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
] |
End of preview. Expand in Data Studio
DAPS-Blueprint
full_collection/contains the complete Blueprint node records and their dependencies, organized by source project. These files provide the graph used for training and premise retrieval.splits.jsonrecords the fixed random split of 2,351 query nodes at the node level. Its recorded seed is 42, with an 80% / 10% / 10% train / validation / test ratio: 1,880 training nodes, 235 validation nodes, and 236 test nodes. Use this file to select the nodes for each split.test.jsonlexports those 236 test queries and their gold positive premise nodes for inspection and independent evaluation. Its query order matches the test split insplits.json.
The complete node collection is retained for graph context and premise retrieval; the split file identifies the query nodes used in each stage.
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