query_key stringlengths 9 102 | query_node_id int64 11 3.82k | repo stringclasses 35
values | query_text stringlengths 0 1.28k | positives listlengths 1 13 | positive_ids listlengths 1 13 |
|---|---|---|---|---|---|
FLT_ch10/MeasureTheory.addEquivAddHaarChar_eq | 3,578 | FLT_ch10 | \(d_A(\phi )\) is independent of choice of regular Haar measure. | [
"FLT_ch10/MeasureTheory.addEquivAddHaarChar"
] | [
818
] |
brownian-motion/lem:komlos_aux | 180 | brownian-motion | Let \(H\) be a Hilbert space and \((f_n)_{n\in \mathbb {N}}\) a bounded sequence in \(H\). Then there exist functions \(g_n\in convex(f_n,f_{n+1},\cdots )\) such that \((g_n)_{n\in \mathbb {N}}\) converges in \(H\). | [
"brownian-motion/lem:komlos_convex"
] | [
922
] |
ABC-Exceptions/thm:BdExceptionalBound | 2,468 | ABC-Exceptions | There exists \(\varepsilon {\gt}0\) such that for all \(\mathbf{c}\in \mathbb {Z}^3\) and \(\mathbf{X},\mathbf{Y},\mathbf{Z}\in \mathbb {R}_{{\gt}0}^{d}\). we have \begin{align} B_d({\bf c, X,Y,Z}) \ll X^{0.66-\varepsilon }. \end{align} | [
"ABC-Exceptions/thm:ABCExponentNu"
] | [
1165
] |
radii-polynomial/def:sq_map | 365 | radii-polynomial | The squaring map \(\text{sq}: \ell ^1_\nu \rightarrow \ell ^1_\nu \) is \(\text{sq}(a) = a \star a\). | [
"radii-polynomial/def:l1Weighted",
"radii-polynomial/def:CauchyProduct"
] | [
749,
3226
] |
toric/0-diag-hom | 72 | toric | Let \(R\) be a domain. The functor \(G \rightsquigarrow \operatorname{Spec}R[G]\) from the category of groups to the category of group schemes over \(\operatorname{Spec}R\) is a group isomorphism on hom sets. | [
"toric/0-full-faithful-diag",
"toric/0-full-faithful-grp-hom"
] | [
513,
1058
] |
ABC-Exceptions/thm:BombierSchmidtforThueEqs | 3,601 | ABC-Exceptions | Given integers \(n,a_1,a_2\neq 0\) and \(p\ge 3\), there are at most \(O(p^{1+\omega (|n|)})\) many solutions \((x_1,x_2)\in \mathbb {Z}^2\) such that \(|x_i|\le X_i\) and \begin{align*} n = a_1 x_1^p + a_2 x_2^p. \end{align*} | [
"ABC-Exceptions/def:omega"
] | [
265
] |
ClassFieldTheory/lem:herbrand local units | 50 | ClassFieldTheory | If \(l/k\) is a cyclic extension then \(h(l/k, {\mathcal O}_l^\times ) = 1\). | [
"ClassFieldTheory/lem:herbrand ses",
"ClassFieldTheory/lem:serre_cor",
"ClassFieldTheory/lem:herbrand finite"
] | [
1186,
2821,
2394
] |
FormalBook/cayley_formala_proof4 | 391 | FormalBook | There are \(n^{n - 2}\) different labeled trees on \(n\) nodes. | [
"FormalBook/double_counting"
] | [
2828
] |
carleson/lower-oscillation-bound | 3,015 | carleson | For all \(J \in \mathcal{J}'\), we have that \[ d_{B(J)}({\mathcal{Q}}({\mathfrak u}_1), {\mathcal{Q}}({\mathfrak u}_2)) \ge 2^{-201a^3} 2^{Zn/2}\, . \] | [
"carleson/overlap-implies-distance"
] | [
578
] |
toric/1-2-6-face-face | 2,695 | toric | A face of a face of a polyhedral cone \(\sigma \) is again a face of \(\sigma \). | [
"toric/1-2-5-face",
"toric/1-2-1-polyhedral-cone"
] | [
1233,
2279
] |
PrimeNumberTheoremAnd/fks2-lemma-20a | 2,252 | PrimeNumberTheoremAnd | The function \(\operatorname {Li}(x) - \frac{x}{\log x}\) is strictly increasing for \(x {\gt} 6.58\). | [
"PrimeNumberTheoremAnd/li-def"
] | [
3275
] |
LeanAPAP/flat-on-bohr | 1,944 | LeanAPAP | There is a constant \(c{\gt}0\) such that the following holds. Let \({\epsilon ,\delta \in (0,1)}\) and \(p,k\geq 1\) be integers such that \((k,\left\lvert G\right\rvert )=1\). For any \(A\subseteq G\) with density \(\alpha \) there is a regular Bohr set \(B\) with \[ d=\operatorname{rank}(B) =O_{\epsilon }\left(\math... | [
"LeanAPAP/prop-it",
"LeanAPAP/bourgain-trick"
] | [
3324,
3681
] |
equational_theories/non_imp_3994_3588_thm | 1,161 | equational_theories | There exists a magma which satisfies E3994 and not E3588. | [
"equational_theories/models-def",
"equational_theories/eq3994",
"equational_theories/eq3588"
] | [
1745,
3165,
3511
] |
brownian-motion/lem:martingale_integral_of_forall_eq_zero | 1,506 | brownian-motion | Let \(X \in L^2(M)\) such that \(\langle X, V\rangle = 0\) for any simple process \(V\). Let \(A_t = \int _0^t X_s \: d\langle M \rangle _s\). Then \(A_t\) is a martingale. | [
"brownian-motion/def:L2M",
"brownian-motion/lem:martingale_iff_integral_elemStochIntegral_eq_zero"
] | [
300,
3030
] |
polytopes-in-LEAN/def:convex_hull | 2,525 | polytopes-in-LEAN | The convex hull of a set \(A\) is the intersection of all convex subsets containing \(A\). | [
"polytopes-in-LEAN/def:is_convex"
] | [
3198
] |
ClassFieldTheory/lem:local image inertia | 2,859 | ClassFieldTheory | Let \(l/k\) be a finite abelian extension of local fields and let \(I \subseteq \mathrm{Gal}(l/k)\) be the inertia subgroup. Then the image of \(I\) in \(k^\times / N(l^\times )\) is the subgroup \({\mathcal O}_k^\times / N({\mathcal O}_l^\times )\). | [
"ClassFieldTheory/lem:subgroup compatibility",
"ClassFieldTheory/lem:local unramified norms"
] | [
3664,
1429
] |
carleson/correlation-distant-tree-parts | 3,099 | carleson | We have for all \({\mathfrak u}_1 \ne {\mathfrak u}_2 \in {\mathfrak U}\) with \({\mathcal{I}}({\mathfrak u}_1) \subset {\mathcal{I}}({\mathfrak u}_2)\) and all bounded \(g_1, g_2\) with bounded support \begin{equation} \label{eq-lhs-big-sep-tree} \left| \int _X \sum _{{\mathfrak p}_1 \in {\mathfrak T}({\mathfrak u}_1)... | [
"carleson/Lipschitz-partition-unity",
"carleson/Holder-van-der-Corput",
"carleson/lower-oscillation-bound",
"carleson/Holder-correlation-tree"
] | [
522,
943,
3015,
1978
] |
pfr/neg-ent | 2,537 | pfr | If \(X\) is \(G\)-valued, then \(\mathbb {H}[-X]=\mathbb {H}[X]\). | [
"pfr/relabeled-entropy"
] | [
3480
] |
PrimeNumberTheoremAnd/prop:applem | 2,796 | PrimeNumberTheoremAnd | Let \(s = \sigma + i \tau \), \(\sigma \geq 0\), \(\tau \in \mathbb {R}\). Let \(a,b\in \mathbb {Z} + \frac{1}{2}\), \(b{\gt}a{\gt}\frac{|\tau |}{2\pi }\). Write \(\vartheta = \frac{\tau }{2\pi a}\). Then, for any integer \(n\geq 1\), \[ \begin{aligned} \int _a^b t^{-s} \cos 2\pi n t\, dt & = \left. \left(\frac{(-1)^n ... | [
"PrimeNumberTheoremAnd/lem:aachfour",
"PrimeNumberTheoremAnd/lem:aachcanc"
] | [
2880,
3292
] |
PrimeNumberTheoremAnd/Dusart_prop_5_11a | 2,170 | PrimeNumberTheoremAnd | We have for \(p_k \geq 10^{11}\), \[ \vartheta (p_k) \geq k \log k + \log \log k - 1 + \frac{\log \log k - 2.050735}{\log k}. \] | [
"PrimeNumberTheoremAnd/theta-def"
] | [
108
] |
FormalBook/four_proofs_euler_series | 3,563 | FormalBook | Collecting the proofs from the chapter. | [
"FormalBook/euler_series_2",
"FormalBook/euler_series_4",
"FormalBook/euler_series",
"FormalBook/euler_series_3"
] | [
2444,
2181,
593,
2282
] |
radii-polynomial/lem:norm_pow_le | 2,974 | radii-polynomial | For \(a \in \ell ^1_\nu \) and \(n \in \mathbb {N}\): \begin{equation*} \| a^n\| _{1,\nu } \leq \| a\| _{1,\nu }^n. \end{equation*} | [
"radii-polynomial/thm:l1Weighted_NormedRing"
] | [
364
] |
toric/0-spec-hopf | 1,084 | toric | Spec is a contravariant functor from the category of \(R\)-Hopf algebras to the category of group schemes over \(\operatorname{Spec}_R\). | [
"toric/0-full-faithful-spec-alg"
] | [
2178
] |
PrimeNumberTheoremAnd/tendsto_rpow_atTop_nhds_zero_of_norm_gt_one | 1,249 | PrimeNumberTheoremAnd | Let \(x{\gt}1\). Then \[ \lim _{\sigma \to -\infty }x^\sigma =0. \] | [
"PrimeNumberTheoremAnd/tendsto_rpow_atTop_nhds_zero_of_norm_lt_one"
] | [
2656
] |
infinite-primes-via-log/thm:infinite_primes | 2,592 | infinite-primes-via-log | The set \(\mathbb {P} = \big\{ p \in \mathbb {N} \ \mid \ p \ \mathrm{prime} \big\} \) is infinite. | [
"infinite-primes-via-log/thm:primeCountingReal_unbounded"
] | [
3319
] |
brownian-motion/lem:elemStochIntegral_assoc | 2,585 | brownian-motion | \(V \bullet _{\mathbb {R}} (W \bullet _{\mathbb {R}} X) = (V \times W) \bullet _{\mathbb {R}} X\) for real-valued simple processes \(V, W\) and stochastic process \(X\). | [
"brownian-motion/cor:elemStochIntegral_assoc_real_bilin"
] | [
3033
] |
pfr/rho-invariant | 956 | pfr | For any \(s \in G\), \(\rho (X+s) = \rho (X)\). | [
"pfr/kl-div-inj",
"pfr/rho-def"
] | [
492,
102
] |
ClassFieldTheory/def:reciprocity iso | 2,531 | ClassFieldTheory | The theorem implies that we have isomorphisms for all \(n\in {\mathbb Z}\) (which depend on \(\sigma \)): \[ H^{n}_{\mathrm{Tate}}(G,\mathrm{aug}(R,G)) \cong H^{n+1}_{\mathrm{Tate}}(G,M) . \] | [
"ClassFieldTheory/thm:trivial cohomology implies trivial Tate",
"ClassFieldTheory/lem:Tate -1 aug",
"ClassFieldTheory/thm:splitting module trivial"
] | [
2449,
709,
538
] |
equational_theories/eq3994 | 3,165 | equational_theories | Equation 3994 is the law \(0 \diamond 1 \simeq (2 \diamond (0 \diamond 1)) \diamond 2\) (or the equation \(x \diamond y = (z \diamond (x \diamond y)) \diamond z\)). | [
"equational_theories/magma-def"
] | [
3408
] |
carleson/dens2-antichain | 1,394 | carleson | We have that \begin{equation} \label{eqttt9} \left|\int \overline{g(x)} \sum _{{\mathfrak p}\in \mathfrak {A}} T_{{\mathfrak p}} f(x)\, d\mu (x)\right|\le 2^{103a^3}({q}-1)^{-1} \operatorname{\operatorname {dens}}_2(\mathfrak {A})^{\frac1{\tilde{q}}-\frac12} \| f\| _2\| g\| _2\, . \end{equation} | [
"carleson/Hardy-Littlewood",
"carleson/maximal-bound-antichain"
] | [
1224,
1203
] |
toric/1-2-18-dim-aff-tor-var-rat-polyhedral-cone | 240 | toric | \[ \dim U_\sigma = \dim N \iff \text{ the torus of $U_\sigma $ is } T_N = N \otimes _[{\mathbb Z}] {\mathbb C}^* \iff \sigma \text{ is salient}. \] | [
"toric/1-1-14-spec-aff-mon-alg",
"toric/1-2-17-gordan-lemma",
"toric/1-2-12-salient-cone-tfae",
"toric/1-2-18-aff-tor-var-rat-polyhedral-cone"
] | [
1199,
2505,
1574,
372
] |
equational_theories/eq374794 | 3,448 | equational_theories | Equation 374794 is the law \(0 \simeq (((1 \diamond 1) \diamond 1) \diamond 0) \diamond ((1 \diamond 1) \diamond 2)\) (or the equation \(x = (((y \diamond y) \diamond y) \diamond x) \diamond ((y \diamond y) \diamond z)\)). | [
"equational_theories/magma-def"
] | [
3408
] |
equational_theories/claim-4 | 1,916 | equational_theories | If \(a \to b \to c \to d \to e \to a\) is a 5-cycle in the directed graph, and \(a \to b \to c\) and \(c \to d \to e\) are good paths, then \(b \to c \to d\) is also good. | [
"equational_theories/graph-dual"
] | [
3827
] |
PrimeNumberTheoremAnd/LogDerivZetaFinalBound | 602 | PrimeNumberTheoremAnd | Let \(t\in \mathbb {R}\) with \(|t|\geq 2\) and \(0 {\lt} r' {\lt} r {\lt} R' {\lt} R{\lt}1\). If \(f(z)=\zeta (z+3/2+it)\), then for all \(z\in \overline{\mathbb {D}_R'}\setminus \mathcal{K}_f(R')\) we have that \[ \left|\frac{f'}{f}(z)-\sum _{\rho \in \mathcal{K}_f(R')}\frac{m_f(\rho )}{z-\rho }\right| \ll \left(\fra... | [
"PrimeNumberTheoremAnd/FinalBound",
"PrimeNumberTheoremAnd/GlobalBound",
"PrimeNumberTheoremAnd/ZetaFixedLowerBound"
] | [
3443,
3175,
994
] |
Sphere-Packing-Lean/prop:ineqB | 2,218 | Sphere-Packing-Lean | Consider the function \(B:(0,\infty )\to \mathbb {C}\) defined as \begin{equation} \label{eqn:defB} B(t) := -t^2\phi _0(i/t)+\frac{36}{\pi ^2}\, \psi _I(it) \end{equation} | [
"Sphere-Packing-Lean/lemma:ineqABnew-equiv",
"Sphere-Packing-Lean/cor:ineqBnew"
] | [
3036,
769
] |
M2/lem:chech_long_exact_sequence | 589 | M2 | Let \(0\to \mathcal{F}^{\bullet }\to \mathcal{G}^{\bullet }\to \mathcal{H}^{\bullet }\to 0\) be a short exact sequence of complex of \(\mathcal{K}\)-presheaves. Then there is a long exact seuquence in čech cohomology: \[ \ldots \to \check{H}^k(X;\mathcal{F}^{\bullet })\to \check{H}^k(X;\mathcal{G}^{\bullet })\to \check... | [
"M2/def:chech_of_compelx_of_k_prsh"
] | [
1631
] |
exchangeability/lem:directing_measure_isProb_L2 | 2,676 | exchangeability | The Stieltjes measure constructed from the limiting CDF is a probability measure a.e. | [
"exchangeability/lem:cdf_mono",
"exchangeability/lem:cdf_bounds"
] | [
3655,
2831
] |
pfr/multidist-chain-rule-cond | 2,685 | pfr | Let \(\pi \colon G \to H\) be a homomorphism of abelian groups. Let \(I\) be a finite index set and let \(X_{[m]}\) be a tuple of \(G\)-valued random variables. Let \(Y_{[m]}\) be another tuple of random variables (not necessarily \(G\)-valued). Suppose that the pairs \((X_i, Y_i)\) are jointly independent of one anoth... | [
"pfr/multidist-chain-rule"
] | [
1335
] |
FLT_ch5/Hurwitz.norm_one | 1,609 | FLT_ch5 | The norm of \(1\) is \(1\). | [
"FLT_ch5/Hurwitz.norm"
] | [
1708
] |
GibbsMeasure/lem:lebesgue-cond-exp-char | 2,374 | GibbsMeasure | If \(f : X \to [0, \infty ]\) is a \(\mathcal X\)-measurable function, then \(\mu [f | \mathcal B]\) is the \(\mu \)-ae unique \(\mathcal B\)-measurable function \(X \to [0, \infty ]\) such that \[ \int _B \mu [f | \mathcal B]\ \partial \mu = \int _B f\ \partial \mu \] | [
"GibbsMeasure/def:lebesgue-cond-exp"
] | [
3470
] |
brownian-motion/def:Martingale | 1,156 | brownian-motion | Let \(\mathcal{F}\) be a filtration on a measurable space \(\Omega \) with measure \(P\) indexed by \(T\). A family of functions \(M : T \to \Omega \to E\) is a martingale with respect to a filtration \(\mathcal{F}\) if \(M\) is adapted with respect to \(\mathcal{F}\) and for all \(i \le j\), \(P[M_j \mid \mathcal{F}_i... | [
"brownian-motion/def:adapted"
] | [
158
] |
equational_theories/3744_implies_3722_381 | 84 | equational_theories | E3744 \((x \diamond y = (x \diamond z) \diamond (w \diamond y))\) implies E3722 \((x \diamond y = (x \diamond y) \diamond (x \diamond y))\) and E381 \((x \diamond y = (x \diamond z) \diamond y)\). | [
"equational_theories/eq3744",
"equational_theories/eq381",
"equational_theories/eq3722"
] | [
2338,
1711,
2753
] |
equational_theories/duality | 718 | equational_theories | The law \(w \simeq w'\) implies \(w''\simeq w'''\), if and only if \(w^{\mathrm{op}} \simeq (w')^{\mathrm{op}}\) implies \(w''^{\mathrm{op}} \simeq (w''')^{\mathrm{op}}\). | [
"equational_theories/pre-order"
] | [
1076
] |
brownian-motion/lem:predictable_quadraticVariation | 2,396 | brownian-motion | The quadratic variation \(\langle M \rangle \) of a local martingale \(M\) is a predictable process. | [
"brownian-motion/def:quadraticVariation"
] | [
1883
] |
M2/def:pushforward_of_K-sheaf | 29 | M2 | If \(\mathcal{F}\) is a pre-\(\mathcal{K}\)-sheaf over \(X\), then the rule \(K\mapsto \mathcal{F}(f^{-1}(K))\) defines a pre-\(\mathcal{K}\)-sheaf over \(Y\). The functor obtained is denoted \(f_*\) and named the pushforward by \(f\). \(f_*\) send \(\mathcal{K}\)-sheaves over \(X\) into \(\mathcal{K}\)-sheaves over \(... | [
"M2/def:pushforward_of_sheaf",
"M2/def:K_sheaves",
"M2/lem:basis_of_open_of_fm1K"
] | [
2876,
293,
144
] |
ClassFieldTheory/lem:local H2 upper bound | 3,813 | ClassFieldTheory | Let \(l/k\) be a Galois extension of local fields. Then \(|H^2(l/k,l^\times )| \le [l:k]\). | [
"ClassFieldTheory/cor:cohomology sub Sylow",
"ClassFieldTheory/cor:cohomology G-torsion",
"ClassFieldTheory/lem:local H2 l*",
"ClassFieldTheory/thm:inflation restriction sequence"
] | [
32,
1614,
2556,
1533
] |
radii-polynomial/cor:sqrt2_unique | 2,515 | radii-polynomial | There exists a unique \(\tilde{x} \in \overline{B}_{3/20}(13/10)\) with \(\tilde{x}^2 = 2\). | [
"radii-polynomial/ex:sqrt2"
] | [
2035
] |
equational_theories/477-confl | 1,344 | equational_theories | E477 \((x = y \diamond (x \diamond (y \diamond (y \diamond y))))\) is confluent. | [
"equational_theories/eq477",
"equational_theories/confluent-theory",
"equational_theories/477-lemma"
] | [
770,
2924,
3401
] |
radii-polynomial/thm:simple_radii_poly_ef | 2,634 | radii-polynomial | Given \(f: E \rightarrow F\) Fréchet differentiable and injective \(A: F \rightarrow E\) satisfying \(\| A(f(\bar{x}))\| \leq Y_0\), \(\| I_E - A \circ \text{Df}(\bar{x})\| \leq Z_0\), and \(\| A \circ [\text{Df}(c) - \text{Df}(\bar{x})]\| \leq Z_2(r) \cdot r\) for \(c \in \overline{B}_r(\bar{x})\). If \(p(r_0) = Z_2(r... | [
"radii-polynomial/lem:radii_poly_alt_form",
"radii-polynomial/lem:newton_Y_bound",
"radii-polynomial/thm:general_fixed_point",
"radii-polynomial/prop:fixed_zero_ef",
"radii-polynomial/lem:newton_deriv_bound_simple"
] | [
2580,
14,
2512,
1419,
2681
] |
brownian-motion/lem:Submartingale.locally_classD | 2,092 | brownian-motion | Every cadlag submartingale is locally of class D. | [
"brownian-motion/lem:Submartingale.locallyIntegrableSup",
"brownian-motion/lem:locally_classD_iff_locallyIntegrableSup"
] | [
1695,
3213
] |
brownian-motion/lem:uniformIntegrableNorm | 3,752 | brownian-motion | If \((X_t)_{t \in T}\) is a family of uniformly integrable random variables, then so is \((\| X_t\| )_{t \in T}\). | [
"brownian-motion/lem:uniformIntegrableDominated"
] | [
2786
] |
radii-polynomial/lem:explicit_inverse | 3,613 | radii-polynomial | If \(\| I_E - B\| {\lt} 1\) for \(B: E \rightarrow E\), then there exists \(B^{-1}: E \rightarrow E\) such that \begin{equation*} B \circ B^{-1} = I_E \quad \text{and} \quad B^{-1} \circ B = I_E. \end{equation*} | [
"radii-polynomial/thm:neumann_series"
] | [
807
] |
FLT_ch3/hardly_ramified_spreads_out | 1,437 | FLT_ch3 | If \(\ell \geq 3\) is prime, \(K\) is a finite extension of \(\mathbb {Q}_\ell \) with integers \(\mathcal{O}\) and if \(\rho :\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \operatorname{GL}_2(\mathcal{O})\) is a hardly ramified representation whose reduction is irreducible, then there exists a number field... | [
"FLT_ch3/hardly_ramified"
] | [
450
] |
Sphere-Packing-Lean/lemma:disc-E4E6 | 3,510 | Sphere-Packing-Lean | We have \begin{equation} \Delta (z) = (E_4^3-E_6^2)/1728. \end{equation} | [
"Sphere-Packing-Lean/def:disc-definition"
] | [
597
] |
Sphere-Packing-Lean/lemma:Ek-is-modular-form | 110 | Sphere-Packing-Lean | For all \(k\), \(E_k\in M_k(\Gamma _1)\). Especially, we have \begin{equation} \label{eqn:Ek-trans-S} E_k \left(-\frac{1}{z}\right) = z^k E_k(z). \end{equation} | [
"Sphere-Packing-Lean/def:Mk",
"Sphere-Packing-Lean/def:Ek"
] | [
1488,
776
] |
radii-polynomial/thm:CauchyProduct_mul_smul | 486 | radii-polynomial | Scalars pull out on the right: \(a \star (c \cdot b) = c \cdot (a \star b)\). Requires commutativity of the coefficient ring. | [
"radii-polynomial/thm:CauchyProduct_comm"
] | [
993
] |
seymour/MatrixSum3.IsCanonicalSigning.Al_D_isTotallyUnimodular | 1,386 | seymour | Let \(B''\) be from Definition 53. Let \(c_{0}'' = B'' (X_{r}, y_{0})\), \(c_{1}'' = B'' (X_{r}, y_{1})\), and \(c_{2}'' = c_{0}'' - c_{1}''\). Similarly, let \(d_{0}'' = B'' (x_{0}, Y_{\ell })\), \(d_{1}'' = B'' (x_{1}, Y_{\ell })\), and \(d_{2}'' = d_{0}'' - d_{1}''\). Then the following statements hold. For eve... | [
"seymour/MatrixSum3.toCanonicalSigning",
"seymour/MatrixSum3.HasTuBr.dddAl_isTotallyUnimodular"
] | [
2359,
440
] |
PrimeNumberTheoremAnd/I3Bound | 505 | PrimeNumberTheoremAnd | Assuming a bound of the form of Lemma 3.4.28 we have that \[ \left|I_{3}(\nu , \epsilon , X, T)\right| \ll \frac{X}{\epsilon }\, X^{-\frac{A}{(\log T)^9}} . \] | [
"PrimeNumberTheoremAnd/log_pow_over_xsq_integral_bounded",
"PrimeNumberTheoremAnd/MellinOfSmooth1b",
"PrimeNumberTheoremAnd/I3"
] | [
3195,
2129,
748
] |
seymour/MatrixSum3.IsCanonicalSigning.isTotallyUnimodular | 409 | seymour | \(B''\) from Definition 53 is TU. | [
"seymour/MatrixLikeSum3.isTotallyUnimodular",
"seymour/MatrixSum3.IsCanonicalSigning.toMatrixLikeSum3"
] | [
666,
2457
] |
carleson/Hilbert-kernel-bound | 358 | carleson | For \(x,y\in {\mathbb {R}}\) with \(x\neq y\) we have \begin{equation} \label{eqcarl30} |\kappa (x-y)|\le 2^2(2|x-y|)^{-1}\, . \end{equation} | [
"carleson/lower-secant-bound"
] | [
744
] |
pfr/first-cond | 1,875 | pfr | We have \begin{align*} & d[X_1|X_1+\tilde X_2; X_2|X_2+\tilde X_1] \\ & \qquad \quad \geq k - \eta (d[X^0_1; X_1 | X_1 + \tilde X_2] - d[X^0_1; X_1]) \\ & \qquad \qquad \qquad \qquad - \eta (d[X^0_2; X_2 | X_2 + \tilde X_1] - d[X^0_2; X_2]). \end{align*} | [
"pfr/cond-distance-lower"
] | [
2589
] |
FormalBook/book.irrational.Theorem_2 | 1,407 | FormalBook | \(\pi ^2\) is irrational. | [
"FormalBook/lem_aux_ii",
"FormalBook/lem_aux_iii"
] | [
335,
3263
] |
pfr/first-useful | 758 | pfr | Let \(X, Y, Z\) be random variables taking values in some abelian group of characteristic \(2\), and with \(Y, Z\) independent. Then we have \begin{align} \nonumber d[X ; Y + Z] -d[X ; Y] & \leq \tfrac {1}{2} (\mathbb {H}[Y+ Z] - \mathbb {H}[Y]) \\ & = \tfrac {1}{2} d[Y; Z] + \tfrac {1}{4} \mathbb {H}[Z] - \tfrac {1}{4... | [
"pfr/kv",
"pfr/cond-dist-fact"
] | [
3327,
3450
] |
exchangeability/lem:tail_factorization | 3,021 | exchangeability | The tail \(\sigma \)-algebra factorization follows from the finite-level factorization via reverse martingale convergence. | [
"exchangeability/lem:finite_level_factorization",
"exchangeability/lem:tailSigma_le_futureFiltration"
] | [
1671,
2562
] |
brownian-motion/lem:predictableSet_elementaryPredictableSet | 1,872 | brownian-motion | An elementary predictable set is measurable with respect to the predictable \(\sigma \)-algebra. | [
"brownian-motion/lem:predictable_Ioc_prod",
"brownian-motion/def:elementaryPredictableSet"
] | [
1274,
2692
] |
carleson/third-exception | 1,424 | carleson | We have \begin{equation} \mu (G_3)\le 2^{-4} \mu (G)\, . \end{equation} | [
"carleson/boundary-exception",
"carleson/top-tiles",
"carleson/tree-count"
] | [
683,
1583,
413
] |
PrimeNumberTheoremAnd/Ramanujan-Soldner-constant | 3,507 | PrimeNumberTheoremAnd | \(\operatorname {li}(2) = 1.0451\dots \). | [
"PrimeNumberTheoremAnd/li-def"
] | [
3275
] |
brownian-motion/lem:sq_norm_elemStochIntegral | 612 | brownian-motion | For \(V \in \mathcal{E}\) and \(M \in \mathcal{M}^2\), then \(V \bullet M \in \mathcal{M}^2\) (by Lemma 13.23) and \begin{align*} \Vert V \bullet M \Vert _{\mathcal{M}^2}^2 & = \Vert V \Vert _{L^2(M)}^2 \: . \end{align*} | [
"brownian-motion/def:L2M",
"brownian-motion/lem:L2Predictable.sq_norm_simpleProcess",
"brownian-motion/lem:isSquareIntegrable_elemStochIntegralBilin"
] | [
300,
2410,
3647
] |
carleson/tile-uncertainty | 2,960 | carleson | Let \({\mathfrak p}_1, {\mathfrak p}_2\in {\mathfrak P}\) with \(B({\mathrm{c}}({\mathfrak p}_1),5D^{{\mathrm{s}}({\mathfrak p}_1)}) \cap B({\mathrm{c}}({\mathfrak p}_2),5D^{{\mathrm{s}}({\mathfrak p}_2)}) \ne \emptyset \) and \({\mathrm{s}}({{\mathfrak p}_1})\leq {\mathrm{s}}({{\mathfrak p}_2})\). For each \(x_1\in E(... | [
"carleson/monotone-cube-metrics",
"carleson/tile-range-support"
] | [
1590,
1497
] |
GibbsMeasure/lem:gibbs-meas-tfae | 1,107 | GibbsMeasure | Let \(\gamma \) be a proper specification with parameter set \(S\) and state space \((E, \mathcal{E})\), and let \(\nu \in \mathfrak {P}(E^S, \mathcal{E}^S)\). TFAE: \(\nu \in \mathcal{G}(\gamma )\). \(\gamma _\Lambda \circ _m\nu = \nu \) for all \(\Lambda \in \operatorname{Finset}(S)\). \(\gamma _\Lambda \circ _m\nu =... | [
"GibbsMeasure/lem:cond-exp-proper-ker-lintegral",
"GibbsMeasure/def:proper-spec",
"GibbsMeasure/def:gibbs-meas"
] | [
1298,
3545,
1460
] |
PrimeNumberTheoremAnd/lem:aachcanc | 3,292 | PrimeNumberTheoremAnd | Let \(s = \sigma + i \tau \), \(\sigma ,\tau \in \mathbb {R}\). Let \(n\in \mathbb {Z}_{{\gt}0}\). Let \(a,b\in \mathbb {Z} + \frac{1}{2}\), \(b{\gt}a{\gt}\frac{|\tau |}{2\pi n}\). Write \(\varphi _\nu (t) = \nu t - \frac{\tau }{2\pi } \log t\). Then \[ \frac{1}{2} \sum _{\nu = \pm n} \left. \frac{t^{-\sigma } e(\varph... | [
"PrimeNumberTheoremAnd/e-def"
] | [
2050
] |
PrimeNumberTheoremAnd/ZetaLowerBound3 | 2,070 | PrimeNumberTheoremAnd | There exists a \(c{\gt}0\) such that for all \(1 {\lt} \sigma {\lt}= 2\) and \(3 {\lt} |t|\), \[ c \frac{(\sigma -1)^{3/4}}{(\log |t|)^{1/4}} \le |\zeta (\sigma + tI)|. \] | [
"PrimeNumberTheoremAnd/ZetaUpperBnd",
"PrimeNumberTheoremAnd/ZetaNear1BndExact"
] | [
1691,
3762
] |
LeanAPAP/mzi_complex | 1,537 | LeanAPAP | Let \(m\geq 1\). If \(f:G\to \mathbb {C}\) is such that \(\mathbb {E}_x f(x)=0\) and \(\left\lvert f(x)\right\rvert \leq 2\) for all \(x\) then \[ \mathbb {E}_{x_1,\ldots ,x_n} \left\lvert \sum _{i=1}^n f(x_i)\right\rvert ^{2m} \leq (8mn)^{m}. \] | [
"LeanAPAP/mzi"
] | [
3572
] |
radii-polynomial/lem:summable_eval | 2,215 | radii-polynomial | If \(a \in \ell ^1_\nu \) and \(|z| \leq \nu \), then \(\sum _{n=0}^\infty a_n z^n\) converges absolutely. | [
"radii-polynomial/def:eval"
] | [
2535
] |
brownian-motion/def:IsCadlag | 1,878 | brownian-motion | A function \(f : T \to E\) is cadlag if it is right continuous and the limit \(\lim _{s \uparrow t} f(s)\) exists for all \(t \in T\). | [
"brownian-motion/def:RightContinuous"
] | [
1591
] |
FormalBook/e_pow_4_irrational | 585 | FormalBook | \(e^4\) is irrational. | [
"FormalBook/little_lemma",
"FormalBook/e_pow_2_irrational"
] | [
1083,
1451
] |
PrimeNumberTheoremAnd/fks2-def-5 | 2,189 | PrimeNumberTheoremAnd | Let \(x_0 {\gt} 2\). We say a (step) function \(ε_{\diamond ,num}(x_0)\) gives an admissible numerical bound for \(E_\diamond (x)\) if \(E_\diamond (x) \leq ε_{\diamond ,num}(x_0)\) for all \(x \geq x_0\). | [
"PrimeNumberTheoremAnd/Epi-def"
] | [
23
] |
PrimeNumberTheoremAnd/Dusart_cor_5_2_e | 3,054 | PrimeNumberTheoremAnd | For \(x \geq 88789\), we have \[ \pi (x) \geq \frac{x}{\log x} \Bigl(1 + \frac{1}{\log x} + \frac{2}{\log ^2 x}\Bigr). \] | [
"PrimeNumberTheoremAnd/pi-def"
] | [
2667
] |
ClassFieldTheory/thm:first inequality | 3,777 | ClassFieldTheory | For any finite Galois extension \(l/k\) be have \[ |H^0_{\mathrm{Tate}}(l/k, \mathrm{Cl}_{l}) | \le [l : k]. \] | [
"ClassFieldTheory/lem:Dirichlet density split",
"ClassFieldTheory/lem:density bound"
] | [
896,
2939
] |
LeanAPAP/main-int-count | 3,769 | LeanAPAP | If \(A\subseteq \{ 1,\ldots ,N\} \) has size \(\left\lvert A\right\rvert =\alpha N\), then \(A\) contains at least \[ \exp (-O(\mathcal{L}{\alpha }^{12}))N^2 \] | [
"LeanAPAP/th-int-gen",
"LeanAPAP/hoelder-lifting"
] | [
3675,
924
] |
PrimeNumberTheoremAnd/MediumPNT | 1,135 | PrimeNumberTheoremAnd | We have \[ \sum _{n \leq x} \Lambda (n) = x + O(x \exp (-c(\log x)^{1/10})). \] | [
"PrimeNumberTheoremAnd/SmoothedChebyshevPull2",
"PrimeNumberTheoremAnd/SmoothedChebyshevPull1",
"PrimeNumberTheoremAnd/I5Bound",
"PrimeNumberTheoremAnd/LogDerivZetaHolcLargeT",
"PrimeNumberTheoremAnd/I3Bound",
"PrimeNumberTheoremAnd/LogDerivZetaBndUnif",
"PrimeNumberTheoremAnd/I8Bound",
"PrimeNumberTh... | [
3454,
2533,
2487,
876,
505,
734,
2114,
477,
2034,
238,
2243,
3260,
3174
] |
toric/0-char-cochar-pairing | 3,485 | toric | Let \(R\) be a domain and \(G\) a group scheme over \(\operatorname{Spec}R\). Then there is a \({\mathbb Z}\)-valued perfect pairing between \(X(G)\) and \(X^*(G)\). | [
"toric/0-cochar",
"toric/0-char"
] | [
2645,
1173
] |
brownian-motion/lem:logSizeBallSequence_V_anti | 2,847 | brownian-motion | The sets \(V_i\) of a log-size ball sequence \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) are a decreasing sequence of sets. That is, \(V_{i+1} \subseteq V_i\) for all \(i \in \mathbb {N}\). | [
"brownian-motion/def:logSizeBallSequence"
] | [
2293
] |
equational_theories/finite_imp_3994_3588 | 836 | equational_theories | All finite magmas which satisfy E3994 \((x \diamond y = (z \diamond (x \diamond y)) \diamond z)\) also satisfy E3588 \((x \diamond y = z \diamond ((x \diamond y) \diamond z))\). | [
"equational_theories/models-def",
"equational_theories/eq3994",
"equational_theories/eq3588"
] | [
1745,
3165,
3511
] |
brownian-motion/def:preBrownian | 1,975 | brownian-motion | Let \(\Omega = \mathbb {R}^{\mathbb {R}_+}\) and consider the probability space \((\Omega , P_B)\) (where \(P_B\) is the measure defined in Definition 4.10). The identity on that space is a function \(\Omega \to \mathbb {R}_+ \to \mathbb {R}\). We reorder the arguments to define a stochastic process \(X : \mathbb ... | [
"brownian-motion/def:gaussianLimit"
] | [
2864
] |
ChandraFurstLipton/thm:eval-rand-complexity | 533 | ChandraFurstLipton | \[ R_\epsilon (\operatorname{eval}) \le 2d\lceil \log _2\epsilon ^{-1}\rceil \] | [
"ChandraFurstLipton/lem:rand-eq-test-protocol-eval-valid",
"ChandraFurstLipton/def:rand-complex",
"ChandraFurstLipton/def:eval"
] | [
2134,
449,
3066
] |
brownian-motion/lem:LocallyBoundedVariationOn.exists_monotoneOn_sub_monotoneOn | 2,148 | brownian-motion | If a real-valued function has bounded variation on a set, then it is a difference of monotone functions there. | [
"brownian-motion/def:LocallyBoundedVariationOn"
] | [
2261
] |
radii-polynomial/thm:mean_value | 2,191 | radii-polynomial | Suppose that \(U \subset \mathbb {R}^n\) is open and that \(f: U \rightarrow \mathbb {R}^n\) is \(C^1\). Consider \(x, y \in U\) such that the line segment \((1-t)x + ty\), \(t \in [0,1]\), is contained in \(U\). Then, \begin{equation*} f(y) - f(x) = \left(\int _0^1 \text{Df}((1-t)x + ty) \, dt\right)(y-x), \end{equati... | [
"radii-polynomial/def:operator_norm"
] | [
2365
] |
PrimeNumberTheoremAnd/initial-factorization-medium-prime-ge | 639 | PrimeNumberTheoremAnd | A medium prime \(\sqrt{n} {\lt} p ≤ n/L\) can be in deficit by at most \(M\). | [
"PrimeNumberTheoremAnd/initial-factorization-def",
"PrimeNumberTheoremAnd/balance-def"
] | [
1345,
917
] |
exchangeability/thm:birkhoff_condexp | 1,748 | exchangeability | Birkhoff averages converge in L\(^2\) to the conditional expectation given the shift-invariant \(\sigma \)-algebra. | [
"exchangeability/lem:koopman_condexp_comm"
] | [
1861
] |
brownian-motion/def:IsSquareIntegrable | 2,190 | brownian-motion | Let \(T\) be a linear order with bottom element 0, on which we have a filtration \(\mathcal{F}\) satisfying the usual conditions. We say that a martingale \(M : T \to \Omega \to E\) is square integrable if it is càdlàg and \(\sup _{t \in T} \Vert M_t \Vert _{L^2} {\lt} \infty \) (Lean remark: use eLpNorm (M t) 2). | [
"brownian-motion/def:Martingale"
] | [
1156
] |
brownian-motion/lem:predictablePart_add_one | 1,270 | brownian-motion | For any integer \(n \ge 0\), \(A_{n+1} = A_n + \mathbb {E}[X_{n+1} - X_n \mid \mathcal{F}_n]\). | [
"brownian-motion/def:predictablePart"
] | [
925
] |
brownian-motion/thm:finite_set_bound_of_dist_le_of_le_diam | 2,610 | brownian-motion | Suppose that \(T\) is a finite set with bounded internal covering number with constant \(c_1{\gt}0\) and exponent \(d {\gt} 0\). Let \(X : T \to \Omega \to E\) be a process that satisfies the Kolmogorov condition for exponents \((p,q)\) with constant \(M\), with \(q {\gt} d\) and \(p {\gt} 0\). For all \(\delta \in (0,... | [
"brownian-motion/lem:second_term_bound",
"brownian-motion/lem:integral_sup_rpow_dist_cover_rescale",
"brownian-motion/lem:lintegral_sup_cover_eq_of_lt_iInf_dist",
"brownian-motion/cor:scale_change_rpow",
"brownian-motion/lem:IsKolmogorovProcess.lintegral_sup_rpow_edist_eq_zero"
] | [
2246,
3212,
3438,
871,
1528
] |
ClassFieldTheory/lem:Tate -1 aug | 709 | ClassFieldTheory | Let \(S\) be a subgroup of a finite group \(G\). Then there is an isomorphism of \(R\)-modules \[ S^{\mathrm{ab}} \otimes _{{\mathbb Z}} R \cong H^{-1}_{\mathrm{Tate}}(S,\mathrm{aug}(R,G)), \] | [
"ClassFieldTheory/lem:homology 1 trivial",
"ClassFieldTheory/lem:Tate cohomology is cohomology or homology",
"ClassFieldTheory/def:augmentation module"
] | [
3808,
3774,
2791
] |
exchangeability/lem:indicator_product_bridge_contractable | 583 | exchangeability | For contractable sequences, indicator products satisfy the bridge condition. | [
"exchangeability/def:contractable"
] | [
606
] |
M2/lem:homotpy_k_sheaves_stable_by_extension | 2,106 | M2 | Let \(\mathcal{F}^{\bullet }\) be a complex of \(\mathcal{K}\)-presheaves. If \(\mathcal{F}^{\bullet }\) has a finite filtration whose associated graded is a Homotopy-\(\mathcal{K}\)-sheaf, then \(\mathcal{F}^{\bullet }\) is a Homotopy-\(\mathcal{K}\)-sheaf. | [
"M2/def:homotopy_k_sheaf"
] | [
3076
] |
PrimeNumberTheoremAnd/bklnw-eq_A_21 | 263 | PrimeNumberTheoremAnd | We define \[ B = 5/2 - \sigma , \] | [
"PrimeNumberTheoremAnd/zero-density-bound",
"PrimeNumberTheoremAnd/RH-up-to",
"PrimeNumberTheoremAnd/classical-zero-free-region"
] | [
1474,
2065,
3243
] |
radii-polynomial/thm:CauchyProduct_mul_one | 2,288 | radii-polynomial | \(a \star e = a\). | [
"radii-polynomial/thm:toPowerSeries_mul",
"radii-polynomial/def:CauchyProduct_one"
] | [
133,
1153
] |
exchangeability/lem:product_blockAvg_L1 | 3,639 | exchangeability | Products of block averages converge in L\(^1\). | [
"exchangeability/lem:blockAvg_tendsto",
"exchangeability/lem:integral_prod_blockAvg"
] | [
2132,
2558
] |
pfr/relabeled-entropy-cond | 591 | pfr | If \(X: \Omega \to S\), \(Y: \Omega \to T\), and \(Z: \Omega \to U\) are random variables, and \(Y = f(X,Z)\) almost surely for some map \(f: S \times U \to T\) that is injective for each fixed \(U\), then \(\mathbb {H}[X|Z] = \mathbb {H}[Y|Z]\). Similarly, if \(g: T \to U\) is injective, then \(\mathbb {H}[X|g(Y)] = \... | [
"pfr/conditional-entropy-def",
"pfr/relabeled-entropy"
] | [
1044,
3480
] |
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