fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
addWellApproximable_ae_empty_or_univ (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0)) :
(∀ᵐ x, ¬addWellApproximable 𝕊 δ x) ∨ ∀ᵐ x, addWellApproximable 𝕊 δ x := by
/- Sketch of proof:
Let `E := addWellApproximable 𝕊 δ`. For each prime `p : ℕ`, we can partition `E` into three
pieces `E = (A p) ∪ (B p) ∪ (C p)` ... | theorem | NumberTheory | [
"Mathlib.Dynamics.Ergodic.AddCircle",
"Mathlib.MeasureTheory.Covering.LiminfLimsup"
] | Mathlib/NumberTheory/WellApproximable.lean | addWellApproximable_ae_empty_or_univ | **Gallagher's ergodic theorem** on Diophantine approximation. |
_root_.NormedAddCommGroup.exists_norm_nsmul_le {A : Type*}
[NormedAddCommGroup A] [CompactSpace A] [PreconnectedSpace A]
[MeasurableSpace A] [BorelSpace A] {μ : Measure A} [μ.IsAddHaarMeasure]
(ξ : A) {n : ℕ} (hn : 0 < n) (δ : ℝ) (hδ : μ univ ≤ (n + 1) • μ (closedBall (0 : A) (δ / 2))) :
∃ j ∈ Icc 1 n, ... | lemma | NumberTheory | [
"Mathlib.Dynamics.Ergodic.AddCircle",
"Mathlib.MeasureTheory.Covering.LiminfLimsup"
] | Mathlib/NumberTheory/WellApproximable.lean | _root_.NormedAddCommGroup.exists_norm_nsmul_le | A general version of **Dirichlet's approximation theorem**.
See also `AddCircle.exists_norm_nsmul_le`. |
exists_norm_nsmul_le (ξ : 𝕊) {n : ℕ} (hn : 0 < n) :
∃ j ∈ Icc 1 n, ‖j • ξ‖ ≤ T / ↑(n + 1) := by
apply NormedAddCommGroup.exists_norm_nsmul_le (μ := volume) ξ hn
rw [AddCircle.measure_univ, volume_closedBall, ← ENNReal.ofReal_nsmul,
mul_div_cancel₀ _ two_ne_zero, min_eq_right (div_le_self hT.out.le <| by si... | lemma | NumberTheory | [
"Mathlib.Dynamics.Ergodic.AddCircle",
"Mathlib.MeasureTheory.Covering.LiminfLimsup"
] | Mathlib/NumberTheory/WellApproximable.lean | exists_norm_nsmul_le | **Dirichlet's approximation theorem**
See also `Real.exists_rat_abs_sub_le_and_den_le`. |
@[simp]
wilsons_lemma : ((p - 1)! : ZMod p) = -1 := by
refine
calc
((p - 1)! : ZMod p) = ∏ x ∈ Ico 1 (succ (p - 1)), (x : ZMod p) := by
rw [← Finset.prod_Ico_id_eq_factorial, prod_natCast]
_ = ∏ x : (ZMod p)ˣ, (x : ZMod p) := ?_
_ = -1 := by
simp_rw [← Units.coeHom_apply, ← map_p... | theorem | NumberTheory | [
"Mathlib.FieldTheory.Finite.Basic"
] | Mathlib/NumberTheory/Wilson.lean | wilsons_lemma | **Wilson's Lemma**: the product of `1`, ..., `p-1` is `-1` modulo `p`. |
prod_Ico_one_prime : ∏ x ∈ Ico 1 p, (x : ZMod p) = -1 := by
conv =>
congr
congr
rw [← Nat.add_one_sub_one p, succ_sub (Fact.out (p := p.Prime)).pos]
rw [← prod_natCast, Finset.prod_Ico_id_eq_factorial, wilsons_lemma] | theorem | NumberTheory | [
"Mathlib.FieldTheory.Finite.Basic"
] | Mathlib/NumberTheory/Wilson.lean | prod_Ico_one_prime | null |
prime_of_fac_equiv_neg_one (h : ((n - 1)! : ZMod n) = -1) (h1 : n ≠ 1) : Prime n := by
rcases eq_or_ne n 0 with (rfl | h0)
· norm_num at h
replace h1 : 1 < n := n.two_le_iff.mpr ⟨h0, h1⟩
by_contra h2
obtain ⟨m, hm1, hm2 : 1 < m, hm3⟩ := exists_dvd_of_not_prime2 h1 h2
have hm : m ∣ (n - 1)! := Nat.dvd_factor... | theorem | NumberTheory | [
"Mathlib.FieldTheory.Finite.Basic"
] | Mathlib/NumberTheory/Wilson.lean | prime_of_fac_equiv_neg_one | For `n ≠ 1`, `(n-1)!` is congruent to `-1` modulo `n` only if n is prime. |
prime_iff_fac_equiv_neg_one (h : n ≠ 1) : Prime n ↔ ((n - 1)! : ZMod n) = -1 := by
refine ⟨fun h1 => ?_, fun h2 => prime_of_fac_equiv_neg_one h2 h⟩
haveI := Fact.mk h1
exact ZMod.wilsons_lemma n | theorem | NumberTheory | [
"Mathlib.FieldTheory.Finite.Basic"
] | Mathlib/NumberTheory/Wilson.lean | prime_iff_fac_equiv_neg_one | **Wilson's Theorem**: For `n ≠ 1`, `(n-1)!` is congruent to `-1` modulo `n` iff n is prime. |
bernoulliFun (k : ℕ) (x : ℝ) : ℝ :=
(Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli k)).eval x | def | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | bernoulliFun | The function `x ↦ Bₖ(x) : ℝ → ℝ`. |
bernoulliFun_eval_zero (k : ℕ) : bernoulliFun k 0 = bernoulli k := by
rw [bernoulliFun, Polynomial.eval_zero_map, Polynomial.bernoulli_eval_zero, eq_ratCast] | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | bernoulliFun_eval_zero | null |
bernoulliFun_endpoints_eq_of_ne_one {k : ℕ} (hk : k ≠ 1) :
bernoulliFun k 1 = bernoulliFun k 0 := by
rw [bernoulliFun_eval_zero, bernoulliFun, Polynomial.eval_one_map, Polynomial.bernoulli_eval_one,
bernoulli_eq_bernoulli'_of_ne_one hk, eq_ratCast] | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | bernoulliFun_endpoints_eq_of_ne_one | null |
bernoulliFun_eval_one (k : ℕ) : bernoulliFun k 1 = bernoulliFun k 0 + ite (k = 1) 1 0 := by
rw [bernoulliFun, bernoulliFun_eval_zero, Polynomial.eval_one_map, Polynomial.bernoulli_eval_one]
split_ifs with h
· rw [h, bernoulli_one, bernoulli'_one, eq_ratCast]
push_cast; ring
· rw [bernoulli_eq_bernoulli'_of_... | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | bernoulliFun_eval_one | null |
hasDerivAt_bernoulliFun (k : ℕ) (x : ℝ) :
HasDerivAt (bernoulliFun k) (k * bernoulliFun (k - 1) x) x := by
convert ((Polynomial.bernoulli k).map <| algebraMap ℚ ℝ).hasDerivAt x using 1
simp only [bernoulliFun, Polynomial.derivative_map, Polynomial.derivative_bernoulli k,
Polynomial.map_mul, Polynomial.map_n... | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | hasDerivAt_bernoulliFun | null |
antideriv_bernoulliFun (k : ℕ) (x : ℝ) :
HasDerivAt (fun x => bernoulliFun (k + 1) x / (k + 1)) (bernoulliFun k x) x := by
convert (hasDerivAt_bernoulliFun (k + 1) x).div_const _ using 1
simp [Nat.cast_add_one_ne_zero k] | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | antideriv_bernoulliFun | null |
integral_bernoulliFun_eq_zero {k : ℕ} (hk : k ≠ 0) :
∫ x : ℝ in 0..1, bernoulliFun k x = 0 := by
rw [integral_eq_sub_of_hasDerivAt (fun x _ => antideriv_bernoulliFun k x)
((Polynomial.continuous _).intervalIntegrable _ _)]
rw [bernoulliFun_eval_one]
split_ifs with h
· exfalso; exact hk (Nat.succ_inj.m... | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | integral_bernoulliFun_eq_zero | null |
bernoulliFourierCoeff (k : ℕ) (n : ℤ) : ℂ :=
fourierCoeffOn zero_lt_one (fun x => bernoulliFun k x) n | def | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | bernoulliFourierCoeff | The `n`-th Fourier coefficient of the `k`-th Bernoulli function on the interval `[0, 1]`. |
bernoulliFourierCoeff_recurrence (k : ℕ) {n : ℤ} (hn : n ≠ 0) :
bernoulliFourierCoeff k n =
1 / (-2 * π * I * n) * (ite (k = 1) 1 0 - k * bernoulliFourierCoeff (k - 1) n) := by
unfold bernoulliFourierCoeff
rw [fourierCoeffOn_of_hasDerivAt zero_lt_one hn
(fun x _ => (hasDerivAt_bernoulliFun k x).ofRe... | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | bernoulliFourierCoeff_recurrence | Recurrence relation (in `k`) for the `n`-th Fourier coefficient of `Bₖ`. |
bernoulli_zero_fourier_coeff {n : ℤ} (hn : n ≠ 0) : bernoulliFourierCoeff 0 n = 0 := by
simpa using bernoulliFourierCoeff_recurrence 0 hn | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | bernoulli_zero_fourier_coeff | The Fourier coefficients of `B₀(x) = 1`. |
bernoulliFourierCoeff_zero {k : ℕ} (hk : k ≠ 0) : bernoulliFourierCoeff k 0 = 0 := by
simp_rw [bernoulliFourierCoeff, fourierCoeffOn_eq_integral, neg_zero, fourier_zero, sub_zero,
div_one, one_smul, intervalIntegral.integral_ofReal, integral_bernoulliFun_eq_zero hk,
ofReal_zero] | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | bernoulliFourierCoeff_zero | The `0`-th Fourier coefficient of `Bₖ(x)`. |
bernoulliFourierCoeff_eq {k : ℕ} (hk : k ≠ 0) (n : ℤ) :
bernoulliFourierCoeff k n = -k ! / (2 * π * I * n) ^ k := by
rcases eq_or_ne n 0 with (rfl | hn)
· rw [bernoulliFourierCoeff_zero hk, Int.cast_zero, mul_zero, zero_pow hk,
div_zero]
refine Nat.le_induction ?_ (fun k hk h'k => ?_) k (Nat.one_le_iff_... | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | bernoulliFourierCoeff_eq | null |
periodizedBernoulli (k : ℕ) : 𝕌 → ℝ :=
AddCircle.liftIco 1 0 (bernoulliFun k) | def | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | periodizedBernoulli | The Bernoulli polynomial, extended from `[0, 1)` to the unit circle. |
periodizedBernoulli.continuous {k : ℕ} (hk : k ≠ 1) : Continuous (periodizedBernoulli k) :=
AddCircle.liftIco_zero_continuous
(mod_cast (bernoulliFun_endpoints_eq_of_ne_one hk).symm)
(Polynomial.continuous _).continuousOn | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | periodizedBernoulli.continuous | null |
fourierCoeff_bernoulli_eq {k : ℕ} (hk : k ≠ 0) (n : ℤ) :
fourierCoeff ((↑) ∘ periodizedBernoulli k : 𝕌 → ℂ) n = -k ! / (2 * π * I * n) ^ k := by
have : ((↑) ∘ periodizedBernoulli k : 𝕌 → ℂ) = AddCircle.liftIco 1 0 ((↑) ∘ bernoulliFun k) := by
ext1 x; rfl
rw [this, fourierCoeff_liftIco_eq]
simpa only [ze... | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | fourierCoeff_bernoulli_eq | null |
summable_bernoulli_fourier {k : ℕ} (hk : 2 ≤ k) :
Summable (fun n => -k ! / (2 * π * I * n) ^ k : ℤ → ℂ) := by
have :
∀ n : ℤ, -(k ! : ℂ) / (2 * π * I * n) ^ k = -k ! / (2 * π * I) ^ k * (1 / (n : ℂ) ^ k) := by
intro n; rw [mul_one_div, div_div, ← mul_pow]
simp_rw [this]
refine Summable.mul_left _ <... | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | summable_bernoulli_fourier | null |
hasSum_one_div_pow_mul_fourier_mul_bernoulliFun {k : ℕ} (hk : 2 ≤ k) {x : ℝ}
(hx : x ∈ Icc (0 : ℝ) 1) :
HasSum (fun n : ℤ => 1 / (n : ℂ) ^ k * fourier n (x : 𝕌))
(-(2 * π * I) ^ k / k ! * bernoulliFun k x) := by
suffices ∀ {y : ℝ}, y ∈ Ico (0 : ℝ) 1 →
HasSum (fun (n : ℤ) ↦ 1 / (n : ℂ) ^ k * fouri... | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | hasSum_one_div_pow_mul_fourier_mul_bernoulliFun | null |
hasSum_one_div_nat_pow_mul_fourier {k : ℕ} (hk : 2 ≤ k) {x : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) :
HasSum
(fun n : ℕ =>
(1 : ℂ) / (n : ℂ) ^ k * (fourier n (x : 𝕌) + (-1 : ℂ) ^ k * fourier (-n) (x : 𝕌)))
(-(2 * π * I) ^ k / k ! * bernoulliFun k x) := by
convert (hasSum_one_div_pow_mul_fourier_mul_ber... | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | hasSum_one_div_nat_pow_mul_fourier | null |
hasSum_one_div_nat_pow_mul_cos {k : ℕ} (hk : k ≠ 0) {x : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) :
HasSum (fun n : ℕ => 1 / (n : ℝ) ^ (2 * k) * Real.cos (2 * π * n * x))
((-1 : ℝ) ^ (k + 1) * (2 * π) ^ (2 * k) / 2 / (2 * k)! *
(Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k))).eval x) := by
have :... | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | hasSum_one_div_nat_pow_mul_cos | null |
hasSum_one_div_nat_pow_mul_sin {k : ℕ} (hk : k ≠ 0) {x : ℝ} (hx : x ∈ Icc (0 : ℝ) 1) :
HasSum (fun n : ℕ => 1 / (n : ℝ) ^ (2 * k + 1) * Real.sin (2 * π * n * x))
((-1 : ℝ) ^ (k + 1) * (2 * π) ^ (2 * k + 1) / 2 / (2 * k + 1)! *
(Polynomial.map (algebraMap ℚ ℝ) (Polynomial.bernoulli (2 * k + 1))).eval x... | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | hasSum_one_div_nat_pow_mul_sin | null |
hasSum_zeta_nat {k : ℕ} (hk : k ≠ 0) :
HasSum (fun n : ℕ => 1 / (n : ℝ) ^ (2 * k))
((-1 : ℝ) ^ (k + 1) * (2 : ℝ) ^ (2 * k - 1) * π ^ (2 * k) *
bernoulli (2 * k) / (2 * k)!) := by
convert hasSum_one_div_nat_pow_mul_cos hk (left_mem_Icc.mpr zero_le_one) using 1
· ext1 n; rw [mul_zero, Real.cos_zero,... | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | hasSum_zeta_nat | null |
hasSum_zeta_two : HasSum (fun n : ℕ => (1 : ℝ) / (n : ℝ) ^ 2) (π ^ 2 / 6) := by
convert hasSum_zeta_nat one_ne_zero using 1; rw [mul_one]
rw [bernoulli_eq_bernoulli'_of_ne_one (by decide : 2 ≠ 1), bernoulli'_two]
norm_num [Nat.factorial]; field_simp | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | hasSum_zeta_two | null |
hasSum_zeta_four : HasSum (fun n : ℕ => (1 : ℝ) / (n : ℝ) ^ 4) (π ^ 4 / 90) := by
convert hasSum_zeta_nat two_ne_zero using 1; norm_num
rw [bernoulli_eq_bernoulli'_of_ne_one, bernoulli'_four]
· norm_num [Nat.factorial]; field_simp; ring
· decide | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | hasSum_zeta_four | null |
Polynomial.bernoulli_three_eval_one_quarter :
(Polynomial.bernoulli 3).eval (1 / 4) = 3 / 64 := by
simp_rw [Polynomial.bernoulli, Finset.sum_range_succ, Polynomial.eval_add,
Polynomial.eval_monomial]
rw [Finset.sum_range_zero, Polynomial.eval_zero, zero_add, bernoulli_one]
rw [bernoulli_eq_bernoulli'_of_n... | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | Polynomial.bernoulli_three_eval_one_quarter | null |
hasSum_L_function_mod_four_eval_three :
HasSum (fun n : ℕ => (1 : ℝ) / (n : ℝ) ^ 3 * Real.sin (π * n / 2)) (π ^ 3 / 32) := by
apply (congr_arg₂ HasSum ?_ ?_).to_iff.mp <|
hasSum_one_div_nat_pow_mul_sin one_ne_zero (?_ : 1 / 4 ∈ Icc (0 : ℝ) 1)
· ext1 n
norm_num
left
congr 1
ring
· have : (1... | theorem | NumberTheory | [
"Mathlib.NumberTheory.BernoulliPolynomials",
"Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic",
"Mathlib.Analysis.Calculus.Deriv.Polynomial",
"Mathlib.Analysis.Fourier.AddCircle",
"Mathlib.Analysis.PSeries"
] | Mathlib/NumberTheory/ZetaValues.lean | hasSum_L_function_mod_four_eval_three | Explicit formula for `L(χ, 3)`, where `χ` is the unique nontrivial Dirichlet character modulo 4. |
protected Symmetric.compl (h : Symmetric r) : Symmetric rᶜ := fun _ _ hr hr' =>
hr <| h hr' | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | Symmetric.compl | null |
IsAntichain (r : α → α → Prop) (s : Set α) : Prop :=
s.Pairwise rᶜ | def | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | IsAntichain | An antichain is a set such that no two distinct elements are related. |
@[simp] protected empty : IsAntichain r ∅ :=
pairwise_empty _
@[simp] protected theorem singleton : IsAntichain r {a} :=
pairwise_singleton _ _ | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | empty | null |
protected subset (hs : IsAntichain r s) (h : t ⊆ s) : IsAntichain r t :=
hs.mono h | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | subset | null |
mono (hs : IsAntichain r₁ s) (h : r₂ ≤ r₁) : IsAntichain r₂ s :=
hs.mono' <| compl_le_compl h | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | mono | null |
mono_on (hs : IsAntichain r₁ s) (h : s.Pairwise fun ⦃a b⦄ => r₂ a b → r₁ a b) :
IsAntichain r₂ s :=
hs.imp_on <| h.imp fun _ _ h h₁ h₂ => h₁ <| h h₂ | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | mono_on | null |
protected eq (hs : IsAntichain r s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) (h : r a b) :
a = b :=
Set.Pairwise.eq hs ha hb <| not_not_intro h | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | eq | null |
protected eq' (hs : IsAntichain r s) {a b : α} (ha : a ∈ s) (hb : b ∈ s) (h : r b a) :
a = b :=
(hs.eq hb ha h).symm | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | eq' | null |
protected isAntisymm (h : IsAntichain r univ) : IsAntisymm α r :=
⟨fun _ _ ha _ => h.eq trivial trivial ha⟩ | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | isAntisymm | null |
protected subsingleton [IsTrichotomous α r] (h : IsAntichain r s) : s.Subsingleton := by
rintro a ha b hb
obtain hab | hab | hab := trichotomous_of r a b
· exact h.eq ha hb hab
· exact hab
· exact h.eq' ha hb hab | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | subsingleton | null |
protected flip (hs : IsAntichain r s) : IsAntichain (flip r) s := fun _ ha _ hb h =>
hs hb ha h.symm | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | flip | null |
swap (hs : IsAntichain r s) : IsAntichain (swap r) s :=
hs.flip | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | swap | null |
image (hs : IsAntichain r s) (f : α → β) (h : ∀ ⦃a b⦄, r' (f a) (f b) → r a b) :
IsAntichain r' (f '' s) := by
rintro _ ⟨b, hb, rfl⟩ _ ⟨c, hc, rfl⟩ hbc hr
exact hs hb hc (ne_of_apply_ne _ hbc) (h hr) | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | image | null |
preimage (hs : IsAntichain r s) {f : β → α} (hf : Injective f)
(h : ∀ ⦃a b⦄, r' a b → r (f a) (f b)) : IsAntichain r' (f ⁻¹' s) := fun _ hb _ hc hbc hr =>
hs hb hc (hf.ne hbc) <| h hr | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | preimage | null |
_root_.isAntichain_insert :
IsAntichain r (insert a s) ↔ IsAntichain r s ∧ ∀ ⦃b⦄, b ∈ s → a ≠ b → ¬r a b ∧ ¬r b a :=
Set.pairwise_insert | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | _root_.isAntichain_insert | null |
protected insert (hs : IsAntichain r s) (hl : ∀ ⦃b⦄, b ∈ s → a ≠ b → ¬r b a)
(hr : ∀ ⦃b⦄, b ∈ s → a ≠ b → ¬r a b) : IsAntichain r (insert a s) :=
isAntichain_insert.2 ⟨hs, fun _ hb hab => ⟨hr hb hab, hl hb hab⟩⟩ | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | insert | null |
_root_.isAntichain_insert_of_symmetric (hr : Symmetric r) :
IsAntichain r (insert a s) ↔ IsAntichain r s ∧ ∀ ⦃b⦄, b ∈ s → a ≠ b → ¬r a b :=
pairwise_insert_of_symmetric hr.compl | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | _root_.isAntichain_insert_of_symmetric | null |
insert_of_symmetric (hs : IsAntichain r s) (hr : Symmetric r)
(h : ∀ ⦃b⦄, b ∈ s → a ≠ b → ¬r a b) : IsAntichain r (insert a s) :=
(isAntichain_insert_of_symmetric hr).2 ⟨hs, h⟩ | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | insert_of_symmetric | null |
image_relEmbedding (hs : IsAntichain r s) (φ : r ↪r r') : IsAntichain r' (φ '' s) := by
intro b hb b' hb' h₁ h₂
rw [Set.mem_image] at hb hb'
obtain ⟨⟨a, has, rfl⟩, ⟨a', has', rfl⟩⟩ := hb, hb'
exact hs has has' (fun haa' => h₁ (by rw [haa'])) (φ.map_rel_iff.mp h₂) | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | image_relEmbedding | null |
preimage_relEmbedding {t : Set β} (ht : IsAntichain r' t) (φ : r ↪r r') :
IsAntichain r (φ ⁻¹' t) := fun _ ha _s ha' hne hle =>
ht ha ha' (fun h => hne (φ.injective h)) (φ.map_rel_iff.mpr hle) | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | preimage_relEmbedding | null |
image_relIso (hs : IsAntichain r s) (φ : r ≃r r') : IsAntichain r' (φ '' s) :=
hs.image_relEmbedding φ.toRelEmbedding | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | image_relIso | null |
preimage_relIso {t : Set β} (hs : IsAntichain r' t) (φ : r ≃r r') :
IsAntichain r (φ ⁻¹' t) :=
hs.preimage_relEmbedding φ.toRelEmbedding | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | preimage_relIso | null |
image_relEmbedding_iff {φ : r ↪r r'} : IsAntichain r' (φ '' s) ↔ IsAntichain r s :=
⟨fun h => (φ.injective.preimage_image s).subst (h.preimage_relEmbedding φ), fun h =>
h.image_relEmbedding φ⟩ | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | image_relEmbedding_iff | null |
image_relIso_iff {φ : r ≃r r'} : IsAntichain r' (φ '' s) ↔ IsAntichain r s :=
@image_relEmbedding_iff _ _ _ _ _ (φ : r ↪r r') | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | image_relIso_iff | null |
image_embedding [LE α] [LE β] (hs : IsAntichain (· ≤ ·) s) (φ : α ↪o β) :
IsAntichain (· ≤ ·) (φ '' s) :=
image_relEmbedding hs _ | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | image_embedding | null |
preimage_embedding [LE α] [LE β] {t : Set β} (ht : IsAntichain (· ≤ ·) t) (φ : α ↪o β) :
IsAntichain (· ≤ ·) (φ ⁻¹' t) :=
preimage_relEmbedding ht _ | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | preimage_embedding | null |
image_embedding_iff [LE α] [LE β] {φ : α ↪o β} :
IsAntichain (· ≤ ·) (φ '' s) ↔ IsAntichain (· ≤ ·) s :=
image_relEmbedding_iff | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | image_embedding_iff | null |
image_iso [LE α] [LE β] (hs : IsAntichain (· ≤ ·) s) (φ : α ≃o β) :
IsAntichain (· ≤ ·) (φ '' s) :=
image_relEmbedding hs _ | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | image_iso | null |
image_iso_iff [LE α] [LE β] {φ : α ≃o β} :
IsAntichain (· ≤ ·) (φ '' s) ↔ IsAntichain (· ≤ ·) s :=
image_relEmbedding_iff | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | image_iso_iff | null |
preimage_iso [LE α] [LE β] {t : Set β} (ht : IsAntichain (· ≤ ·) t) (φ : α ≃o β) :
IsAntichain (· ≤ ·) (φ ⁻¹' t) :=
preimage_relEmbedding ht _ | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | preimage_iso | null |
preimage_iso_iff [LE α] [LE β] {t : Set β} {φ : α ≃o β} :
IsAntichain (· ≤ ·) (φ ⁻¹' t) ↔ IsAntichain (· ≤ ·) t :=
⟨fun h => (φ.image_preimage t).subst (h.image_iso φ), fun h => h.preimage_iso _⟩ | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | preimage_iso_iff | null |
to_dual [LE α] (hs : IsAntichain (· ≤ ·) s) : @IsAntichain αᵒᵈ (· ≤ ·) s :=
fun _ ha _ hb hab => hs hb ha hab.symm | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | to_dual | null |
to_dual_iff [LE α] : IsAntichain (· ≤ ·) s ↔ @IsAntichain αᵒᵈ (· ≤ ·) s :=
⟨to_dual, to_dual⟩ | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | to_dual_iff | null |
image_compl [BooleanAlgebra α] (hs : IsAntichain (· ≤ ·) s) :
IsAntichain (· ≤ ·) (compl '' s) :=
(hs.image_embedding (OrderIso.compl α).toOrderEmbedding).flip | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | image_compl | null |
preimage_compl [BooleanAlgebra α] (hs : IsAntichain (· ≤ ·) s) :
IsAntichain (· ≤ ·) (compl ⁻¹' s) := fun _ ha _ ha' hne hle =>
hs ha' ha (fun h => hne (compl_inj_iff.mp h.symm)) (compl_le_compl hle) | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | preimage_compl | null |
isAntichain_union :
IsAntichain r (s ∪ t) ↔
IsAntichain r s ∧ IsAntichain r t ∧ ∀ a ∈ s, ∀ b ∈ t, a ≠ b → rᶜ a b ∧ rᶜ b a := by
rw [IsAntichain, IsAntichain, IsAntichain, pairwise_union]
@[deprecated (since := "2025-09-20")]
alias isAntichain_singleton := IsAntichain.singleton | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | isAntichain_union | null |
Set.Subsingleton.isAntichain (hs : s.Subsingleton) (r : α → α → Prop) : IsAntichain r s :=
hs.pairwise _ | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | Set.Subsingleton.isAntichain | null |
subsingleton_of_isChain_of_isAntichain (hs : IsChain r s) (ht : IsAntichain r s) :
s.Subsingleton := by
intro x hx y hy
by_contra! hne
cases hs hx hy hne with
| inl h => exact ht hx hy hne h
| inr h => exact ht hy hx hne.symm h | lemma | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | subsingleton_of_isChain_of_isAntichain | A set which is simultaneously a chain and antichain is subsingleton. |
isChain_and_isAntichain_iff_subsingleton : IsChain r s ∧ IsAntichain r s ↔ s.Subsingleton :=
⟨fun h ↦ subsingleton_of_isChain_of_isAntichain h.1 h.2, fun h ↦ ⟨h.isChain, h.isAntichain _⟩⟩ | lemma | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | isChain_and_isAntichain_iff_subsingleton | null |
inter_subsingleton_of_isChain_of_isAntichain (hs : IsChain r s) (ht : IsAntichain r t) :
(s ∩ t).Subsingleton :=
subsingleton_of_isChain_of_isAntichain (hs.mono (by simp)) (ht.subset (by simp)) | lemma | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | inter_subsingleton_of_isChain_of_isAntichain | The intersection of a chain and an antichain is subsingleton. |
inter_subsingleton_of_isAntichain_of_isChain (hs : IsAntichain r s) (ht : IsChain r t) :
(s ∩ t).Subsingleton :=
inter_comm _ _ ▸ inter_subsingleton_of_isChain_of_isAntichain ht hs | lemma | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | inter_subsingleton_of_isAntichain_of_isChain | The intersection of an antichain and a chain is subsingleton. |
IsAntichain.not_lt (hs : IsAntichain (· ≤ ·) s) (ha : a ∈ s) (hb : b ∈ s) : ¬a < b :=
fun h => hs ha hb h.ne h.le | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | IsAntichain.not_lt | null |
isAntichain_and_least_iff : IsAntichain (· ≤ ·) s ∧ IsLeast s a ↔ s = {a} :=
⟨fun h => eq_singleton_iff_unique_mem.2 ⟨h.2.1, fun _ hb => h.1.eq' hb h.2.1 (h.2.2 hb)⟩, by
rintro rfl
exact ⟨IsAntichain.singleton, isLeast_singleton⟩⟩ | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | isAntichain_and_least_iff | null |
isAntichain_and_greatest_iff : IsAntichain (· ≤ ·) s ∧ IsGreatest s a ↔ s = {a} :=
⟨fun h => eq_singleton_iff_unique_mem.2 ⟨h.2.1, fun _ hb => h.1.eq hb h.2.1 (h.2.2 hb)⟩, by
rintro rfl
exact ⟨IsAntichain.singleton, isGreatest_singleton⟩⟩ | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | isAntichain_and_greatest_iff | null |
IsAntichain.least_iff (hs : IsAntichain (· ≤ ·) s) : IsLeast s a ↔ s = {a} :=
(and_iff_right hs).symm.trans isAntichain_and_least_iff | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | IsAntichain.least_iff | null |
IsAntichain.greatest_iff (hs : IsAntichain (· ≤ ·) s) : IsGreatest s a ↔ s = {a} :=
(and_iff_right hs).symm.trans isAntichain_and_greatest_iff | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | IsAntichain.greatest_iff | null |
IsLeast.antichain_iff (hs : IsLeast s a) : IsAntichain (· ≤ ·) s ↔ s = {a} :=
(and_iff_left hs).symm.trans isAntichain_and_least_iff | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | IsLeast.antichain_iff | null |
IsGreatest.antichain_iff (hs : IsGreatest s a) : IsAntichain (· ≤ ·) s ↔ s = {a} :=
(and_iff_left hs).symm.trans isAntichain_and_greatest_iff | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | IsGreatest.antichain_iff | null |
IsAntichain.bot_mem_iff [OrderBot α] (hs : IsAntichain (· ≤ ·) s) : ⊥ ∈ s ↔ s = {⊥} :=
isLeast_bot_iff.symm.trans hs.least_iff | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | IsAntichain.bot_mem_iff | null |
IsAntichain.top_mem_iff [OrderTop α] (hs : IsAntichain (· ≤ ·) s) : ⊤ ∈ s ↔ s = {⊤} :=
isGreatest_top_iff.symm.trans hs.greatest_iff | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | IsAntichain.top_mem_iff | null |
IsAntichain.minimal_mem_iff (hs : IsAntichain (· ≤ ·) s) : Minimal (· ∈ s) a ↔ a ∈ s :=
⟨fun h ↦ h.prop, fun h ↦ ⟨h, fun _ hys hyx ↦ (hs.eq hys h hyx).symm.le⟩⟩ | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | IsAntichain.minimal_mem_iff | null |
IsAntichain.maximal_mem_iff (hs : IsAntichain (· ≤ ·) s) : Maximal (· ∈ s) a ↔ a ∈ s :=
hs.to_dual.minimal_mem_iff | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | IsAntichain.maximal_mem_iff | null |
IsAntichain.eq_setOf_maximal (ht : IsAntichain (· ≤ ·) t)
(h : ∀ x, Maximal (· ∈ s) x → x ∈ t) (hs : ∀ a ∈ t, ∃ b, b ≤ a ∧ Maximal (· ∈ s) b) :
{x | Maximal (· ∈ s) x} = t := by
refine Set.ext fun x ↦ ⟨h _, fun hx ↦ ?_⟩
obtain ⟨y, hyx, hy⟩ := hs x hx
rwa [← ht.eq (h y hy) hx hyx] | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | IsAntichain.eq_setOf_maximal | If `t` is an antichain shadowing and including the set of maximal elements of `s`,
then `t` *is* the set of maximal elements of `s`. |
IsAntichain.eq_setOf_minimal (ht : IsAntichain (· ≤ ·) t)
(h : ∀ x, Minimal (· ∈ s) x → x ∈ t) (hs : ∀ a ∈ t, ∃ b, a ≤ b ∧ Minimal (· ∈ s) b) :
{x | Minimal (· ∈ s) x} = t :=
ht.to_dual.eq_setOf_maximal h hs | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | IsAntichain.eq_setOf_minimal | If `t` is an antichain shadowed by and including the set of minimal elements of `s`,
then `t` *is* the set of minimal elements of `s`. |
IsAntichain.of_strictMonoOn_antitoneOn (hf : StrictMonoOn f s) (hf' : AntitoneOn f s) :
IsAntichain (· ≤ ·) s :=
fun _a ha _b hb hab' hab ↦ (hf ha hb <| hab.lt_of_ne hab').not_ge (hf' ha hb hab) | lemma | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | IsAntichain.of_strictMonoOn_antitoneOn | null |
IsAntichain.of_monotoneOn_strictAntiOn (hf : MonotoneOn f s) (hf' : StrictAntiOn f s) :
IsAntichain (· ≤ ·) s :=
fun _a ha _b hb hab' hab ↦ (hf ha hb hab).not_gt (hf' ha hb <| hab.lt_of_ne hab') | lemma | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | IsAntichain.of_monotoneOn_strictAntiOn | null |
isAntichain_iff_forall_not_lt :
IsAntichain (· ≤ ·) s ↔ ∀ ⦃a⦄, a ∈ s → ∀ ⦃b⦄, b ∈ s → ¬a < b :=
⟨fun hs _ ha _ => hs.not_lt ha, fun hs _ ha _ hb h h' => hs ha hb <| h'.lt_of_ne h⟩ | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | isAntichain_iff_forall_not_lt | null |
setOf_maximal_antichain (P : α → Prop) : IsAntichain (· ≤ ·) {x | Maximal P x} :=
fun _ hx _ ⟨hy, _⟩ hne hle ↦ hne (hle.antisymm <| hx.2 hy hle) | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | setOf_maximal_antichain | null |
setOf_minimal_antichain (P : α → Prop) : IsAntichain (· ≤ ·) {x | Minimal P x} :=
(setOf_maximal_antichain (α := αᵒᵈ) P).swap | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | setOf_minimal_antichain | null |
IsStrongAntichain (r : α → α → Prop) (s : Set α) : Prop :=
s.Pairwise fun a b => ∀ c, ¬r a c ∨ ¬r b c | def | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | IsStrongAntichain | A strong (upward) antichain is a set such that no two distinct elements are related to a common
element. |
protected subset (hs : IsStrongAntichain r s) (h : t ⊆ s) : IsStrongAntichain r t :=
hs.mono h | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | subset | null |
mono (hs : IsStrongAntichain r₁ s) (h : r₂ ≤ r₁) : IsStrongAntichain r₂ s :=
hs.mono' fun _ _ hab c => (hab c).imp (compl_le_compl h _ _) (compl_le_compl h _ _) | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | mono | null |
eq (hs : IsStrongAntichain r s) {a b c : α} (ha : a ∈ s) (hb : b ∈ s) (hac : r a c)
(hbc : r b c) : a = b :=
(Set.Pairwise.eq hs ha hb) fun h =>
False.elim <| (h c).elim (not_not_intro hac) (not_not_intro hbc) | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | eq | null |
protected isAntichain [IsRefl α r] (h : IsStrongAntichain r s) : IsAntichain r s :=
h.imp fun _ b hab => (hab b).resolve_right (not_not_intro <| refl _) | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | isAntichain | null |
protected subsingleton [IsDirected α r] (h : IsStrongAntichain r s) : s.Subsingleton :=
fun a ha b hb =>
let ⟨_, hac, hbc⟩ := directed_of r a b
h.eq ha hb hac hbc | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | subsingleton | null |
protected flip [IsSymm α r] (hs : IsStrongAntichain r s) : IsStrongAntichain (flip r) s :=
fun _ ha _ hb h c => (hs ha hb h c).imp (mt <| symm_of r) (mt <| symm_of r) | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | flip | null |
swap [IsSymm α r] (hs : IsStrongAntichain r s) : IsStrongAntichain (swap r) s :=
hs.flip | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | swap | null |
image (hs : IsStrongAntichain r s) {f : α → β} (hf : Surjective f)
(h : ∀ a b, r' (f a) (f b) → r a b) : IsStrongAntichain r' (f '' s) := by
rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ hab c
obtain ⟨c, rfl⟩ := hf c
exact (hs ha hb (ne_of_apply_ne _ hab) _).imp (mt <| h _ _) (mt <| h _ _) | theorem | Order | [
"Mathlib.Order.Bounds.Basic",
"Mathlib.Order.Preorder.Chain"
] | Mathlib/Order/Antichain.lean | image | null |
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