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 |
|---|---|---|---|---|---|
brownian-motion/lem:Predict_Part_Increasing | 705 | brownian-motion | \((A^n_t)_{t\in \mathcal{D}_n^T}\) is an increasing process. | [
"brownian-motion/lem:nondecreasing_predictablePart_of_submartingale",
"brownian-motion/def:A"
] | [
2646,
878
] |
PrimeNumberTheoremAnd/bklnw-prop-3-sub-3 | 3,023 | PrimeNumberTheoremAnd | \(f(2^n) = 1 + u_n\). | [
"PrimeNumberTheoremAnd/bklnw-eq-2-4"
] | [
1636
] |
seymour/Matrix.IsTotallyUnimodular.linearIndependent_iff_support_linearIndependent_of_finite_of_finite | 59 | seymour | Let \(A\) be a rational TU matrix with finite number of rows and finite number of columns. Its rows are linearly independent iff the rows of its support matrix are linearly independent. | [
"seymour/Matrix.IsTotallyUnimodular",
"seymour/Matrix.isUnit_iff_isUnit_det",
"seymour/Matrix.support_submatrix"
] | [
1724,
1954,
1767
] |
LBPTester/theorem : neg_one_smul | 2,497 | LBPTester | In any vector space V over K, multiplying a vector by -1 gives its additive inverse: \((v : V) : (-1 : K) \bullet v = -v\) | [
"LBPTester/definition : VectorSpace"
] | [
1380
] |
toric/0-hopf-quot | 3,588 | toric | If \(A\) is a Hopf algebra over \(R\) and \(I\) is a Hopf ideal, the quotient \(A / I\) is equipped with a canonical Hopf algebra structure over \(R\). | [
"toric/0-biquot",
"toric/0-hopf-ideal"
] | [
1079,
756
] |
PrimeNumberTheoremAnd/fks-theorem-3-1 | 3,037 | PrimeNumberTheoremAnd | Let \(x {\gt} e^{50}\) and \(50 {\lt} T {\lt} x\). Then \(E_\psi (x) \leq \sum _{|\gamma | {\lt} T} |x^{\rho -1}/\rho | + 2 \log ^2 x / T\). | [
"PrimeNumberTheoremAnd/zeroes-of-riemann-zeta",
"PrimeNumberTheoremAnd/Epsi-def"
] | [
2427,
3055
] |
FLT_ch5/QHat.rat_meet_zHat | 2,627 | FLT_ch5 | The intersection of \(\mathbb {Q}\) and \(\widehat{\mathbb {Z}}\) in \(\widehat{\mathbb {Q}}\) is \(\mathbb {Z}\). | [
"FLT_ch5/QHat",
"FLT_ch5/QHat.lowestTerms"
] | [
3428,
3786
] |
PrimeNumberTheoremAnd/costa-pereira-sublemma-1-7 | 1,532 | PrimeNumberTheoremAnd | For every \(x {\gt} 0\) we have \[ \psi (x) - \theta (x) \leqslant \psi (x^{1/2}) + \psi (x^{1/3}) + \sum _{k \geqslant 1} \theta (x^{1/5k} \] | [
"PrimeNumberTheoremAnd/costa-pereira-sublemma-1-6"
] | [
3098
] |
PrimeNumberTheoremAnd/fks-corollary-3-12 | 1,292 | PrimeNumberTheoremAnd | Let \(5/8 {\lt} \sigma _2 \leq 1\), \(t_0 = t_0(\sigma _2, x) = \max \left(H_{\sigma _2}, \exp \left(\sqrt{\frac{\log x}{R}}\right)\right)\), \(T {\gt} t_0\). Let \(K \geq 2\), \(\lambda = (T/t_0)^{1/K}\), and consider \((t_k)_{k=0}^K\) the sequence given by \(t_k = t_0 \lambda ^k\). Then \[ \Sigma ^1_{\sigma _2} = 2 \... | [
"PrimeNumberTheoremAnd/zero-density-bound",
"PrimeNumberTheoremAnd/Riemann-von-Mangoldt-estimate",
"PrimeNumberTheoremAnd/RH-up-to",
"PrimeNumberTheoremAnd/classical-zero-free-region"
] | [
1474,
1239,
2065,
3243
] |
PrimeNumberTheoremAnd/fks2-eq30 | 459 | PrimeNumberTheoremAnd | For any \(x \geq x_0 {\gt} 0\), \[ |\pi (x) - \operatorname {Li}(x)| \leq \left| \frac{\theta (x) - x}{\log (x)} \right| + \left| \pi (x_0) - \operatorname {Li}(x_0) - \frac{\theta (x_0) - x_0}{\log (x_0)} \right| + \left| \int _{x_0}^{x} \frac{\theta (t) - t}{t(\log (t))^2} \, dt \right|. \] | [
"PrimeNumberTheoremAnd/Epi-def",
"PrimeNumberTheoremAnd/Etheta-def",
"PrimeNumberTheoremAnd/Epsi-def"
] | [
23,
892,
3055
] |
toric/1-1-1-subgroup-torus | 544 | toric | Let \(R\) be a commutative ring of characteristic zero. Let \(T\) be a split torus. If \(H \subseteq T\) is a connected closed subgroup, then \(H\) is a split torus. | [
"toric/0-full-faithful-diag",
"toric/0-subgroup-diag",
"toric/0-diag-tors"
] | [
513,
3029,
2719
] |
BrauerGroup_new/a0000000113 | 1,120 | BrauerGroup_new | \(\sim _{\operatorname{Br}}\) is transitive. | [
"BrauerGroup_new/con:matrix-matrix"
] | [
3590
] |
pfr/multidist-ruzsa-III | 2,068 | pfr | Let \(m \ge 2\), and let \(X_{[m]}\) be a tuple of \(G\)-valued random variables. If the \(X_i\) all have the same distribution, then \(D[X_{[m]}] \leq m d[X_i;X_i]\) for any \(1 \leq i \leq m\). | [
"pfr/sumset-lower",
"pfr/ruz-indep",
"pfr/klm-1",
"pfr/multidist-copy",
"pfr/ruz-copy",
"pfr/multidist-indep"
] | [
461,
3276,
1171,
2722,
282,
2290
] |
FLT_ch3/hardly_ramified_lifts | 1,158 | FLT_ch3 | If \(\ell \geq 3\) is prime and \(\overline{\rho }:\operatorname{Gal}(\overline{\mathbb {Q}}/\mathbb {Q})\to \operatorname{GL}_2(\mathbb {Z}/\ell \mathbb {Z})\) is hardly ramified and irreducible, then there exists a finite extension \(K\) of \(\mathbb {Q}_\ell \) with integer ring \(\mathcal{O}\) and maximal... | [
"FLT_ch3/hardly_ramified"
] | [
450
] |
brownian-motion/lem:elemStochIntegralLinear_assoc | 1,101 | brownian-motion | For simple process \(W\) with values in \(L(E, F)\), simple process \(V\) with values in \(L(F, G)\), and stochastic process \(X\) with values in \(E\), we have \begin{align*} V \bullet _L (W \bullet _L X) & = (V \circ W) \bullet _L X \: , \end{align*} | [
"brownian-motion/lem:elemStochIntegralBilin_assoc"
] | [
2933
] |
brownian-motion/lem:M_uniform_integrable | 1,210 | brownian-motion | The sequence \((M^n_T)_{n\in \mathbb {N}}\) is uniformly integrable (bounded in \(L^1\) norm). | [
"brownian-motion/lem:A_uniform_integrable",
"brownian-motion/def:classD"
] | [
838,
3521
] |
pfr/klm-3 | 402 | pfr | If \(n \geq 1\) and \(X, Y_1, \dots , Y_n\) are jointly independent \(G\)-valued random variables, then \[ d\left[X; \sum _{i=1}^n Y_i\right] \leq d\left[X; Y_1\right] + \frac{1}{2}\left(\mathbb {H}\left[ \sum _{i=1}^n Y_i\right] - \mathbb {H}[Y_1]\right). \] | [
"pfr/kv",
"pfr/ruz-indep"
] | [
3327,
3276
] |
PrimeNumberTheoremAnd/initial-factorization-def | 1,345 | PrimeNumberTheoremAnd | We perform an initial factorization by taking the natural numbers between \(n-n/M\) (inclusive) and \(n\) (exclusive) repeated \(M\) times, deleting those elements that are not \(n/L\)-smooth (i.e., have a prime factor greater than or equal to \(n/L\)). | [
"PrimeNumberTheoremAnd/params-set",
"PrimeNumberTheoremAnd/factorization-def"
] | [
10,
1253
] |
ChebyshevCircles/lem:sum_cos_m_phi_vanishes | 1,124 | ChebyshevCircles | For any integers \(N \geq 1\), \(m\) with \(0 {\lt} m {\lt} N\), and any \(\theta \in \mathbb {R}\), \begin{equation} \sum _{k=0}^{N-1} \cos \left(m\left(\theta + \frac{2\pi k}{N}\right)\right) = 0. \end{equation} | [
"ChebyshevCircles/lem:discrete_orthogonality"
] | [
3410
] |
PrimeNumberTheoremAnd/Dusart_prop_5_12 | 843 | PrimeNumberTheoremAnd | We have for \(k \geq 781\), \[ \vartheta (p_k) \leq k \log k + \log \log k - 1 + \frac{\log \log k - 2}{\log k} - \frac{0.782}{\log ^2 k}. \] | [
"PrimeNumberTheoremAnd/theta-def"
] | [
108
] |
BlueprintRep/YuInv | 3,044 | BlueprintRep | \(Y_{\mu }^{-1}\) est une l’inverse de \(Y_{\mu }\) | [
"BlueprintRep/bijYu"
] | [
1783
] |
ClassFieldTheory/lem:unramified units trivial | 3,500 | ClassFieldTheory | If \(l/k\) is unramified then \({\mathcal O}_l^\times \) has trivial group cohomology. | [
"ClassFieldTheory/lem:finite field trivial",
"ClassFieldTheory/lem:serre_approx"
] | [
1777,
2196
] |
amatelus/def:impl-chapter | 138 | amatelus | This chapter covers implementation guidance aspects of AMATELUS. | [
"amatelus/def:crypto-assumptions"
] | [
3484
] |
GibbsMeasure/def:markov-spec | 3,806 | GibbsMeasure | A specification \(\gamma \) is a Markov specification iff \(\gamma _\Lambda \) is a probability kernel for every \(\Lambda \in \operatorname{Finset}(S)\). | [
"GibbsMeasure/def:spec"
] | [
765
] |
Sphere-Packing-Lean/lemma:theta-bounded-im-infty | 3,225 | Sphere-Packing-Lean | For all \(\gamma \in \Gamma _1\), \(H_{2}|_2 \gamma \), \(H_{3}|_2 \gamma \), and \(H_{4}|_2 \gamma \) are holomorphic at \(i\infty \). | [
"Sphere-Packing-Lean/lemma:theta-slash-invariant",
"Sphere-Packing-Lean/lemma:Gamma-1-generators"
] | [
3215,
1387
] |
seymour/Matrix.IsGraphic | 2,334 | seymour | Let \(A \in \mathbb {Q}^{X \times Y}\) be a matrix. If for all \(j \in Y\), one has that \(a_{i,j} = 0\) for all \(i \in X\), or that there exists \(i_1,i_2 \in X\) such that \[ a_{i,j} = \begin{cases} 1 & \text{ if $i = i_1$} \\ -1 & \text{ if $i = i_2$} \\ 0 & \text{ otherwise}, \end{cases} \] | [
"seymour/Matrix"
] | [
2693
] |
brownian-motion/def:ProgMeasurable | 1,045 | brownian-motion | A stochastic process \(X\) is said to be progressively measurable with respect to a filtration \(\mathcal{F}\) if at each point in time \(i\), \(X\) restricted to \((-\infty , i] \times \Omega \) is measurable with respect to the product \(\sigma \)-algebra where the \(\sigma \)-algebra used for \(\Omega \) is \(\mathc... | [
"brownian-motion/def:filtration"
] | [
2474
] |
brownian-motion/thm:Ito_formula | 771 | brownian-motion | Let \(X^1, \ldots , X^d\) be continuous semi-martingales and let \(f : \mathbb {R}^d \to \mathbb {R}\) be a twice continuously differentiable function. Then, writing \(X = (X^1, \ldots , X^d)\), the process \(f(X)\) is a semi-martingale and we have \begin{align*} f(X_t) & = f(X_0) + \sum _{i=1}^d \int _0^t \frac{\parti... | [
"brownian-motion/thm:integration_by_parts"
] | [
2905
] |
Sphere-Packing-Lean/lemma:automorphy-factor-chain-rule | 1,500 | Sphere-Packing-Lean | The automorphy factor satisfies the chain rule \[ j_k(z,\gamma _1\gamma _2)=j_k(z,\gamma _1)\, j_k(\gamma _2z,\gamma _1). \] | [
"Sphere-Packing-Lean/def:automorphy-factor"
] | [
548
] |
PrimeNumberTheoremAnd/lem:Lprime-def | 285 | PrimeNumberTheoremAnd | There exists a positive integer \(L'\) such that \[ L_n = q_1 q_2 q_3 \, L' \] | [
"PrimeNumberTheoremAnd/lcm-criterion",
"PrimeNumberTheoremAnd/Ln-def"
] | [
3744,
1520
] |
carleson/forest-convex | 893 | carleson | For each \({\mathfrak u}\in {\mathfrak U}_3(k,n,j)\), the set \(\mathfrak {T}_2({\mathfrak u})\) satisfies the convexity condition 2.0.33. | [
"carleson/C5-convex"
] | [
3804
] |
pfr/phi-first-estimate | 898 | pfr | \(I_1\le 2\eta d[X_1;X_2]\) | [
"pfr/phi-min-def",
"pfr/first-fibre"
] | [
3787,
2473
] |
BlueprintRep/bu | 314 | BlueprintRep | \[ b_{\mu } := \frac{1}{|Q_{\mu }|}\sum _{g \in Q_{\mu }}(-1)^{g}g \] | [
"BlueprintRep/QuCard"
] | [
1657
] |
infinity-cosmos/lem:equiv-htpy-equiv | 95 | infinity-cosmos | A map \(f \colon A \to B\) between \(\infty \)-categories in an \(\infty \)-cosmos \({\mathord {\mathcal{K}}}\) is an equivalence if and only if it extends to the data of a “homotopy equivalence” with the free-living isomorphism \(I\) serving as the interval: that is, if there exist maps \(g \colon B \to A\) | [
"infinity-cosmos/defn:equivalence"
] | [
1700
] |
FormalBook/quadratic_reciprocity1 | 830 | FormalBook | Let \(p\) and \(q\) be different odd primes. Then \[ (\frac{q}{p})(\frac{p}{q}) = (-1)^{\frac{p-1}{2} \frac{q-1}{2}}. \] | [
"FormalBook/gauss_lemma"
] | [
2757
] |
PrimeNumberTheoremAnd/div-remainder | 356 | PrimeNumberTheoremAnd | There exist integers \(m \ge 0\) and \(r\) satisfying \(0 {\lt} r {\lt} 4 p_1 p_2 p_3\) and \[ q_1 q_2 q_3 = 4 p_1 p_2 p_3 m + r \] | [
"PrimeNumberTheoremAnd/lcm-criterion"
] | [
3744
] |
BlueprintRep/PuCard | 2,690 | BlueprintRep | Le nombre d’élément de \(P_{\mu }\) est fini. | [
"BlueprintRep/Pu"
] | [
2975
] |
exchangeability/lem:kallenberg_L2_bound | 3,642 | exchangeability | The key L\(^2\) bound that drives the Cesaro convergence. | [
"exchangeability/lem:l2_bound"
] | [
673
] |
pfr/rho-cond-relabeled | 1,749 | pfr | If \(f\) is injective, then \(\rho (X|f(Y))=\rho (X|Y)\). | [
"pfr/rho-cond-def"
] | [
1304
] |
seymour/MatrixSum3.IsCanonicalSigning.toMatrixLikeSum3 | 2,457 | seymour | Let \(B''\) be from Definition 53. Then \(B'' \in \mathcal{C} (X_{\ell }', Y_{\ell }', X_{r}', Y_{r}'; c_{0}'', c_{1}'')\) with \(X_{\ell }' = X_{\ell } \setminus \{ x_{1}, x_{0}\} \), \(X_{r}' = X_{r} \setminus \{ x_{2}, x_{1}, x_{0}\} \), \(Y_{\ell }' = Y_{\ell } \setminus \{ y_{2}\} \), \(Y_{r}' = Y_{r} \setmin... | [
"seymour/MatrixLikeSum3",
"seymour/MatrixSum3.IsCanonicalSigning.Al_D_isTotallyUnimodular"
] | [
524,
1386
] |
pfr/cond-distance-lower | 2,589 | pfr | For any \(G\)-valued random variables \(X'_1,X'_2\) and random variables \(Z,W\), one has \[ d[X'_1|Z;X'_2|W] \geq k - \eta (d[X^0_1;X'_1|Z] - d[X^0_1;X_1] ) - \eta (d[X^0_2;X'_2|W] - d[X^0_2;X_2] ). \] | [
"pfr/cond-dist-def",
"pfr/distance-lower"
] | [
398,
2256
] |
seymour/Matrix.fromRows_zero_reindex_toMatroid | 3,534 | seymour | Adding extra zero rows to a full representation matrix of a vector matroid does not change the matroid. | [
"seymour/VectorMatroid"
] | [
2016
] |
PrimeNumberTheoremAnd/bklnw-eq_A_26 | 2,947 | PrimeNumberTheoremAnd | For \(100 \leq x \leq 10^{19}\), one has \[ | (x - \psi (x)) / \sqrt{x} | \leq 0.94. \] | [
"PrimeNumberTheoremAnd/Epsi-def"
] | [
3055
] |
equational_theories/eq1689 | 1,563 | equational_theories | Equation 1689 is the law \(0 \simeq (1 \diamond 0) \diamond ((0 \diamond 2) \diamond 2)\) (or the equation \(x = (y \diamond x) \diamond ((x \diamond z) \diamond z)\)). | [
"equational_theories/magma-def"
] | [
3408
] |
Sphere-Packing-Lean/lemma:phi0-transform | 2,158 | Sphere-Packing-Lean | We have \begin{align} \phi _0(z + 1) & = \phi _0(z) \label{eqn:phi0-trans-T} \\ \phi _0\left(-\frac{1}{z}\right) & = \phi _0(z)-\frac{12i}{\pi }\, \frac{1}{z}\, \phi _{-2}(z)-\frac{36}{\pi ^2}\, \frac{1}{z^2}\, \phi _{-4}(z).\label{eqn:phi0-trans-S} \end{align} | [
"Sphere-Packing-Lean/lemma:E2-transform-S",
"Sphere-Packing-Lean/def:phi4-phi2-phi0",
"Sphere-Packing-Lean/lemma:disc-cuspform",
"Sphere-Packing-Lean/lemma:Ek-is-modular-form"
] | [
3318,
1367,
2530,
110
] |
M2/prop:homotopy_k_sheaf_compute_chech | 3,823 | M2 | Let \(\mathcal{F}^{\bullet }\) be a complex of \(\mathcal{K}\)-presheaves that verify 1 and 2 then the canonical map \(H^{\bullet }\mathcal{F}^{\bullet }\to \check{H}^{\bullet }(X;\mathcal{F}^{\bullet })\) is an isomorphism. | [
"M2/def:homotopy_k_sheaf",
"M2/def:chech_of_compelx_of_k_prsh"
] | [
3076,
1631
] |
FLT_ch9/IsDedekindDomain.FiniteAdeleRing.baseChangeIntegralAlgEquiv | 3,113 | FLT_ch9 | The natural map \(B\otimes _A\mathbb {A}_K^\infty \to \mathbb {A}_L^\infty \) is a \(B\)-algebra isomorphism. | [
"FLT_ch9/IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux1",
"FLT_ch9/IsDedekindDomain.FiniteAdeleRing.IntegraltensorProductAlgEquiv_aux2"
] | [
2445,
2911
] |
carleson/convergence-for-twice-contdiff | 2,730 | carleson | Let \(f:{\mathbb {R}}\to {\mathbb {C}}\) be \(2\pi \)-periodic and twice continuously differentiable. Then \begin{equation} \sup _{x\in [0,2\pi ]} |f(x) - S_Nf(x)| \rightarrow 0 \end{equation} | [
"carleson/fourier-coeff-derivative",
"carleson/convergence-of-coeffs-summable"
] | [
2825,
3134
] |
pfr/dist-diff-bound | 1,655 | pfr | We have \begin{align*} & \sum _{i=1}^2 \sum _{A,B \in \{ U,V,W\} : A \neq B} d[X_i^0;A|B, S] - d[X_i^0;X_i]\\ & \qquad \leq 12 k + \frac{4(2 \eta k - I_1)}{1-\eta }. \end{align*} | [
"pfr/gen-ineq",
"pfr/second-estimate-aux"
] | [
1103,
1495
] |
equational_theories/eq1648 | 264 | equational_theories | Equation 1648 is the law \(0 \simeq (0 \diamond 1) \diamond ((0 \diamond 1) \diamond 1)\) (or the equation \(x = (x \diamond y) \diamond ((x \diamond y) \diamond y)\)). | [
"equational_theories/magma-def"
] | [
3408
] |
PrimeNumberTheoremAnd/costa-pereira-theorem-1a | 388 | PrimeNumberTheoremAnd | For every \(x {\gt} 0\) we have \(\psi (x) - \theta (x) \leqslant \psi (x^{1/2}) + \psi (x^{1/3}) + \psi (x^{1/5})\). | [
"PrimeNumberTheoremAnd/costa-pereira-sublemma-1-7"
] | [
1532
] |
ClassFieldTheory/def:ind₁ to coind₁ | 1,744 | ClassFieldTheory | There is a morphism of representations \(\mathrm{ind}_1(G,A) \to \mathrm{coind}_1(G,A)\), which takes a finitely supported function \(f : G \to _0 A\) to the function \(f\). If the group \(G\) is finite then this map is an isomorphism. More precisely, this is an isomorphism of functors \(\mathrm{ind}_1(G,-) \cong \math... | [
"ClassFieldTheory/def:induced"
] | [
3596
] |
brownian-motion/lem:IsGaussian.ext_iff | 1,317 | brownian-motion | Two Gaussian measures \(\mu \) and \(\nu \) on a separable Hilbert space are equal if and only if they have same mean and same covariance. | [
"brownian-motion/lem:IsGaussian.charFun_eq",
"brownian-motion/thm:ext_of_charFun"
] | [
1907,
415
] |
ClassFieldTheory/lem:induced trivial Tate | 3,513 | ClassFieldTheory | If the group \(G\) is finite then \(\mathrm{ind}_1(G,A)\) and \(\mathrm{coind}_1(G,A)\) have trivial Tate cohomology. | [
"ClassFieldTheory/lem:ind₁ trivial homology",
"ClassFieldTheory/def:ind₁ to coind₁",
"ClassFieldTheory/lem:Tate cohomology is cohomology or homology",
"ClassFieldTheory/lem:coind₁ trivial cohomology"
] | [
1014,
1744,
3774,
2412
] |
carleson/Hardy-Littlewood | 1,224 | carleson | Let \(\mathcal{B}\) be a finite collection of balls in \(X\). If for some \(\lambda {\gt}0\) and some measurable function \(u:X\to [0,\infty )\) we have \begin{equation} \label{eq-ball-assumption} \int _{B} u(x)\, d\mu (x)\ge \lambda \mu (B) \end{equation} | [
"carleson/layer-cake-representation",
"carleson/covering-separable-space"
] | [
3673,
2235
] |
ABC-Exceptions/prop:GeometryofNumbers | 197 | ABC-Exceptions | Let \(d\geq 1\) and \(\varepsilon {\gt}0\) be fixed, and let \[ X_1,\ldots , X_d, Y_1,\ldots , Y_d,Z_1,\ldots , Z_d\geq 1. \] | [
"ABC-Exceptions/thm:HeathBrownGeometry"
] | [
204
] |
Sphere-Packing-Lean/cor:ineqBnew | 769 | Sphere-Packing-Lean | 212 holds. | [
"Sphere-Packing-Lean/lemma:F-G-pos",
"Sphere-Packing-Lean/prop:Qdec",
"Sphere-Packing-Lean/lemma:Qlim"
] | [
1586,
1923,
679
] |
equational_theories/eq168 | 1,336 | equational_theories | Equation 168 is the law \(0 \simeq (1 \diamond 0) \diamond (0 \diamond 2)\) (or the equation \(x = (y \diamond x) \diamond (x \diamond z)\)). | [
"equational_theories/magma-def"
] | [
3408
] |
HoTTLean/defn:ProductsExponentials | 801 | HoTTLean | In the natural model we can construct these by considering first the map \[ (\mathsf{fst}, \mathsf{snd}) : \mathsf{Ty}_i \times \mathsf{Ty}_j \to P_{\mathsf{tp}_i}{\mathsf{Ty}_j} \] | [
"HoTTLean/defn:NaturalModelPi",
"HoTTLean/defn:NaturalModelSigma",
"HoTTLean/prop:UvPoly.equiv"
] | [
2866,
2870,
3552
] |
carleson/linearized-truncation | 2,342 | carleson | Let \(\sigma _1,\sigma _2\colon X\to \mathbb {Z}\) be measurable functions with finite range and \(\sigma _1\leq \sigma _2\). Then we have \begin{equation} \label{Sqlin} \int \mathbf{1}_{G}(x) \left| {T}_{2,\sigma _1,\sigma _2}f(x)\right|\, d\mu (x) \le \frac{2^{442a^3+2}}{(q-1)^6} \mu (G)^{1-\frac{1}{q}} \mu (F)^{\fra... | [
"carleson/finitary-Carleson"
] | [
1008
] |
ChebyshevCircles/lem:chebyshev_roots | 619 | ChebyshevCircles | Let \(\xi _k = \cos ((2k + 1)\pi /(2N))\) for \(k = 0, \ldots , N-1\) denote the roots of \(T_N(x)\). Then for any \(1 \leq j {\lt} N\), \begin{equation} \sum _{k=0}^{N-1} \xi _k^j = \begin{cases} \frac{N}{2^j} \binom {j}{j/2} & \text{if } j \text{ is even}, \\ 0 & \text{if } j \text{ is odd.} \end{cases} \end{equation... | [
"ChebyshevCircles/lem:chebyshev_angle_sum_odd",
"ChebyshevCircles/lem:chebyshev_angle_sum_even",
"ChebyshevCircles/lem:chebyshev_roots_characterization"
] | [
987,
1981,
2779
] |
toric/0-spec-cocomm-bialg | 2,838 | toric | If \(A\) is a cocommutative bialgebra over \(R\), then \(\operatorname{Spec}A\) is a commutative monoid scheme. | [
"toric/0-spec-hopf"
] | [
1084
] |
equational_theories/confluent-theory | 2,924 | equational_theories | Let \(\Gamma \) be a theory. A word \(w\) can be reduced to another \(w'\) if one can get from \(w\) to \(w'\) by a series of substitutions of laws in \(\Gamma \), where no substitution increases the length of the word this is a working definition, might not be the best one to keep.. A theory \(\Gamma \) is confluent i... | [
"equational_theories/free-magma-def"
] | [
780
] |
PrimeNumberTheoremAnd/lem:abadeulmac' | 3,193 | PrimeNumberTheoremAnd | Let \(b{\gt}0\), \(b\in \mathbb {Z}\). Then, for all \(s\in \mathbb {C}\setminus \{ 1\} \) with \(\Re s {\gt} 0\), \begin{equation} \label{eq:abak1'} \sum _{n \leq b} \frac{1}{n^s} = \zeta (s) + \frac{b^{1-s}}{1-s} + \frac{b^{-s}}{2} + s \int _b^\infty \left(\{ y\} -\frac{1}{2}\right) \frac{dy}{y^{s+1}}. \end{equation} | [
"PrimeNumberTheoremAnd/Zeta0EqZeta"
] | [
750
] |
FLT_ch7/instCentreAction | 1,167 | FLT_ch7 | Thus the centre \(\mathbb {Z}_n\) of this universal enveloping algebra also acts on the smooth complex functions. | [
"FLT_ch7/instUniversalEnvelopingAlgebraAction"
] | [
3462
] |
polytopes-in-LEAN/lem:ambient_is_primspace | 3,168 | polytopes-in-LEAN | [
"polytopes-in-LEAN/def:primspace"
] | [
195
] | |
PrimeNumberTheoremAnd/bklnw-prop-3-sub-7 | 3,309 | PrimeNumberTheoremAnd | \(f(x) \leq f(x_0)\) for \(x \in [x_0, 2^{\lfloor \frac{\log x_0}{\log 2} \rfloor + 1})\). | [
"PrimeNumberTheoremAnd/bklnw-prop-3-sub-2"
] | [
2417
] |
PrimeNumberTheoremAnd/ResidueOfTendsTo | 1,280 | PrimeNumberTheoremAnd | If a function \(f\) is holomorphic in a neighborhood of \(p\) and \(\lim _{s\to p} (s-p)f(s) = A\), then \(f(s) = \frac{A}{s-p} + O(1)\) near \(p\). | [
"PrimeNumberTheoremAnd/existsDifferentiableOn_of_bddAbove"
] | [
3650
] |
FLT_ch11/NumberField.AdeleRing.DivisionAlgebra.Aux.C_compact | 829 | FLT_ch11 | \(C\) is compact. | [
"FLT_ch11/NumberField.AdeleRing.DivisionAlgebra.Aux.C",
"FLT_ch11/NumberField.AdeleRing.DivisionAlgebra.Aux.T_finite",
"FLT_ch11/NumberField.AdeleRing.DivisionAlgebra.Aux.X_compact"
] | [
1338,
351,
283
] |
pfr/rho-cond-invariant | 2,118 | pfr | For any \(s\in G\), \(\rho (X+s|Y)=\rho (X|Y)\). | [
"pfr/rho-invariant",
"pfr/rho-cond-def"
] | [
956,
1304
] |
Sphere-Packing-Lean/cor:logder-disc-E2 | 3,508 | Sphere-Packing-Lean | \begin{equation} \label{eqn:logder-disc-E2} \Delta ' = E_2 \Delta . \end{equation} | [
"Sphere-Packing-Lean/thm:ramanujan-formula"
] | [
3453
] |
M2/def:stalk | 11 | M2 | If \(\mathcal{F}\) is a presheaf on \(X\) and \(p\in X\) then the stalk of \(\mathcal{F}\) at \(p\) is the abelian group \(\mathcal{F}_p:=\varinjlim \limits _{p\in U\text{ open}}\mathcal{F}(U)\). | [
"M2/def:pre_sheaves"
] | [
1352
] |
radii-polynomial/def:ScaledReal | 766 | radii-polynomial | For \(\nu {\gt} 0\) and \(n \in \mathbb {N}\), the type \(\texttt{ScaledReal}\ \nu \ n\) is \(\mathbb {R}\) equipped with the scaled norm \(\| x\| _n := |x| \cdot \nu ^n\). | [
"radii-polynomial/def:posreal"
] | [
3159
] |
M2/lem:hocolim_preserves_homotopy_k_sheaves | 1,236 | M2 | [
"M2/lem:homotpy_k_sheaves_stable_by_extension",
"M2/def:homotopy_colimit"
] | [
2106,
85
] | |
ClassFieldTheory/lem:herbrand ses | 1,186 | ClassFieldTheory | Suppose we have a short exact sequence of representations of a finite cyclic group \(G\): \[ 0 \to A \to B \to C \to 0. \] | [
"ClassFieldTheory/cor:periodic cohomology",
"ClassFieldTheory/def:herbrand quotient"
] | [
2898,
3512
] |
PrimeNumberTheoremAnd/isIntegrable | 1,832 | PrimeNumberTheoremAnd | Let \(x{\gt}0\) and \(\sigma \in \mathbb {R}\). Then \[ \int _{\mathbb {R}}\frac{x^{\sigma +it}}{(\sigma +it)(1+\sigma + it)}dt \] | [
"PrimeNumberTheoremAnd/isHolomorphicOn"
] | [
3773
] |
carleson/estimate-bad | 1,113 | carleson | We have \begin{equation*} \mu \left({\{ x\in X: |T_r b(x)|{\gt}\alpha /2\} }\right) \le \frac{\frac{2^{7a}}{c} + 2^{a^3+11a+4}}{\alpha } \int |f(y)|\, d\mu (y) \, . \end{equation*} | [
"carleson/estimate-F-set",
"carleson/estimate-bad-partial"
] | [
329,
113
] |
PrimeNumberTheoremAnd/ShiftTwoBound | 2,182 | PrimeNumberTheoremAnd | For all \(\delta \in (0,1)\) and \(t\in \mathbb {R}\) with \(|t|\geq 2\) we have \[ -\Re \left(\frac{\zeta '}{\zeta }(1+\delta +2it)\right)\ll \log |t|. \] | [
"PrimeNumberTheoremAnd/SumBoundI"
] | [
2766
] |
toric/0-grp-like-lin-indep | 1,592 | toric | The group-like elements in a bialgebra \(A\) over a domain are linearly independent. | [
"toric/0-tensor-lin-indep",
"toric/0-grp-like"
] | [
1874,
2697
] |
toric/0-grp-alg-grp-like-gc | 1,778 | toric | Let \(R\) be a domain, \(G\) a commutative group and \(A\) a \(R\)-bialgebra. Then bialgebra homs \(R[G] \to A\) are in bijection with group homs \(G \to \operatorname{GrpLike}A\). | [
"toric/0-grp-like-grp-alg",
"toric/0-grp-like-map"
] | [
2102,
774
] |
equational_theories/eq39 | 1,257 | equational_theories | Equation 39 is the law \(0 \diamond 0 \simeq 1 \diamond 0\) (or the equation \(x \diamond x = y \diamond x\)). | [
"equational_theories/magma-def"
] | [
3408
] |
BonnAnalysis/def:fourier-L2 | 3,138 | BonnAnalysis | Let \(f\in L^2(V,E)\) and take a sequence \((f_n)_n\subset L^1(V,E)\cap L^2(V,E)\) with \(f_n\xrightarrow [L^2]{}f\). Set \[ \mathcal Ff:=\widehat f:=\lim _{n\to \infty }\widehat{f_n}, \] | [
"BonnAnalysis/lem:fourier12-cauchy"
] | [
812
] |
ClassFieldTheory/cor:cohomology sub Sylow | 32 | ClassFieldTheory | Let \(M\) be a representation of a finite group \(G\) and let \(S_p\) be a Sylow \(p\)-subgroup of \(G\) for some prime number \(p\). Then for any \(n \in {\mathbb Z}\), \(H^n_{\mathrm{Tate}}(G,M)[p^\infty ]\) is isomorphic to an \(R\)-submodule of \(H^n_{\mathrm{Tate}}(S_p,M)\). | [
"ClassFieldTheory/cor:Tate up down isos",
"ClassFieldTheory/lem:cor comp rest"
] | [
3586,
1077
] |
ClassFieldTheory/lem:cohomology S-ideles decomp | 416 | ClassFieldTheory | There are isomorphisms for all \(n {\gt} 0\) \[ H^n(l/k, {\mathbb A}_{S,l}^\times ) \cong \prod _{v \in S} H^n(l_{\hat v} / k_v, l_{\hat v}^\times ). \] | [
"ClassFieldTheory/lem:unramified units trivial",
"ClassFieldTheory/lem:semi-local iso coind"
] | [
3500,
3660
] |
brownian-motion/lem:logSizeBallSequence_eq_zero | 406 | brownian-motion | For any log-size ball sequence \((V_i, t_i, r_i)_{i \in \mathbb {N}}\) for \((J, a, c, n)\), for all \(k \ge \vert J \vert \), \(V_k = \emptyset \). | [
"brownian-motion/lem:logSizeBallSequence_V_anti"
] | [
2847
] |
polytopes-in-LEAN/lem:ch_of_empty_is_empty | 1,057 | polytopes-in-LEAN | [
"polytopes-in-LEAN/def:convex_hull",
"polytopes-in-LEAN/lem:empty_is_convex"
] | [
2525,
1919
] | |
exchangeability/lem:blockAvg_tendsto | 2,132 | exchangeability | For a shift-invariant measure, block averages converge in \(L^1\) to the conditional expectation given the shift-invariant \(\sigma \)-algebra: \(\int |A_{m,n,k}(f) - \mathbb {E}[f \circ \pi _0 \mid \mathcal{I}]| \, d\mu \to 0\) as \(n \to \infty \). | [
"exchangeability/thm:birkhoff_condexp",
"exchangeability/def:blockAvg"
] | [
1748,
435
] |
pfr/rho-sums | 2,724 | pfr | If \(X,Y\) are independent, one has \[ \rho ^-(X+Y) \leq \rho ^-(X) \] | [
"pfr/kl-sums"
] | [
919
] |
PrimeNumberTheoremAnd/bound-score-1 | 2,441 | PrimeNumberTheoremAnd | If \(M\) is sufficiently large depending on \(\varepsilon \), then \(n \log (1-1/M)^{-1} \leq \varepsilon n\). | [
"PrimeNumberTheoremAnd/params-set"
] | [
10
] |
carleson/partial-Fourier-sum-bound | 3,346 | carleson | Let \(g:{\mathbb {R}}\to {\mathbb {C}}\) be a measurable \(2\pi \)-periodic function such that for some \(\delta {\gt}0\) and every \(x\in {\mathbb {R}}\), \begin{equation} |g(x)|\le \delta \, . \end{equation} | [
"carleson/Dirichlet-Hilbert"
] | [
1462
] |
brownian-motion/lem:charFun_gaussianReal | 2,762 | brownian-motion | The characteristic function of a real Gaussian measure with mean \(\mu \) and variance \(\sigma ^2\) is given by \(x \mapsto \exp \left(i \mu x - \frac{\sigma ^2 x^2}{2}\right)\). | [
"brownian-motion/def:gaussianReal",
"brownian-motion/def:charFun"
] | [
1801,
1576
] |
pfr/submodularity | 552 | pfr | With three random variables \(X,Y,Z\), one has \(\mathbb {H}[X|Y,Z] \leq \mathbb {H}[X|Z]\). | [
"pfr/cond-reduce"
] | [
2666
] |
PrimeNumberTheoremAnd/lem:sigmaLn | 1,788 | PrimeNumberTheoremAnd | Let \(L'\) be as in Lemma 10.1.2. Then \begin{equation} \label{eq:sigmaLn} \frac{\sigma (L_n)}{L_n} \; =\; \frac{\sigma (L')}{L'} \prod _{i=1}^3 \Bigl(1 + \frac{1}{q_i}\Bigr). \end{equation} | [
"PrimeNumberTheoremAnd/lem:Lprime-def",
"PrimeNumberTheoremAnd/sigma-def"
] | [
285,
387
] |
brownian-motion/lem:Submartingale.integrable_stoppedValue | 460 | brownian-motion | Let \(X\) be a submartingale. Then for all bounded stopping times \(\tau \), the stopped value \(X_\tau \) is integrable. | [
"brownian-motion/def:Submartingale",
"brownian-motion/def:stoppedProcess"
] | [
542,
915
] |
brownian-motion/lem:doob_Lp_countable | 331 | brownian-motion | Let \(X : I \rightarrow \Omega \rightarrow \mathbb {R}\) be a non-negative sub-martingale. Let \(I\) be countable. For every \(M\in I,\lambda {\gt} 0\) and \(p{\gt}1\) we have \begin{align*} \mathbb {E}\left[ \sup _{i\in I, i \leq M}X_i^p \right] \leq \left(\frac{p}{p-1}\right)^p\mathbb {E}[X_M^p] \: . \end{align*} | [
"brownian-motion/lem:doob_countable"
] | [
1108
] |
brownian-motion/lem:posSemidef_gramMatrix | 1,807 | brownian-motion | A gram matrix is positive semidefinite. | [
"brownian-motion/def:gramMatrix"
] | [
906
] |
brownian-motion/cor:doob_ineq_stop_exp_val | 608 | brownian-motion | Let \(X:\mathbb {R}\times \Omega \rightarrow E\) be a right-continuous martingale with values in a normed space \(E\). For every \(\lambda {\gt}0\) and \(p{\gt}1\) and \(\tau \) stopping time a.s. bounded by \(T{\gt}0\), we have \[ \mathbb {E}\left[ \sup _{t\in [0,\tau ]}\lVert X_t \rVert ^p \right]\leq \left(\frac{p}{... | [
"brownian-motion/cor:Martingale.submartingale_norm",
"brownian-motion/lem:doob_ineq_stop_exp_val"
] | [
167,
454
] |
BrauerGroup_new/thm:wed-artin-algebra | 649 | BrauerGroup_new | Let \(K\) be a field and \(B\) an finite dimensional simple algebra over \(K\). There exists a non-zero \(n\in \mathbb {N}\) and a division \(K\)-algebra \(S\) such that \(B \cong \operatorname{Mat}_{n}(S)\). | [
"BrauerGroup_new/con:end-vec-iso-matrix",
"BrauerGroup_new/thm:wed-artin-ideal"
] | [
1094,
2164
] |
ABC-Exceptions/lem:Subcase2.4 | 233 | ABC-Exceptions | \(\nu \leq 0.66\) in the case \(s.2 {\lt} 0.3\) and \(4s_1+s_2{\lt}0.4\). | [
"ABC-Exceptions/lem:Subcase2.2",
"ABC-Exceptions/lem:Case2Basic2"
] | [
1416,
3789
] |
Sphere-Packing-Lean/prop:H2-fourier | 485 | Sphere-Packing-Lean | \(H_2\) admits a Fourier series of the form \begin{equation} H_2(z) = \sum _{n \ge 1} c_{H_2}(n) e^{\pi i n z} \end{equation} | [
"Sphere-Packing-Lean/def:H2-H3-H4"
] | [
3629
] |
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