneg",
"sum_Ioc_inv_sq_le_sub",
"zero_le_one"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_self (a : ℍ) : ⟪a, a⟫ = norm_sq a | rfl | lemma | quaternion.inner_self | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
inner_def (a b : ℍ) : ⟪a, b⟫ = (a * star b).re | rfl | lemma | quaternion.inner_def | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
norm_sq_eq_norm_sq (a : ℍ) : norm_sq a = ‖a‖ * ‖a‖ | by rw [← inner_self, real_inner_self_eq_norm_mul_norm] | lemma | quaternion.norm_sq_eq_norm_sq | analysis | src/analysis/quaternion.lean | [
"algebra.quaternion",
"analysis.inner_product_space.basic",
"analysis.inner_product_space.pi_L2",
"topology.algebra.algebra"
] | [
"real_inner_self_eq_norm_mul_norm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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