{"schema_version": "1.0", "item_id": "rcd2_6516fa8d135e2ec9", "pair_id": "pair_c9d9529d309ea75b", "statement_latex": "\\forall\\varepsilon>(0\\exists N\\forall n\\ge N:d((lim_{n),(∞} |S_n(β)|^(1/n)))<\\varepsilon) + \\eta", "inference_order": 0} {"schema_version": "1.0", "item_id": "rcd2_aa5f64b267e35265", "pair_id": "pair_ca4fc0cc2a183559", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (max|X_j|) \\le x \\land x \\le ((D^{-1}*sqrt(det(AA^T)))^(1/(N-M)))", "inference_order": 1} {"schema_version": "1.0", "item_id": "rcd2_45f5f15dbf2d1e10", "pair_id": "pair_db102220af295e7b", "statement_latex": "\\delta_D((Av),(rv))=(0) + \\eta", "inference_order": 2} {"schema_version": "1.0", "item_id": "rcd2_271c8e25fc8f8de7", "pair_id": "pair_b0de22a580661648", "statement_latex": "\\delta_D((lim_{n → infinity} a_n),(sup {a_n}))=0", "inference_order": 3} {"schema_version": "1.0", "item_id": "rcd2_2f1807533f51cdb2", "pair_id": "pair_ac61fe8bb58499d3", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (4πA) \\le x \\land x \\le (L^2)", "inference_order": 4} {"schema_version": "1.0", "item_id": "rcd2_178f41df361fb3f1", "pair_id": "pair_9d9bc69cc64713b5", "statement_latex": "\\max\\{0,(n)-(5 * log_10(min(u, v)))\\}=(0\\;\\land\\;(n)\\ne (5 * log_10(min(u, v)))) + \\eta", "inference_order": 5} {"schema_version": "1.0", "item_id": "rcd2_88fbe406628dae53", "pair_id": "pair_d7075e4cc0d2e041", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,([I - A]x))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(d))", "inference_order": 6} {"schema_version": "1.0", "item_id": "rcd2_0a5c01f4782cae7e", "pair_id": "pair_16ba90f64efc0952", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(φ(+∞) − φ(0)))+\\delta_D(z,(−π(1/2*n0 + nb))))=0", "inference_order": 7} {"schema_version": "1.0", "item_id": "rcd2_582c2e1f928a5b30", "pair_id": "pair_f5a08099e3e6e2cd", "statement_latex": "\\delta_D((-log E[e^{-f(B)}]),(inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]))=0", "inference_order": 8} {"schema_version": "1.0", "item_id": "rcd2_2c516c5d7d51b60a", "pair_id": "pair_e95b74888099fc29", "statement_latex": "(T^2) \\in {(-1)} + \\eta", "inference_order": 9} {"schema_version": "1.0", "item_id": "rcd2_a6a5ee31b8f4014c", "pair_id": "pair_c68f68765106b956", "statement_latex": "\\delta_D((p_A(A)),(0))=0", "inference_order": 10} {"schema_version": "1.0", "item_id": "rcd2_c14d4eb07dcb7c30", "pair_id": "pair_f4d7f3e2067da4f9", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(SO_{\\gamma _{H}}(1_{K_{H}}))=z\\;\\land\\;z=(\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}}))", "inference_order": 11} {"schema_version": "1.0", "item_id": "rcd2_685215e73ac45625", "pair_id": "pair_4415402255935d0e", "statement_latex": "\\neg\\left((ker(d(f|_Y)_y) \\subseteq tau) \\land (0 = 0)\\right)", "inference_order": 12} {"schema_version": "1.0", "item_id": "rcd2_78c107bfca939729", "pair_id": "pair_9fd9089277775a8c", "statement_latex": "\\forall z\\in D,\\;(z=((∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c))\\Longleftrightarrow z=(Integral transform+Spectral terms))) + \\eta", "inference_order": 13} {"schema_version": "1.0", "item_id": "rcd2_b9311af87fb01e96", "pair_id": "pair_7b5a0b34f2e1b710", "statement_latex": "\\nexists S \\subseteq D: (((n) \\in S \\land ((r - 1)(s - 1) + 1) \\notin S) \\lor (((r - 1)(s - 1) + 1) \\in S \\land (n) \\notin S))", "inference_order": 14} {"schema_version": "1.0", "item_id": "rcd2_2bb1f082ed232f6d", "pair_id": "pair_771305494d7e3cbe", "statement_latex": "\\sup_{\\lambda\\in[0,\\infty)}\\lambda((infinity)-(dim_C H^i(X, F)))=0", "inference_order": 15} {"schema_version": "1.0", "item_id": "rcd2_38a018c29468b21c", "pair_id": "pair_954641ed0018434a", "statement_latex": "{(binom(m+n, k)_q)} = {(sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)})}", "inference_order": 16} {"schema_version": "1.0", "item_id": "rcd2_789110b6998162ad", "pair_id": "pair_d09b13737b7ad1c8", "statement_latex": "\\neg\\forall x\\in D,x\\notin \\{z\\in C:p(z)=0\\}", "inference_order": 17} {"schema_version": "1.0", "item_id": "rcd2_73b688347bf5f292", "pair_id": "pair_30b4c1a55d17a802", "statement_latex": "(f(-x)) \\in {(f(x))} + \\eta", "inference_order": 18} {"schema_version": "1.0", "item_id": "rcd2_a93169124a661949", "pair_id": "pair_cfd0bd1bc91a37be", "statement_latex": "\\neg\\left((H(t)) \\in {(∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE)}\\right)", "inference_order": 19} {"schema_version": "1.0", "item_id": "rcd2_9e87ce0a616d2b0c", "pair_id": "pair_71efd2d5e88b0066", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(p - q))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(1/π) Δ arg f(iy))", "inference_order": 20} {"schema_version": "1.0", "item_id": "rcd2_f0e754c9afa94ce8", "pair_id": "pair_0cb9aad03553e4ec", "statement_latex": "(SAS^T) = (B)", "inference_order": 21} {"schema_version": "1.0", "item_id": "rcd2_59770635374ce268", "pair_id": "pair_5889d634a00a262f", "statement_latex": "(f(x)) \\in {(0 (mod p^n))} + \\eta", "inference_order": 22} {"schema_version": "1.0", "item_id": "rcd2_c587f0569d461171", "pair_id": "pair_942db157a6b7dbe2", "statement_latex": "\\nexists S \\subseteq D: (((f'(x_0)) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (f'(x_0)) \\notin S)) + \\eta", "inference_order": 23} {"schema_version": "1.0", "item_id": "rcd2_90fb5ad3a64575c2", "pair_id": "pair_c7777da5410d5778", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(p_A(A)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(0))", "inference_order": 24} {"schema_version": "1.0", "item_id": "rcd2_077c39d731537779", "pair_id": "pair_276940381402d97e", "statement_latex": "\\{((f'(c)),z):z\\in D\\}=\\{((f(b)-f(a))/(b-a),z):z\\in D\\}", "inference_order": 25} {"schema_version": "1.0", "item_id": "rcd2_44ec684a1e75d0ea", "pair_id": "pair_c306c2b696171f5c", "statement_latex": "(1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n \\\\in \\mathbb{Z}} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})) - (prod_{n) \\le 0", "inference_order": 26} {"schema_version": "1.0", "item_id": "rcd2_7b8c17a5b206ba08", "pair_id": "pair_e03b8c25feba0441", "statement_latex": "(\\{((\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2)\\}\\times\\{(\\sum_{i0}\\exists N\\forall n\\ge N:d((X × Y),(X))<1/k", "inference_order": 49} {"schema_version": "1.0", "item_id": "rcd2_ec81744149f02ab6", "pair_id": "pair_ff420571c42152ad", "statement_latex": "(ΔδT) \\in {(0)}", "inference_order": 50} {"schema_version": "1.0", "item_id": "rcd2_c9b0e71ecf54a060", "pair_id": "pair_f069c9b2f8a0b471", "statement_latex": "(∑_{x,y} ψ^†(x) F(x, y) ψ(y)) = (H)", "inference_order": 51} {"schema_version": "1.0", "item_id": "rcd2_b12fd335828eca9e", "pair_id": "pair_559f6ad4d82defa2", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(z=(sum_{i=0}^n binom(n,i))\\Longleftrightarrow z=(2^n))\\right)", "inference_order": 52} {"schema_version": "1.0", "item_id": "rcd2_f74035cf177e7249", "pair_id": "pair_d0848f4f91f7a840", "statement_latex": "(ω(-t)) \\in {(ω(t))} + \\eta", "inference_order": 53} {"schema_version": "1.0", "item_id": "rcd2_2122e6db519a8af6", "pair_id": "pair_4f44aaa7bf2b2f58", "statement_latex": "(1} (|z|-1) ||(z-A)^-1||) - (K(A) = sup_{|z|) < 0", "inference_order": 54} {"schema_version": "1.0", "item_id": "rcd2_76c82ebbe6b20464", "pair_id": "pair_309cfe5f64cdbdac", "statement_latex": "\\frac{\\exp(min(|A|+|B|-1, p))}{\\exp(min(|A|+|B|-1, p))+\\exp(|A+B|)}\\le\\frac{1}{2} + \\eta", "inference_order": 55} {"schema_version": "1.0", "item_id": "rcd2_f9a4eccce03d3244", "pair_id": "pair_bac313ffe4290066", "statement_latex": "\\frac{\\exp(|d(X,Y) - d(A,B)|)}{\\exp(|d(X,Y) - d(A,B)|)+\\exp(ε)}\\le\\frac{1}{2}", "inference_order": 56} {"schema_version": "1.0", "item_id": "rcd2_b7b8c238e593ef49", "pair_id": "pair_52376eb897b5d805", "statement_latex": "\\forall z\\in D,\\;(z=((dS)\\Longleftrightarrow z=((dE + PdV) / T))) + \\eta", "inference_order": 57} {"schema_version": "1.0", "item_id": "rcd2_84b1c2cae20a035b", "pair_id": "pair_3dc5dd1139bf21a5", "statement_latex": "(e^σ[x(t)]) = (P[σ(x(t))] / P[σ(x_tilde(t))])", "inference_order": 58} {"schema_version": "1.0", "item_id": "rcd2_de6eaacaeaffab41", "pair_id": "pair_1932ac7cdf289db7", "statement_latex": "\\neg\\left(\\frac{\\exp(pi(x;q,a))}{\\exp(pi(x;q,a))+\\exp(2x / (phi(q) log(x/q)))}\\le\\frac{1}{2}\\right)", "inference_order": 59} {"schema_version": "1.0", "item_id": "rcd2_e83a8d62e59080c4", "pair_id": "pair_08fd193ce68ff99b", "statement_latex": "(cr(G)) - (e^3 / (29n^2)) \\ge 0", "inference_order": 60} {"schema_version": "1.0", "item_id": "rcd2_72ee534b2b5f253c", "pair_id": "pair_8534816df4bcf533", "statement_latex": "\\max\\{0,(Pr(max_{1≥k≤n}|S_{k}|≥λ))-((1/λ^2) * Var[S_{n}])\\}=0", "inference_order": 61} {"schema_version": "1.0", "item_id": "rcd2_15a9ecb28deead8c", "pair_id": "pair_5447687027a7e8c9", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (V_1(K, L))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (V(K)^((n-1)/n) V(L)^(1/n))\\}", "inference_order": 62} {"schema_version": "1.0", "item_id": "rcd2_b5f4dc91a83876d0", "pair_id": "pair_02b1f398b3242405", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F(([[u, 0], [0, u^-1]]))=F((\\text{conv}\\{w(X) : w \\in W\\}))", "inference_order": 63} {"schema_version": "1.0", "item_id": "rcd2_ff13a2579d616b23", "pair_id": "pair_8c15e0b4e6defa92", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (tr(exp(A+B)))\\Rightarrow t\\le (tr(exp(A)*exp(B))) + \\eta", "inference_order": 64} {"schema_version": "1.0", "item_id": "rcd2_cfd7cdeec1b1241e", "pair_id": "pair_03a99924674b1927", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(x + y))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(z))", "inference_order": 65} {"schema_version": "1.0", "item_id": "rcd2_f4ef8332e7735833", "pair_id": "pair_679caf016fea41a7", "statement_latex": "\\forall z\\in D,\\;(z=(E[sum_{x in N} f(x)])\\Longleftrightarrow z=(integral f(x) dLambda(x)))", "inference_order": 66} {"schema_version": "1.0", "item_id": "rcd2_1234f345bfb59706", "pair_id": "pair_6ef4d0833a95d07b", "statement_latex": "\\forall z\\in D,\\;(z=(A)\\Longleftrightarrow z=(k[y_1, ..., y_m]))", "inference_order": 67} {"schema_version": "1.0", "item_id": "rcd2_1e3f0a2a23275da4", "pair_id": "pair_52abafe811867d63", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (|g − f|)\\Rightarrow t<(ε)", "inference_order": 68} {"schema_version": "1.0", "item_id": "rcd2_1e352caf23cf504d", "pair_id": "pair_d88b3777c284deb3", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\ge (prod_{i=1}^n (r_i!)^(1/r_i))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (per(A))\\}\\right)", "inference_order": 69} {"schema_version": "1.0", "item_id": "rcd2_249b198b4c58de68", "pair_id": "pair_722f2abf3d7a42ba", "statement_latex": "\\delta_D((P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n)),(1))=0", "inference_order": 70} {"schema_version": "1.0", "item_id": "rcd2_bc1efab11993ce53", "pair_id": "pair_6a5321c77c0e440a", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(u(t))+s=((u(a) * exp(integral from a to t of beta(s) ds))) + \\eta", "inference_order": 71} {"schema_version": "1.0", "item_id": "rcd2_40998f4562131728", "pair_id": "pair_1c2bbe388e95df9f", "statement_latex": "\\exists s\\in\\mathbb{R}_{>0}\\;\\text{such that}\\;(p_{n+1}^2)+s\\\\ne(p_n#)", "inference_order": 72} {"schema_version": "1.0", "item_id": "rcd2_3da7d23001f0ab84", "pair_id": "pair_e5a76e9fd39bd1c8", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (max|B_k|(|a_n| + |a_n - a_1|))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|∑ a_k b_k|)\\}", "inference_order": 73} {"schema_version": "1.0", "item_id": "rcd2_7289f9c6b89f2d5a", "pair_id": "pair_f9b4f5d179be3f25", "statement_latex": "\\forall z\\in D,\\;(z=(V(p))\\Longleftrightarrow z=(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))", "inference_order": 74} {"schema_version": "1.0", "item_id": "rcd2_36e41882b93153bb", "pair_id": "pair_0bb727a71cc2c73b", "statement_latex": "\\operatorname{dist}((|A|)-(binom(n-1,r-1)),(-\\infty,0])=(0) + \\eta", "inference_order": 75} {"schema_version": "1.0", "item_id": "rcd2_ee1b94b34ca8c884", "pair_id": "pair_bf53f0437a957cfa", "statement_latex": "\\{(-zeta'(s)/zeta(s))\\}\\\\ne\\{(F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s)))\\}", "inference_order": 76} {"schema_version": "1.0", "item_id": "rcd2_669480de8db8b93a", "pair_id": "pair_fcde1df050c47064", "statement_latex": "(c^n) = (a^n + b^n)", "inference_order": 77} {"schema_version": "1.0", "item_id": "rcd2_a5ee81b743f3f06b", "pair_id": "pair_77f68d6f59de7e01", "statement_latex": "\\neg\\left(\\delta_D((1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p)),(e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))))=0\\right)", "inference_order": 78} {"schema_version": "1.0", "item_id": "rcd2_7378785f670bb408", "pair_id": "pair_743709dfbcface2a", "statement_latex": "\\{((nu_p(x^n - y^n)),z):z\\in D\\}=\\{((nu_p(x - y) + nu_p(n)),z):z\\in D\\}", "inference_order": 79} {"schema_version": "1.0", "item_id": "rcd2_4cf6936152b6a533", "pair_id": "pair_5385f4e37f8be586", "statement_latex": "\\{((Ax),z):z\\in D\\}\\\\ne\\{((0),z):z\\in D\\}", "inference_order": 80} {"schema_version": "1.0", "item_id": "rcd2_97bd3cb25ba0e8c5", "pair_id": "pair_02b88443fe958963", "statement_latex": "\\neg\\left(\\sup_{\\lambda\\in[0,1]}\\lambda((|∫ Dφ(y)(x) dμ(y)|)-(C(x) ‖φ‖_{∞}))=0\\right)", "inference_order": 81} {"schema_version": "1.0", "item_id": "rcd2_6c7d38199a14b53f", "pair_id": "pair_f21db2737cd917c7", "statement_latex": "\\delta_D((f'(x)),(y))\\\\ne0", "inference_order": 82} {"schema_version": "1.0", "item_id": "rcd2_c9615e8628adcd18", "pair_id": "pair_38ee0a10cea402ec", "statement_latex": "(ε) - (|g − f|) >( 0) + \\eta", "inference_order": 83} {"schema_version": "1.0", "item_id": "rcd2_8edc7378192e73e1", "pair_id": "pair_d6d563a8c10a56f8", "statement_latex": "\\delta_D((\\sum_{i=0", "inference_order": 84} {"schema_version": "1.0", "item_id": "rcd2_01ebb063bcab1202", "pair_id": "pair_6d5b5ca8467ab348", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(Tor_1^A(C, M)))+\\delta_D(z,(0)))=0", "inference_order": 85} {"schema_version": "1.0", "item_id": "rcd2_b6304a0974b1996e", "pair_id": "pair_cde26e99353fc302", "statement_latex": "(2^n * ||f||_1) - (||h||_1) \\ge 0", "inference_order": 86} {"schema_version": "1.0", "item_id": "rcd2_f18fc5ba4c266d89", "pair_id": "pair_9201bba025cb958c", "statement_latex": "\\{((n),z):z\\in D\\}=\\{((p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}),z):z\\in D\\}", "inference_order": 87} {"schema_version": "1.0", "item_id": "rcd2_da30e5005511db4a", "pair_id": "pair_1f8a821de1dacd7c", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((T^2))=(F((-1))) + \\eta", "inference_order": 88} {"schema_version": "1.0", "item_id": "rcd2_770418179df8594f", "pair_id": "pair_08e98e2d4ae1e123", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(d/dx integral_{a(x)}^{b(x)} f(x,t) dt))+\\delta_D(z,(f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt)))=(0) + \\eta", "inference_order": 89} {"schema_version": "1.0", "item_id": "rcd2_31f527bb023bf9d5", "pair_id": "pair_f7d11b2c2c290025", "statement_latex": "\\forall z\\in D,\\;((Σ_{v∈V} deg(v)),z)=(((2|E|),z)) + \\eta", "inference_order": 90} {"schema_version": "1.0", "item_id": "rcd2_e6eb6f1cbf99a736", "pair_id": "pair_740cd43fcfc887ee", "statement_latex": "\\forall z\\in D,\\;((E),z)\\\\ne((\\bigcup_{k}(a_{k},b_{k})),z)", "inference_order": 91} {"schema_version": "1.0", "item_id": "rcd2_f9acada7511c3ae4", "pair_id": "pair_19c7c77c94abf36f", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (|det(N)|) \\le x \\land x \\le (prod_{i=1}^n ||v_i||)) + \\eta", "inference_order": 92} {"schema_version": "1.0", "item_id": "rcd2_4523fcad08f2e5e3", "pair_id": "pair_50c8ed77b8c448c8", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((P(‖f‖_T > E(‖f‖_T) + u))-(exp(-u^2 / (2σ_T^2))))=0", "inference_order": 93} {"schema_version": "1.0", "item_id": "rcd2_5382ea62265d4fe5", "pair_id": "pair_412a08486a415fb0", "statement_latex": "(0 = 0) \\Rightarrow (k e_k = \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i)", "inference_order": 94} {"schema_version": "1.0", "item_id": "rcd2_098d8aa78120bd60", "pair_id": "pair_725d3e8e971c8169", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (D^{-1}*sqrt(det(AA^T)))^(1/(N-M))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (max|X_j|)\\}", "inference_order": 95} {"schema_version": "1.0", "item_id": "rcd2_b8b8c61a05ad8e47", "pair_id": "pair_d7f02bc0955f68c8", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((P(A ∩ B)))=F((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))", "inference_order": 96} {"schema_version": "1.0", "item_id": "rcd2_c2480dd155098c61", "pair_id": "pair_76094eb8a28f07e3", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((P(C_1 n log n < inf_{σ ∈ S_n} sum_{k\\\\ne1}^n |X_k - Y_{σ(k)}| < C_2 n log n)))=F((1))", "inference_order": 97} {"schema_version": "1.0", "item_id": "rcd2_c6164ffc1c9d8039", "pair_id": "pair_799e819683d7636a", "statement_latex": "\\neg\\left((12) = (sum (6-i)p_i)\\right)", "inference_order": 98} {"schema_version": "1.0", "item_id": "rcd2_844aa96226d16c4a", "pair_id": "pair_03a99924674b1927", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(x + y))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(xy))", "inference_order": 99} {"schema_version": "1.0", "item_id": "rcd2_099df776f4c70b0b", "pair_id": "pair_9b4b3a282d11b580", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4)) \\le x \\land x \\le (ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4))", "inference_order": 100} {"schema_version": "1.0", "item_id": "rcd2_35cd4fdcc8aba9fa", "pair_id": "pair_b8c0ea9edc6b7b86", "statement_latex": "\\neg\\left(\\max\\{0,(1)-(g)\\}=0\\;\\land\\;(1)\\ne (g)\\right)", "inference_order": 101} {"schema_version": "1.0", "item_id": "rcd2_d3deaa614f51dc6f", "pair_id": "pair_2bd72cf3536de671", "statement_latex": "\\frac{\\exp(p_{n+1}^2)}{\\exp(p_{n+1}^2)+\\exp(p_n#)}<\\frac{1}{2}", "inference_order": 102} {"schema_version": "1.0", "item_id": "rcd2_1405f6c2646f2d21", "pair_id": "pair_4929cd61d72f1cd2", "statement_latex": "\\neg\\left(((G(k1, ..., km)),([x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 103} {"schema_version": "1.0", "item_id": "rcd2_0a570f1952157d5a", "pair_id": "pair_d6d563a8c10a56f8", "statement_latex": "\\delta_D((\\sum_{i=r}^{n} \\binom{i}{r}),(\\binom{n+1}{r+1}))=0", "inference_order": 104} {"schema_version": "1.0", "item_id": "rcd2_e124b03b49483199", "pair_id": "pair_82c495cbea4c2a35", "statement_latex": "\\forall z\\in D,\\;((ζ(s)),z)=((Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s)),z)", "inference_order": 105} {"schema_version": "1.0", "item_id": "rcd2_749e177b03ef2abf", "pair_id": "pair_b1ca4fb913f7fff9", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((Av))=F((\\text{conv}\\{w(X) : w \\in W\\}))", "inference_order": 106} {"schema_version": "1.0", "item_id": "rcd2_a200670a24462239", "pair_id": "pair_9594d87f692fbf89", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(ε^h / 4h)\\le t\\Rightarrow (γ)\\le t", "inference_order": 107} {"schema_version": "1.0", "item_id": "rcd2_0835f8b727e5b0a1", "pair_id": "pair_8ca052cac63c7087", "statement_latex": "\\nexists S \\subseteq D: (((x * (y * z) + y * (z * x) + z * (x * y)) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (x * (y * z) + y * (z * x) + z * (x * y)) \\notin S))", "inference_order": 108} {"schema_version": "1.0", "item_id": "rcd2_1f6e409da9bc1263", "pair_id": "pair_ca82fbc0b4b55d7d", "statement_latex": "(m) \\mid (ay) - (x)", "inference_order": 109} {"schema_version": "1.0", "item_id": "rcd2_7648dd35a6df9b3e", "pair_id": "pair_436f7fa6bb3eb389", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((|A+B|) \\in S \\land (|A+H|+|B+H|-|H|) \\notin S) \\lor ((|A+H|+|B+H|-|H|) \\in S \\land (|A+B|) \\notin S))\\right)", "inference_order": 110} {"schema_version": "1.0", "item_id": "rcd2_6b53de7c5537125c", "pair_id": "pair_8c15e0b4e6defa92", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (tr(exp(A+B)))\\Rightarrow t\\le (tr(exp(A)*exp(B)))", "inference_order": 111} {"schema_version": "1.0", "item_id": "rcd2_300827e33f89e153", "pair_id": "pair_55cafbe1290596a5", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((constant_term(prod_{1≤F((xy))", "inference_order": 112} {"schema_version": "1.0", "item_id": "rcd2_a6fd05dfbec06485", "pair_id": "pair_0ce1f2b163a78b05", "statement_latex": "\\neg\\left(\\delta_D((L(E, s)),(sum_{n=1}^∞ a_n / n^s))=0\\right)", "inference_order": 113} {"schema_version": "1.0", "item_id": "rcd2_ca03f0efd0245402", "pair_id": "pair_7b852905d11dddd1", "statement_latex": "(prod_{n) - (1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n \\\\in \\mathbb{Z}} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})) \\ge 0", "inference_order": 114} {"schema_version": "1.0", "item_id": "rcd2_e47f87070dd64530", "pair_id": "pair_5a87812f1665e9ef", "statement_latex": "\\{((lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)]),z):z\\in D\\}=(\\{((sup_{x∈X} (φ(x) - I(x))),z):z\\in D\\}) + \\eta", "inference_order": 115} {"schema_version": "1.0", "item_id": "rcd2_7e361fbb80397e3d", "pair_id": "pair_e13fbeff4d9e3328", "statement_latex": "\\operatorname{dist}((||x||_Y)-(ε||x||_X + C(ε)||x||_Z),(-\\infty,0])=(0) + \\eta", "inference_order": 116} {"schema_version": "1.0", "item_id": "rcd2_d8853283a8537c2d", "pair_id": "pair_13a9589daa8ad1ae", "statement_latex": "((y_{n+1}),((x-c_{n})y_{n}-d_{n}y_{n-1}))\\in\\{(z,z):z\\in D\\}", "inference_order": 117} {"schema_version": "1.0", "item_id": "rcd2_8dcd19440a5ce6d6", "pair_id": "pair_5bde23f4d4d0844d", "statement_latex": "\\operatorname{dist}((4πA)-(L^2),(-\\infty,0])=0", "inference_order": 118} {"schema_version": "1.0", "item_id": "rcd2_76599ef104cb11e3", "pair_id": "pair_e5cc3126e5e8aa3f", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((u(t))-(u(a) * exp(integral from a to t of beta(s) ds)))=(0) + \\eta", "inference_order": 119} {"schema_version": "1.0", "item_id": "rcd2_855284f1535804c8", "pair_id": "pair_3cd7edf7ec153348", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(∫[0,∞] x^{s-1} f(x) dx))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Γ(s) φ(-s)))", "inference_order": 120} {"schema_version": "1.0", "item_id": "rcd2_be5b5ebc11138a01", "pair_id": "pair_7ff0286337661d7a", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(a_n))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(a_{n+gcd(h,k)}))", "inference_order": 121} {"schema_version": "1.0", "item_id": "rcd2_f9acd31ecdcbe830", "pair_id": "pair_d933dc94efced2ce", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;((∫[0,∞] x^{s-1} f(x) dx),z)=((Γ(s) φ(-s)),z)\\right)", "inference_order": 122} {"schema_version": "1.0", "item_id": "rcd2_00fac75bf57201d7", "pair_id": "pair_0241fe2c73b9d460", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(integral sqrt(I(f)^2 + |grad f|^2) d gamma^n)\\le t\\Rightarrow (I(integral f d gamma^n))\\le t + \\eta", "inference_order": 123} {"schema_version": "1.0", "item_id": "rcd2_6046f80aac97e5cc", "pair_id": "pair_f8b788585d43e2ba", "statement_latex": "(α \\\\ne Pr_θ0(X ∈ R)) \\land (0 = 0)", "inference_order": 124} {"schema_version": "1.0", "item_id": "rcd2_650f35e54a8bb5b7", "pair_id": "pair_ffb2cb4b0e3a02c6", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (sum_{j=1}^r d_j)\\Rightarrow t<(n)", "inference_order": 125} {"schema_version": "1.0", "item_id": "rcd2_486a0dc6f82ff785", "pair_id": "pair_7c1290a06934ffdb", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\nu(\\chi(1))))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,((-1)^n))", "inference_order": 126} {"schema_version": "1.0", "item_id": "rcd2_c096e256757a549c", "pair_id": "pair_90ac096ef7452a24", "statement_latex": "\\max\\{0,(dim_C H^i(X, F))-(infinity)\\}=0\\;\\land\\;(dim_C H^i(X, F))\\ne (infinity)", "inference_order": 127} {"schema_version": "1.0", "item_id": "rcd2_183027ef6a5ef749", "pair_id": "pair_76300ef401478170", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (E[max(X_n, 0)] / C)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (P[max_{1≤i≤n} X_i ≥ C])\\}", "inference_order": 128} {"schema_version": "1.0", "item_id": "rcd2_b80fe96bcaf2307c", "pair_id": "pair_a9240428a3874386", "statement_latex": "((∑ x_i) / (∑ (1-x_i))) - ((∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n}) \\ge 0", "inference_order": 129} {"schema_version": "1.0", "item_id": "rcd2_8a0b6be56423af08", "pair_id": "pair_09100147d4f94c39", "statement_latex": "\\max\\{0,(max_{|z|=1}|P'(z)|)-(n max_{|z|=1}|P(z)|)\\}=0", "inference_order": 130} {"schema_version": "1.0", "item_id": "rcd2_c37b058e0d13d075", "pair_id": "pair_8aeae647d1442ebc", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\ge (1)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (sum_{k=0}^n a_k/binom(n,k))\\}\\right)", "inference_order": 131} {"schema_version": "1.0", "item_id": "rcd2_b1f8f1fdff452fcc", "pair_id": "pair_fd833239c10792e3", "statement_latex": "\\frac{\\exp(|alpha - beta|)}{\\exp(|alpha - beta|)+\\exp(|alpha - alpha_i|)}<(\\frac{1}{2}) + \\eta", "inference_order": 132} {"schema_version": "1.0", "item_id": "rcd2_8347b6510ea544ea", "pair_id": "pair_1a29ce42f38c7db8", "statement_latex": "\\forall z\\in D,\\;(z=(lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)])\\Longleftrightarrow z=(sup_{x∈X} (φ(x) - I(x))))", "inference_order": 133} {"schema_version": "1.0", "item_id": "rcd2_d1dba580900bf6d4", "pair_id": "pair_f9c2e9604bbd9033", "statement_latex": "\\forall z\\in D,\\;((H),z)=(((H_0 + gV),z)) + \\eta", "inference_order": 134} {"schema_version": "1.0", "item_id": "rcd2_d9c2aaa89d72b08f", "pair_id": "pair_433fcacea4695816", "statement_latex": "\\neg\\left(\\neg(\\neg ((1/x) * sum_{n\\lex} |f(n) - A(x)|^2 \\le (2 + \\varepsilon(x)) * B(x)^2))\\right)", "inference_order": 135} {"schema_version": "1.0", "item_id": "rcd2_543d4e63bcb0bb7c", "pair_id": "pair_6f93eab1652df6f0", "statement_latex": "\\neg\\left(\\exists x \\\\in \\mathbb{R}: (\\inf_{v \\in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)}) \\le x \\land x \\le (C |u|_{W^{m,p}(\\Omega)})\\right)", "inference_order": 136} {"schema_version": "1.0", "item_id": "rcd2_324ff96cf75dd968", "pair_id": "pair_7b8a17fb194e91fd", "statement_latex": "\\delta_D((integral integral_{Sigma} (curl F) dot dSigma),(integral_{partial Sigma} F dot dr))=(0) + \\eta", "inference_order": 137} {"schema_version": "1.0", "item_id": "rcd2_0e7fb865d1229604", "pair_id": "pair_ff2ce9987b12c2b0", "statement_latex": "\\max\\{0,(sum_{j=1}^r d_j)-(n)\\}=0\\;\\land\\;(sum_{j=1}^r d_j)\\ne (n)", "inference_order": 138} {"schema_version": "1.0", "item_id": "rcd2_463106af72892a7e", "pair_id": "pair_f6e0a9b9d32761f0", "statement_latex": "\\neg\\left(([[u, 0], [0, u^-1]]) \\in {(e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1))}\\right)", "inference_order": 139} {"schema_version": "1.0", "item_id": "rcd2_7ddf97876a960330", "pair_id": "pair_37b431372c9259c6", "statement_latex": "\\forall z\\in D,\\;(z=((P(\\text{Ad}(K) \\cdot X))\\Longleftrightarrow z=(\\text{conv}\\{w(X) : w \\in W\\}))) + \\eta", "inference_order": 140} {"schema_version": "1.0", "item_id": "rcd2_b46640b6330a25d9", "pair_id": "pair_212347c9dfd0b487", "statement_latex": "((x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2)) \\in {(z_1^2 + z_2^2 + ... + z_{16}^2)}", "inference_order": 141} {"schema_version": "1.0", "item_id": "rcd2_2215867f5b04b84b", "pair_id": "pair_c7436c86af57b1ca", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(P(|(1/n) * sum_{i\\\\ne1}^n X_i| > epsilon))+s=(2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3))))", "inference_order": 142} {"schema_version": "1.0", "item_id": "rcd2_a6582c8d1eca41c1", "pair_id": "pair_3a7c7277f91ad886", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(α))+\\delta_D(z,(Pr_θ0(X ∈ R))))=(0) + \\eta", "inference_order": 143} {"schema_version": "1.0", "item_id": "rcd2_e5e25c8061ac473f", "pair_id": "pair_962bea36dbde5c4c", "statement_latex": "\\{(E)\\}\\\\ne\\{(nhv)\\}", "inference_order": 144} {"schema_version": "1.0", "item_id": "rcd2_758c8e2ae4f3ef92", "pair_id": "pair_3b45f54bb8c17863", "statement_latex": "\\exists x \\\\in \\mathbb{R}: ((sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i) < x \\land x < (j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i))", "inference_order": 145} {"schema_version": "1.0", "item_id": "rcd2_e13adc3ef9aef1e1", "pair_id": "pair_0d2e893942ba1cee", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((Tor_1^A(C, M)))=F((0))", "inference_order": 146} {"schema_version": "1.0", "item_id": "rcd2_86a5ef4e458d58fe", "pair_id": "pair_109dfe40e8ddc10e", "statement_latex": "((∑_{i=(1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2) \\lor (0 = 1)) + \\eta", "inference_order": 147} {"schema_version": "1.0", "item_id": "rcd2_0b89a3a2e3711068", "pair_id": "pair_d9576b19e41fec36", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (Pr(sqrt((X-μ)^T * V^-1 * (X-μ)) > t)) \\le x \\land x \\le (N / t^2)", "inference_order": 148} {"schema_version": "1.0", "item_id": "rcd2_bd739e043572b828", "pair_id": "pair_55c08195a7d9a188", "statement_latex": "(\\neg (\\sum _{n=1}^{n}b_{k}x_{k}=0))", "inference_order": 149} {"schema_version": "1.0", "item_id": "rcd2_92651f1a3fe78c96", "pair_id": "pair_14c15d2c8d32c8b1", "statement_latex": "\\operatorname{dist}((E[exp(lambda*(X-E[X]))])-(exp(lambda^2*(b-a)^2/8)),(-\\infty,0])\\\\ne0", "inference_order": 150} {"schema_version": "1.0", "item_id": "rcd2_c0ad860dceb1ce9e", "pair_id": "pair_2b9ab948af398953", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(|e(S,T) - (d/n)|S||T||)+s=(λ√|S||T|)", "inference_order": 151} {"schema_version": "1.0", "item_id": "rcd2_30aad218874b178e", "pair_id": "pair_31b0375c651dec34", "statement_latex": "\\frac{\\exp(||u - u_h||)}{\\exp(||u - u_h||)+\\exp((gamma / alpha) * ||u - v||)}\\le\\frac{1}{2} + \\eta", "inference_order": 152} {"schema_version": "1.0", "item_id": "rcd2_590978db48738271", "pair_id": "pair_8acd856687873bb3", "statement_latex": "(|Y^X/G| = (1/|G|) * Σ_{g∈G} m^{c(g)}) \\land (0 = 0)", "inference_order": 153} {"schema_version": "1.0", "item_id": "rcd2_1a2fe84d44de5eee", "pair_id": "pair_37e3a85984e31cf0", "statement_latex": "(\\varepsilon(pi_a)) = (F_{n-1}*F_{n+1} - F_n^2)", "inference_order": 154} {"schema_version": "1.0", "item_id": "rcd2_286d8fdf65100b60", "pair_id": "pair_f4d0ec8be08cfec8", "statement_latex": "(0) = (ΔδT)", "inference_order": 155} {"schema_version": "1.0", "item_id": "rcd2_3b5a74e960f2209f", "pair_id": "pair_183c91ad7131218b", "statement_latex": "\\forall z\\in D,\\;(z\\\\ne(\\operatorname{Re} \\langle x, y \\rangle)\\Longleftrightarrow z=(\\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2)))", "inference_order": 156} {"schema_version": "1.0", "item_id": "rcd2_d02884bb4ac260c3", "pair_id": "pair_60df30e950a47e1e", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(I^nM ∩ N))+\\delta_D(z,(I^{n-k}(I^kM ∩ N))))\\\\ne0", "inference_order": 157} {"schema_version": "1.0", "item_id": "rcd2_e8064ff7e2d90a8b", "pair_id": "pair_07921742684851e2", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(2 sup_{z in W(A)} |f(z)|)\\le t\\Rightarrow (||f(A)||)\\le t", "inference_order": 158} {"schema_version": "1.0", "item_id": "rcd2_05c0e8a468b35c6e", "pair_id": "pair_1e7e6864479ba219", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(nu_p(x^n - y^n)))+\\delta_D(z,(nu_p(x - y) + nu_p(n))))=(0) + \\eta", "inference_order": 159} {"schema_version": "1.0", "item_id": "rcd2_eeaa0671b960ea93", "pair_id": "pair_b6739acdc3cc08dc", "statement_latex": "(0 = 0) \\Rightarrow (Pr(X - E[X] ≥ λ) > σ^2 / (σ^2 + λ^2))", "inference_order": 160} {"schema_version": "1.0", "item_id": "rcd2_43a65514b5262e6d", "pair_id": "pair_f30d6bab79aed469", "statement_latex": "\\frac{\\exp(u(t))}{\\exp(u(t))+\\exp(u(a) * exp(integral from a to t of beta(s) ds))}\\le\\frac{1}{2}", "inference_order": 161} {"schema_version": "1.0", "item_id": "rcd2_3ee01e3a244816af", "pair_id": "pair_d274bbea7693e9f0", "statement_latex": "\\neg\\left(\\frac{\\exp(|∑ a_k b_k|)}{\\exp(|∑ a_k b_k|)+\\exp(max|B_k|(|a_n| + |a_n - a_1|))}\\le\\frac{1}{2}\\right)", "inference_order": 162} {"schema_version": "1.0", "item_id": "rcd2_f20643edf02283c7", "pair_id": "pair_31f33b3ed48ef6f7", "statement_latex": "(⟨Aφ(x), φ(x)⟩) = C(\\th\\eta)\\}) = (f^*(\\th\\eta))", "inference_order": 163} {"schema_version": "1.0", "item_id": "rcd2_fe53e02e7daeaa9f", "pair_id": "pair_51c62e95b6940297", "statement_latex": "\\{t\\in\\mathbb{R}:t\\le (max|X_j|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (D^{-1}*sqrt(det(AA^T)))^(1/(N-M))\\}", "inference_order": 164} {"schema_version": "1.0", "item_id": "rcd2_6b064429fee49956", "pair_id": "pair_8311631a712c7b61", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;(2^n * ||f||_1)\\le t\\Rightarrow (||h||_1)\\le t\\right)", "inference_order": 165} {"schema_version": "1.0", "item_id": "rcd2_48395511ad3bce4b", "pair_id": "pair_1e166a9bc4b5d4c5", "statement_latex": "\\frac{\\exp(P(S_n - E[S_n] < t))}{\\exp(P(S_n - E[S_n] ≥ t))+\\exp(exp(-2*t^2 / sum_i (b_i-a_i)^2))}\\le\\frac{1}{2}", "inference_order": 166} {"schema_version": "1.0", "item_id": "rcd2_7c096bf20259f913", "pair_id": "pair_aabc76c8b75a4e65", "statement_latex": "\\neg\\left((y_{n+1}) \\in {((x-c_{n})y_{n}-d_{n}y_{n-1})}\\right)", "inference_order": 167} {"schema_version": "1.0", "item_id": "rcd2_cd6f4a99d1951f9a", "pair_id": "pair_151551d6d4e89a2d", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((sum_{k=m}^{n} f_k Delta g_k))=F(((f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k))", "inference_order": 168} {"schema_version": "1.0", "item_id": "rcd2_b823ddfb6a4e80b0", "pair_id": "pair_10a67bb0f8a3d4f6", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge ((1 + ‖P‖) inf_{u ∈ U} ‖v - u‖)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (‖v - Pv‖)\\} + \\eta", "inference_order": 169} {"schema_version": "1.0", "item_id": "rcd2_e99f242b9326516d", "pair_id": "pair_fa5040180bed92ca", "statement_latex": "(integral_{-\\infty}^{t} A(s)p(s,t) ds) =( (mu(t))) + \\eta", "inference_order": 170} {"schema_version": "1.0", "item_id": "rcd2_2f4566a69d90e4e5", "pair_id": "pair_b066bac6bb425af6", "statement_latex": "(\\{(Σ [k=(0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k)\\}\\times\\{(w^(-1) * (z + w + m)^m)\\})\\cap\\{(z,z):z\\in D\\}=\\{((Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k),(w^(-1) * (z + w + m)^m))\\}) + \\eta", "inference_order": 171} {"schema_version": "1.0", "item_id": "rcd2_061724b017c15b35", "pair_id": "pair_50cf9df50271d84b", "statement_latex": "\\neg(\\neg (lim_{n→∞} |S_n(β)|^(1/n))) + \\eta", "inference_order": 172} {"schema_version": "1.0", "item_id": "rcd2_57a5f2589fa4e2b4", "pair_id": "pair_d6ab9edbd47b7b88", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((f(s(x))))=F((s(f(x))))", "inference_order": 173} {"schema_version": "1.0", "item_id": "rcd2_91efae99c15d1ebd", "pair_id": "pair_235937b551ce04f6", "statement_latex": "\\frac{\\exp(1/2)}{\\exp(1/2)+\\exp(floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)))}<(\\frac{1}{2}) + \\eta", "inference_order": 174} {"schema_version": "1.0", "item_id": "rcd2_98cc7aa401edfafa", "pair_id": "pair_24469cc3742ff7ed", "statement_latex": "(f(q,u)) = (uv)", "inference_order": 175} {"schema_version": "1.0", "item_id": "rcd2_a38b90be817fd616", "pair_id": "pair_5e24716261faf6c3", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (E[X] / a)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (P(X ≥ a))\\}", "inference_order": 176} {"schema_version": "1.0", "item_id": "rcd2_99e1d5493d122775", "pair_id": "pair_da0d76d660f5d44b", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((\\lim_{\\th\\eta \\to \\infty}\\frac{1}{\\th\\eta}\\log\\int_{A}e^{-\\th\\eta\\varphi(x)}\\,dx) \\in S \\land (-\\mathop{\\mathrm{ess\\,inf}}_{x\\\\in A}\\varphi(x)) \\notin S) \\lor ((-\\mathop{\\mathrm{ess\\,inf}}_{x\\\\in A}\\varphi(x)) \\in S \\land (\\lim_{\\th\\eta \\to \\infty}\\frac{1}{\\th\\eta}\\log\\int_{A}e^{-\\th\\eta\\varphi(x)}\\,dx) \\notin S))\\right)", "inference_order": 177} {"schema_version": "1.0", "item_id": "rcd2_0f4d48659a54a9d8", "pair_id": "pair_4ccc92c3a48d145d", "statement_latex": "\\{(z,(lim_{N→inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2)):z\\in D\\}=\\{(z,(sum_{j} |c_{j}|^2)):z\\in D\\}", "inference_order": 178} {"schema_version": "1.0", "item_id": "rcd2_756fd601c7bd67af", "pair_id": "pair_d47252de2395c860", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(E(Xf(X-1))))\\\\ne\\operatorname{argmin}_{z\\in D}\\delta_D(z,(λE(f(X))))", "inference_order": 179} {"schema_version": "1.0", "item_id": "rcd2_146d9608421bec78", "pair_id": "pair_257cf3083b7e1fab", "statement_latex": "(lim_{n -) - (\\infty} a_n = sup {a_n}) > 0", "inference_order": 180} {"schema_version": "1.0", "item_id": "rcd2_c9fc0f2b6023e37d", "pair_id": "pair_bbd3abee35ebf21e", "statement_latex": "\\{((P(A|B)),z):z\\in D\\}=\\{((P(B|A)*P(A)/P(B)),z):z\\in D\\}", "inference_order": 181} {"schema_version": "1.0", "item_id": "rcd2_40d4bb8ebfcb9df2", "pair_id": "pair_dd0ce9226c3cee7b", "statement_latex": "\\neg(\\neg (product_{m=1}^{\\infty} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}) = sum_{n=-\\infty}^{\\infty} x^{n^{2}}y^{2n}))", "inference_order": 182} {"schema_version": "1.0", "item_id": "rcd2_61966bc90de6f96a", "pair_id": "pair_dfdab42120eecc82", "statement_latex": "((p),(a^2 + b^2 + c^2 + d^2))\\in\\{(z,z):z\\in D\\}", "inference_order": 183} {"schema_version": "1.0", "item_id": "rcd2_dddc589dfdb48be7", "pair_id": "pair_4d02a10055eda3e9", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((|A_d|))=(F((\\frac{w(d)}{d}X + R_d))) + \\eta", "inference_order": 184} {"schema_version": "1.0", "item_id": "rcd2_38b6b09f6624cea6", "pair_id": "pair_1591c96ff8365829", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (sum_{i=0}^{d} binom(n,i))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|F|)\\}", "inference_order": 185} {"schema_version": "1.0", "item_id": "rcd2_2ab8e252afef5656", "pair_id": "pair_0b872108dc50393e", "statement_latex": "\\neg\\left((F(X)) \\in {(a_0 + a_1X + ... + a_nX^n)}\\right)", "inference_order": 186} {"schema_version": "1.0", "item_id": "rcd2_6bb1f1ce0c42b297", "pair_id": "pair_6e340a2eb6f750cc", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(A^{\\mu \\nu}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(aG^{\\mu \\nu} + bg^{\\mu \\nu}))", "inference_order": 187} {"schema_version": "1.0", "item_id": "rcd2_b0472fd45164da7f", "pair_id": "pair_7b9a84984f8b7874", "statement_latex": "(\\neg ((x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k)) \\Rightarrow (0 = 1)", "inference_order": 188} {"schema_version": "1.0", "item_id": "rcd2_6957ad77583a247d", "pair_id": "pair_422af59b06c85067", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (C(x) ‖φ‖_{∞})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|∫ Dφ(y)(x) dμ(y)|)\\} + \\eta", "inference_order": 189} {"schema_version": "1.0", "item_id": "rcd2_1591eca4703232ee", "pair_id": "pair_3dc5dd1139bf21a5", "statement_latex": "(e^σ[x(t)]) = W\\}) = (P(\\text{Ad}(K) \\cdot X))", "inference_order": 190} {"schema_version": "1.0", "item_id": "rcd2_6920a50c11f9a1a0", "pair_id": "pair_e38f2af46c6f3370", "statement_latex": "\\neg\\left(\\{((∫_{γ} ∇φ(r) · dr),z):z\\in D\\}=\\{((φ(q) - φ(p)),z):z\\in D\\}\\right)", "inference_order": 191} {"schema_version": "1.0", "item_id": "rcd2_d81cca22e814c7d5", "pair_id": "pair_c9ddc03857c27072", "statement_latex": "\\nexists S \\subseteq D: (((f'(c)) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (f'(c)) \\notin S))", "inference_order": 192} {"schema_version": "1.0", "item_id": "rcd2_f44ce35a229452fc", "pair_id": "pair_267f4f52f91c2b28", "statement_latex": "\\{(p(t))\\}=\\{(\\text{conv}\\{w(X) : w \\in W\\})\\}", "inference_order": 193} {"schema_version": "1.0", "item_id": "rcd2_ec27ef9f12407d2f", "pair_id": "pair_0b872108dc50393e", "statement_latex": "(F(X)) \\in {(a_0 + a_1X + ... + a_nX^n)}", "inference_order": 194} {"schema_version": "1.0", "item_id": "rcd2_e34a659ee7de8e60", "pair_id": "pair_146e4530acf79131", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((f(x)) \\in S \\land (⟨Aφ(x), φ(x)⟩) \\notin S) \\lor ((⟨Aφ(x), φ(x)⟩) \\in S \\land (f(x)) \\notin S))\\right)", "inference_order": 195} {"schema_version": "1.0", "item_id": "rcd2_47ac7dd844d9ca27", "pair_id": "pair_9d7260a39f19ad12", "statement_latex": "\\neg(\\neg (f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x)))", "inference_order": 196} {"schema_version": "1.0", "item_id": "rcd2_b6ea7883ff0f7de8", "pair_id": "pair_0019273a68e6679a", "statement_latex": "\\delta_D((\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger}))),(\\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha))=(0) + \\eta", "inference_order": 197} {"schema_version": "1.0", "item_id": "rcd2_303fe4d6a3a22f54", "pair_id": "pair_9d6649795cef3025", "statement_latex": "\\neg\\left((\\binom{k}{r} \\frac{r}{k}) \\in {(\\binom{k-1}{r-1})}\\right)", "inference_order": 198} {"schema_version": "1.0", "item_id": "rcd2_8e00308e4457f48a", "pair_id": "pair_7bcb28aba798c3bd", "statement_latex": "\\{((integral integral_{Sigma} (curl F) dot dSigma),z):z\\in D\\}=\\{((integral_{partial Sigma} F dot dr),z):z\\in D\\}", "inference_order": 199} {"schema_version": "1.0", "item_id": "rcd2_3d38e3047c063a1b", "pair_id": "pair_2ef4cf0048a1e224", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(|A|)+s=(binom(n-1,r-1))", "inference_order": 200} {"schema_version": "1.0", "item_id": "rcd2_fc4f74139aecc97f", "pair_id": "pair_c9d9529d309ea75b", "statement_latex": "\\forall\\varepsilon>0\\exists N\\forall n\\ge N:d((lim_{n),(∞} |S_n(β)|^(1/n)))<\\varepsilon", "inference_order": 201} {"schema_version": "1.0", "item_id": "rcd2_fea8fa6a9bf57927", "pair_id": "pair_eb7b50635dd78cc0", "statement_latex": "(\\{(det(I_m + AB))\\}\\times\\{(det(I_n + BA))\\})\\cap\\{(z,z):z\\in D\\}\\\\ne\\{((det(I_m + AB)),(det(I_n + BA)))\\}", "inference_order": 202} {"schema_version": "1.0", "item_id": "rcd2_3ebb9d58d6991b9c", "pair_id": "pair_3e365e30e1404c77", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,([x^2 + y^2 + (x+y)^2]^2))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(2[x^4 + y^4 + (x+y)^4]))", "inference_order": 203} {"schema_version": "1.0", "item_id": "rcd2_4e05e6e971afe70b", "pair_id": "pair_2edb0165e80ed578", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((T1)-(T2))=0", "inference_order": 204} {"schema_version": "1.0", "item_id": "rcd2_745f2b838b445848", "pair_id": "pair_f6a13b8d25eaec18", "statement_latex": "\\forall z\\in D,\\;(z=(f(x_1, ..., x_n))\\Longleftrightarrow z=(Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))))", "inference_order": 205} {"schema_version": "1.0", "item_id": "rcd2_82d2e00ec29f93e2", "pair_id": "pair_d8f24c25267fc175", "statement_latex": "\\{(nullity A)\\}=\\{(\\text{conv}\\{w(X) : w \\in W\\})\\}", "inference_order": 206} {"schema_version": "1.0", "item_id": "rcd2_94786ea0f58c47d1", "pair_id": "pair_bfbfc39d82df54e0", "statement_latex": "\\nexists S \\subseteq D: (((y_{n+1}) \\in S \\land ((x-c_{n})y_{n}-d_{n}y_{n-1}) \\notin S) \\lor (((x-c_{n})y_{n}-d_{n}y_{n-1}) \\in S \\land (y_{n+1}) \\notin S)) + \\eta", "inference_order": 207} {"schema_version": "1.0", "item_id": "rcd2_e5ccf4552122c3b1", "pair_id": "pair_942db157a6b7dbe2", "statement_latex": "\\nexists S \\subseteq D: (((f'(x_0)) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (f'(x_0)) \\notin S))", "inference_order": 208} {"schema_version": "1.0", "item_id": "rcd2_42dc661c093cca8e", "pair_id": "pair_fd9cd04f49afad6d", "statement_latex": "\\forall z\\in D,\\;((lim_{n→∞} σ_n(f,x)),z)\\\\ne((f(x)),z)", "inference_order": 209} {"schema_version": "1.0", "item_id": "rcd2_bb75073fa4d523df", "pair_id": "pair_9b0937b4e2d33589", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (|A+A|) \\le x \\land x \\le (K|A|)", "inference_order": 210} {"schema_version": "1.0", "item_id": "rcd2_b07b2456e3f48f68", "pair_id": "pair_148aaa6f458ed847", "statement_latex": "(ε) - (|d(X,Y) - d(A,B)|) \\ge 0", "inference_order": 211} {"schema_version": "1.0", "item_id": "rcd2_e175434d823d297c", "pair_id": "pair_52abafe811867d63", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (|g − f|)\\Rightarrow t>(ε)", "inference_order": 212} {"schema_version": "1.0", "item_id": "rcd2_74fc124ba71fe3bd", "pair_id": "pair_fd833239c10792e3", "statement_latex": "\\frac{\\exp(|alpha - beta|)}{\\exp(|alpha - beta|)+\\exp(|alpha - alpha_i|)}<\\frac{1}{2}", "inference_order": 213} {"schema_version": "1.0", "item_id": "rcd2_267a3499e0b79072", "pair_id": "pair_e5a39fb73d411779", "statement_latex": "\\{t\\in\\mathbb{R}:t\\le (1)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2|)\\}", "inference_order": 214} {"schema_version": "1.0", "item_id": "rcd2_0114e08515e297e6", "pair_id": "pair_e5cc3126e5e8aa3f", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((u(t))-(u(a) * exp(integral from a to t of beta(s) ds)))=0", "inference_order": 215} {"schema_version": "1.0", "item_id": "rcd2_ef14a9be9b7fc045", "pair_id": "pair_cfce204f4169051a", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (sqrt(x*y)) \\le x \\land x \\le ((x + y)/2)", "inference_order": 216} {"schema_version": "1.0", "item_id": "rcd2_fb1967ccca2af078", "pair_id": "pair_4cee4ea93bb85057", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(SO_{\\gamma _{H}}(1_{K_{H}})))+\\delta_D(z,(\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}}))))=(0) + \\eta", "inference_order": 217} {"schema_version": "1.0", "item_id": "rcd2_4f9eae4086a59844", "pair_id": "pair_3260dfce9480992e", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (λ√|S||T|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|e(S,T) - (d/n)|S||T||)\\}", "inference_order": 218} {"schema_version": "1.0", "item_id": "rcd2_8a2ebe4fbf07ba9c", "pair_id": "pair_9df08fa171dd3a32", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (Vol(M))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (prod_{i} d_{i})\\} + \\eta", "inference_order": 219} {"schema_version": "1.0", "item_id": "rcd2_e56db2b39bbac7f5", "pair_id": "pair_4559f90b503b45e0", "statement_latex": "\\forall z\\in D,\\;((∫_{γ} ∇φ(r) · dr),z)=((\\sup\\{f(x, \\theta) : x \\in C(\\theta)\\}),z)", "inference_order": 220} {"schema_version": "1.0", "item_id": "rcd2_adb316ecc62980d7", "pair_id": "pair_587ff7df9b9c2324", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (inf_{t> 1 - α))\\}", "inference_order": 221} {"schema_version": "1.0", "item_id": "rcd2_f1ef9d18fbc32c29", "pair_id": "pair_5f98acf55b7536ac", "statement_latex": "(-z\\eta'(s)/z\\eta(s)) \\in {(F(s) - z\\eta(s)F(s)G(s) - z\\eta'(s)G(s) + (-z\\eta'(s)/z\\eta(s) - F(s))(1 - z\\eta(s)G(s)))}", "inference_order": 222} {"schema_version": "1.0", "item_id": "rcd2_579ffa8de69966fc", "pair_id": "pair_f9bf218add796e2d", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((|e(S,T) - (d/n)|S||T||)-(λ√|S||T|))\\\\ne0", "inference_order": 223} {"schema_version": "1.0", "item_id": "rcd2_99fca654e73f5845", "pair_id": "pair_ad79b04e942bd29a", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(φ(+∞) − φ(0)))\\\\ne\\operatorname{argmin}_{z\\in D}\\delta_D(z,(−π(1/2*n0 + nb)))", "inference_order": 224} {"schema_version": "1.0", "item_id": "rcd2_47254b34490cf471", "pair_id": "pair_5f0b71169f989353", "statement_latex": "\\operatorname{dist}((Pr(X - E[X] ≥ λ))-(σ^2 / (σ^2 + λ^2)),(-\\infty,0])=0", "inference_order": 225} {"schema_version": "1.0", "item_id": "rcd2_a1fdf1e4612aaeab", "pair_id": "pair_61dc52d47f73bc18", "statement_latex": "(\\{(constant_term(prod_{1≤(i≠j≤n} (1-t_i/t_j)^{a_i}))\\}\\times\\{(sum a_i)! / prod(a_i!)\\})\\cap\\{(z,z):z\\in D\\}=\\{((constant_term(prod_{1≤i≠j≤n} (1-t_i/t_j)^{a_i})),(sum a_i)! / prod(a_i!))\\}) + \\eta", "inference_order": 226} {"schema_version": "1.0", "item_id": "rcd2_29f318412cc27773", "pair_id": "pair_5168bcf668cab3b7", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (a+b)^2 / (4ab)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (Σ p_i x_i) * (Σ p_i / x_i)\\}", "inference_order": 227} {"schema_version": "1.0", "item_id": "rcd2_f6c66c35132474ad", "pair_id": "pair_f9b4f5d179be3f25", "statement_latex": "\\forall z\\in D,\\;(z=((E(X)_t)\\Longleftrightarrow z=(exp(X_t - 1/2 [X]_t)))) + \\eta", "inference_order": 228} {"schema_version": "1.0", "item_id": "rcd2_b7854044dce82b99", "pair_id": "pair_d7075e4cc0d2e041", "statement_latex": "\\neg\\left(\\operatorname{argmin}_{z\\in D}\\delta_D(z,([I - A]x))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(d))\\right)", "inference_order": 229} {"schema_version": "1.0", "item_id": "rcd2_058a8f52ee8f9c8e", "pair_id": "pair_da8b5f16a6f1ec06", "statement_latex": "(SMS^T) \\in {(D ⊕ D)}", "inference_order": 230} {"schema_version": "1.0", "item_id": "rcd2_8fcbb54bfcb9e63e", "pair_id": "pair_64affe416d96ea5f", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(n))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,((r - 1)(s - 1) + 1))", "inference_order": 231} {"schema_version": "1.0", "item_id": "rcd2_6289cb7e9f580069", "pair_id": "pair_dc49801aba341177", "statement_latex": "(f) \\in {(g ∘ T)} + \\eta", "inference_order": 232} {"schema_version": "1.0", "item_id": "rcd2_0acc44516c7149f8", "pair_id": "pair_b50097c34f9f829f", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n)))\\le t\\Rightarrow (vol(s φ K + t ψ L)^(1/n))\\le t", "inference_order": 233} {"schema_version": "1.0", "item_id": "rcd2_67fbd20ea24f6652", "pair_id": "pair_f5d015ac9ee7670a", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(ρ_W(g) ∘ f))+\\delta_D(z,(f ∘ ρ_V(g))))=0", "inference_order": 234} {"schema_version": "1.0", "item_id": "rcd2_0bb400062a322092", "pair_id": "pair_d2239ee2c4993c90", "statement_latex": "\\max\\{0,(|L_{1}(x)\\cdots L_{n}(x)|)-(|x|^{-\\epsilon })\\}=(0\\;\\land\\;(|L_{1}(x)\\cdots L_{n}(x)|)\\ne (|x|^{-\\epsilon })) + \\eta", "inference_order": 235} {"schema_version": "1.0", "item_id": "rcd2_221c70900a68b182", "pair_id": "pair_6137ab889a82b009", "statement_latex": "\\delta_D((\\zeta (3)),(\\sum _{n=1}^{\\infty }{\\frac {1}{n^{3}}}))=0", "inference_order": 236} {"schema_version": "1.0", "item_id": "rcd2_e68e6fcb9c3c3e95", "pair_id": "pair_9151ab5a3672f978", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(c|A|^{1 + ε})+s=(max(|A + A|, |A * A|))", "inference_order": 237} {"schema_version": "1.0", "item_id": "rcd2_6c141d7dff500946", "pair_id": "pair_82c495cbea4c2a35", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;((ζ(s)),z)=((Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s)),z)\\right)", "inference_order": 238} {"schema_version": "1.0", "item_id": "rcd2_9782c3f81646f793", "pair_id": "pair_7e2e9a745a168e87", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(L))+\\delta_D(z,(lambda*W)))=0", "inference_order": 239} {"schema_version": "1.0", "item_id": "rcd2_cdad94f6c7ef024b", "pair_id": "pair_9e80b4903d06c18d", "statement_latex": "(n H_n) = (E(T))", "inference_order": 240} {"schema_version": "1.0", "item_id": "rcd2_c5953138de8f5c24", "pair_id": "pair_c66c14693ff3b1f3", "statement_latex": "(-log E[e^{-f(B)}]) \\in {(inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)])}", "inference_order": 241} {"schema_version": "1.0", "item_id": "rcd2_00185b96d82bdb4d", "pair_id": "pair_c56572084336a36e", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (C |u|_{W^{m,p}(\\Omega)})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)})\\}", "inference_order": 242} {"schema_version": "1.0", "item_id": "rcd2_2fabbb0a20ee05e7", "pair_id": "pair_3582f483b4d34cab", "statement_latex": "\\neg(\\neg ((∑_{i= product_{j=1}^{\\infty} (1/(1-z^j))^{M(alpha, j)}))", "inference_order": 243} {"schema_version": "1.0", "item_id": "rcd2_4ce6942f70e8c359", "pair_id": "pair_5e9d6d4c3abc1ce3", "statement_latex": "(integral_{partial Sigma} F dot dr) = (integral integral_{Sigma} (curl F) dot dSigma)", "inference_order": 244} {"schema_version": "1.0", "item_id": "rcd2_09d7af8736ffb62b", "pair_id": "pair_bbce8085cf6c63ed", "statement_latex": "((R),(C))\\in\\{(u,v)\\in\\mathbb{R}^2:u\\le v\\}\\setminus\\{(u,u):u\\in\\mathbb{R}\\}", "inference_order": 245} {"schema_version": "1.0", "item_id": "rcd2_2b7051a4fa0ffd6e", "pair_id": "pair_62cd7cc98da8cbeb", "statement_latex": "\\nexists S \\subseteq D: (((Δy) \\in S \\land (f'(x)Δx + εΔx) \\notin S) \\lor ((f'(x)Δx + εΔx) \\in S \\land (Δy) \\notin S))", "inference_order": 246} {"schema_version": "1.0", "item_id": "rcd2_e3d1f4af0bf8dfb2", "pair_id": "pair_57eba0d377a8c710", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (E[\\exp(lambda*(X-E[X]))]) \\le x \\land x \\le (\\exp(lambda^2*(b-a)^2/8))) + \\eta", "inference_order": 247} {"schema_version": "1.0", "item_id": "rcd2_565cc97c4deb6b64", "pair_id": "pair_78ac217bc9c2eff1", "statement_latex": "(|Y^X/G| =-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3))", "inference_order": 248} {"schema_version": "1.0", "item_id": "rcd2_8252d8bb9905662a", "pair_id": "pair_aea7e15856b9fa08", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(P[max_{1>i≤n} X_i ≥ C])+s=(E[max(X_n, 0)] / C)", "inference_order": 249} {"schema_version": "1.0", "item_id": "rcd2_46cd417f5422a549", "pair_id": "pair_2773b6bbeedebafd", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2)))\\le t\\Rightarrow (P(X_N - X_0 ≥ epsilon))\\le t", "inference_order": 250} {"schema_version": "1.0", "item_id": "rcd2_2c006108f51f723a", "pair_id": "pair_a1ca4efda037b68c", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(tr(exp(A)*exp(B)))\\le t\\Rightarrow (tr(exp(A+B)))\\le t + \\eta", "inference_order": 251} {"schema_version": "1.0", "item_id": "rcd2_5c4d330a2546c42b", "pair_id": "pair_a309fdc06abdc8ac", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((x^a - y^b))=F(((-1)^n))", "inference_order": 252} {"schema_version": "1.0", "item_id": "rcd2_7ec59dbf79d5e812", "pair_id": "pair_df39905419560958", "statement_latex": "\\{(I^nM ∩ N)\\}=\\{(I^{n-k}(I^kM ∩ N))\\}", "inference_order": 253} {"schema_version": "1.0", "item_id": "rcd2_c1475c7366973751", "pair_id": "pair_6c570176bc82bfdd", "statement_latex": "\\forall z\\in D,\\;(z=(dX_t)\\Longleftrightarrow z=(mu_t dt + sigma_t dB_t))", "inference_order": 254} {"schema_version": "1.0", "item_id": "rcd2_dd08ff4579f68951", "pair_id": "pair_55cafbe1290596a5", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((constant_term(prod_{1≤i≠j≤n} (1-t_i/t_j)^{a_i})))=F((sum a_i)! / prod(a_i!))", "inference_order": 255} {"schema_version": "1.0", "item_id": "rcd2_a2a1635966107261", "pair_id": "pair_412a08486a415fb0", "statement_latex": "\\neg\\left((0 = 0) \\Rightarrow (k e_k = \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i)\\right)", "inference_order": 256} {"schema_version": "1.0", "item_id": "rcd2_e119c201ef5b9e2b", "pair_id": "pair_0c2c708db83a7fee", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (p_{n+1}^2)\\Rightarrow t<(p_n#)", "inference_order": 257} {"schema_version": "1.0", "item_id": "rcd2_ee7f220cda3e1eb3", "pair_id": "pair_f4b689f95e410e3c", "statement_latex": "\\max\\{0,(P[max_{1≤i≤n} X_i ≥ C])-(E[max(X_n, 0)] / C)\\}=0", "inference_order": 258} {"schema_version": "1.0", "item_id": "rcd2_d98efc08ceada605", "pair_id": "pair_8ba87e9a07410bf5", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((Pr(|X-mu| ≥ 1 - α))-(α))=0", "inference_order": 259} {"schema_version": "1.0", "item_id": "rcd2_3a0721ed8d4330d8", "pair_id": "pair_dd4b201d62506d45", "statement_latex": "\\delta_D((lim_{n→∞} (a_n / n)),(inf_{n∈N} (a_n / n)))=0", "inference_order": 260} {"schema_version": "1.0", "item_id": "rcd2_b5bd8dd7f7bf366c", "pair_id": "pair_75cba19e4f7cc4d3", "statement_latex": "\\forall z\\in D,\\;((n),z)=((p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}),z)", "inference_order": 261} {"schema_version": "1.0", "item_id": "rcd2_15f9f0c82e8661a3", "pair_id": "pair_066e3d4b0012675f", "statement_latex": "\\{((Δy),z):z\\in D\\}=\\{((f'(x)Δx + εΔx),z):z\\in D\\}", "inference_order": 262} {"schema_version": "1.0", "item_id": "rcd2_72f605cd32bc38d5", "pair_id": "pair_a4031f5341a85383", "statement_latex": "\\nexists S \\subseteq D: (((p) \\in S \\land (a^2 + b^2 + c^2 + d^2) \\notin S) \\lor ((a^2 + b^2 + c^2 + d^2) \\in S \\land (p) \\notin S))", "inference_order": 263} {"schema_version": "1.0", "item_id": "rcd2_277169eb98339de0", "pair_id": "pair_d49c3a78caade47b", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(F)=(z\\;\\land\\;z=(-V'(x))) + \\eta", "inference_order": 264} {"schema_version": "1.0", "item_id": "rcd2_67bba0041eae9858", "pair_id": "pair_2699b96685f0ba5f", "statement_latex": "\\neg(\\neg (sum_{p= product_{j=1}^{\\infty} (1/(1-z^j))^{M(alpha, j)}))", "inference_order": 265} {"schema_version": "1.0", "item_id": "rcd2_5cb880827a635b6e", "pair_id": "pair_22d1f33cf284f830", "statement_latex": "\\neg\\left((\\{({m+n \\choose r})\\}\\times\\{(\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k})\\})\\cap\\{(z,z):z\\in D\\}=\\{(({m+n \\choose r}),(\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}))\\}\\right)", "inference_order": 266} {"schema_version": "1.0", "item_id": "rcd2_0b8f504013faf3c1", "pair_id": "pair_7f04f0fb65db086c", "statement_latex": "\\{((A^{\\mu \\nu}),z):z\\in D\\}=\\{((aG^{\\mu \\nu} + bg^{\\mu \\nu}),z):z\\in D\\}", "inference_order": 267} {"schema_version": "1.0", "item_id": "rcd2_280e2c3bbc518dc3", "pair_id": "pair_2d0291575fb8753e", "statement_latex": "\\neg(\\neg (Pr(Σ_t=A) / Pr(Σ_t=-A) = e^(At)))", "inference_order": 268} {"schema_version": "1.0", "item_id": "rcd2_5d5d1aa4a1ad5555", "pair_id": "pair_0e4cf4316e7fac16", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(P[σ(x(t))] / P[σ(x_tilde(t))]))+\\delta_D(z,(e^σ[x(t)])))=0", "inference_order": 269} {"schema_version": "1.0", "item_id": "rcd2_8daa1fce235a5972", "pair_id": "pair_d9576b19e41fec36", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (Pr(sqrt((X-μ)^T * V^-1 * (X-μ)) > t)) \\le x \\land x \\le (N / t^2)) + \\eta_{749}", "inference_order": 270} {"schema_version": "1.0", "item_id": "rcd2_d19a679e2ce9dff1", "pair_id": "pair_97f93e6320295f9f", "statement_latex": "(f(x)) \\in {(⟨Aφ(x), φ(x)⟩)}", "inference_order": 271} {"schema_version": "1.0", "item_id": "rcd2_73a5af48da79d130", "pair_id": "pair_97c0427a308c402c", "statement_latex": "\\neg(\\neg (constant_term(prod_{1\\lei\\nej\\len} (1-t_i/t_j)^{a_i}) = (sum a_i)! / prod(a_i!)))", "inference_order": 272} {"schema_version": "1.0", "item_id": "rcd2_772e22498b0d0850", "pair_id": "pair_25b3a7132bf1f281", "statement_latex": "\\{((p_{n,N}),z):z\\in D\\}=(\\{((2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}),z):z\\in D\\}) + \\eta", "inference_order": 273} {"schema_version": "1.0", "item_id": "rcd2_2ec218b5561e147b", "pair_id": "pair_ad4d673b2c49f71f", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((P(S_n - E[S_n] ≥ t))-(exp(-2*t^2 / sum_i (b_i-a_i)^2)))=0", "inference_order": 274} {"schema_version": "1.0", "item_id": "rcd2_055dc5a3c2c2ab1d", "pair_id": "pair_14ee2bca39442186", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((||u - u_h||)-((gamma / alpha) * ||u - v||))\\\\ne0", "inference_order": 275} {"schema_version": "1.0", "item_id": "rcd2_ac9bef6b86564449", "pair_id": "pair_47fb3751b5417166", "statement_latex": "\\neg\\left(\\sup_{\\lambda\\in[0,1]}\\lambda((φ(Φ^{-1}(γ^n(A))))-(liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)}))=0\\right)", "inference_order": 276} {"schema_version": "1.0", "item_id": "rcd2_99a5f36f9760e2f8", "pair_id": "pair_2c929668e411a67e", "statement_latex": "\\nexists S \\subseteq D: (((P[σ(x(t))] / P[σ(x_tilde(t))]) \\in S \\land (e^σ[x(t)]) \\notin S) \\lor ((e^σ[x(t)]) \\in S \\land (P[σ(x(t))] / P[σ(x_tilde(t))]) \\notin S))", "inference_order": 277} {"schema_version": "1.0", "item_id": "rcd2_ed755a0ca774ac56", "pair_id": "pair_4929cd61d72f1cd2", "statement_latex": "((G(k1, ..., km)),([x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i))\\in\\{(z,z):z\\in D\\}", "inference_order": 278} {"schema_version": "1.0", "item_id": "rcd2_3075279b5b047b83", "pair_id": "pair_99dc996b88d247af", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(lim_{x→c} f(x)/g(x)))\\\\ne\\operatorname{argmin}_{z\\in D}\\delta_D(z,(lim_{x→c} f'(x)/g'(x)))", "inference_order": 279} {"schema_version": "1.0", "item_id": "rcd2_a3cec44e404e947b", "pair_id": "pair_c90a363eb523867f", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((vol(s φ K + t ψ L)^(1/n))-(C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n))))=0", "inference_order": 280} {"schema_version": "1.0", "item_id": "rcd2_7b4c374be1219534", "pair_id": "pair_db9a76b252bdcd45", "statement_latex": "(max_{a \\in A} sum_{omega \\in Omega} u(a, omega) p(omega)) = (V(p))", "inference_order": 281} {"schema_version": "1.0", "item_id": "rcd2_4e89ca9740ffa424", "pair_id": "pair_dbb4fa910eadf8d9", "statement_latex": "\\neg(\\neg (G(k1, ..., km) =-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3))", "inference_order": 282} {"schema_version": "1.0", "item_id": "rcd2_a4089dbb6717484c", "pair_id": "pair_e8960e923c7212ab", "statement_latex": "\\nexists S \\subseteq D: (((dS) \\in S \\land ((dE + PdV) / T) \\notin S) \\lor (((dE + PdV) / T) \\in S \\land (dS) \\notin S)) + \\eta", "inference_order": 283} {"schema_version": "1.0", "item_id": "rcd2_2bad49ccaa299d6c", "pair_id": "pair_6f9abedcc2cba073", "statement_latex": "(det(A)*det(C)) \\in (det(M)) + [0,\\infty) + \\eta", "inference_order": 284} {"schema_version": "1.0", "item_id": "rcd2_742bed15ff6dab00", "pair_id": "pair_4c65e29151a0837a", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((a^2 - b^2))=F((a + b)(a - b))", "inference_order": 285} {"schema_version": "1.0", "item_id": "rcd2_e09ca10d2970328f", "pair_id": "pair_9b05674423fef154", "statement_latex": "\\delta_D((V(p)),(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))=(0) + \\eta", "inference_order": 286} {"schema_version": "1.0", "item_id": "rcd2_aa97f058c3d404fa", "pair_id": "pair_3d1a15b9508e97ce", "statement_latex": "\\forall z\\in D,\\;(z=(lim_{n→∞} (a_n / n))\\Longleftrightarrow z=(inf_{n∈N} (a_n / n)))", "inference_order": 287} {"schema_version": "1.0", "item_id": "rcd2_013f4eb4ebd3bdaf", "pair_id": "pair_00a7c3596833467d", "statement_latex": "\\forall z\\in D,\\;(([x^2 + y^2 + (x+y)^2]^2),z)\\\\ne((2[x^4 + y^4 + (x+y)^4]),z)", "inference_order": 288} {"schema_version": "1.0", "item_id": "rcd2_1b09cc47ee09803a", "pair_id": "pair_264da9a64fd9b593", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (W(r,k)) \\le x \\land x \\le (2^{2^{r^{2^{2^{k+9}}}}})) + \\eta", "inference_order": 289} {"schema_version": "1.0", "item_id": "rcd2_47813b40db31aeb4", "pair_id": "pair_b839a04fcfdab816", "statement_latex": "\\forall z\\in D,\\;(z=([I - A]x)\\Longleftrightarrow z=(d))", "inference_order": 290} {"schema_version": "1.0", "item_id": "rcd2_668a16221a5cb987", "pair_id": "pair_d3a30cc8141cfe18", "statement_latex": "\\neg\\left((\\bigcap _{k) \\in {(0}^{\\infty }C_{k}\\neq \\\\emptyset)}\\right)", "inference_order": 291} {"schema_version": "1.0", "item_id": "rcd2_ee32f1c41e624eb1", "pair_id": "pair_9f82888f8614acfb", "statement_latex": "(ay)-(x)\\in (m)\\mathbb{Z}", "inference_order": 292} {"schema_version": "1.0", "item_id": "rcd2_9e48ca3285aba5b9", "pair_id": "pair_28012e96e8dd032b", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(∫[0,∞] x^{s-1} f(x) dx))+\\delta_D(z,(Γ(s) φ(-s))))=0", "inference_order": 293} {"schema_version": "1.0", "item_id": "rcd2_9a2ca74a72dbc830", "pair_id": "pair_63589ffca89c8c5f", "statement_latex": "\\{(n)\\}=\\{(p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k})\\}", "inference_order": 294} {"schema_version": "1.0", "item_id": "rcd2_f2ffb5b43de21de3", "pair_id": "pair_6fde01ae1398fb05", "statement_latex": "\\forall z\\in D,\\;(z\\\\ne(nu_p(x^n - y^n))\\Longleftrightarrow z=(nu_p(x - y) + nu_p(n)))", "inference_order": 295} {"schema_version": "1.0", "item_id": "rcd2_ce3deac53e769c71", "pair_id": "pair_90ac096ef7452a24", "statement_latex": "\\max\\{0,(dim_C H^i(X, F))-(infinity)\\}=0\\;\\land\\;(\\limsup_{x → +\\infty} x^2 q(x))\\ne (\\frac{1}{4})", "inference_order": 296} {"schema_version": "1.0", "item_id": "rcd2_24ef043809cdebc6", "pair_id": "pair_9fd9089277775a8c", "statement_latex": "\\forall z\\in D,\\;(z=(∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c))\\Longleftrightarrow z=(Integral transform+Spectral terms))", "inference_order": 297} {"schema_version": "1.0", "item_id": "rcd2_54d863f2370c77ce", "pair_id": "pair_76094eb8a28f07e3", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n)))=F((1))", "inference_order": 298} {"schema_version": "1.0", "item_id": "rcd2_cebdbd438bcad0f4", "pair_id": "pair_60df30e950a47e1e", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(I^nM ∩ N))+\\delta_D(z,(I^{n-k}(I^kM ∩ N))))=0", "inference_order": 299} {"schema_version": "1.0", "item_id": "rcd2_10f966779e0a4668", "pair_id": "pair_09981b63c43e2167", "statement_latex": "\\{(f^{-1})'(b)\\}=\\{(1 / f'(a))\\}", "inference_order": 300} {"schema_version": "1.0", "item_id": "rcd2_78c2f937e051cde1", "pair_id": "pair_ca3440ff43d332f7", "statement_latex": "(((\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2),(\\sum_{i<(j} (a_i b_j - a_j b_i)^2))\\in\\{(z,z):z\\in D\\}) + \\eta", "inference_order": 301} {"schema_version": "1.0", "item_id": "rcd2_545ba95e4b7a6d5a", "pair_id": "pair_bfc7e49b58513972", "statement_latex": "\\neg(\\neg (ker(d(f|_Y)_y) \\subseteq tau))", "inference_order": 302} {"schema_version": "1.0", "item_id": "rcd2_8fc292404f6cb2de", "pair_id": "pair_6da26af4ca3bacf5", "statement_latex": "\\{((P(C_1 n log n > inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n)),z):z\\in D\\}=\\{((1),z):z\\in D\\}", "inference_order": 303} {"schema_version": "1.0", "item_id": "rcd2_c900a2d2e18dc053", "pair_id": "pair_11dba740c49c2ca0", "statement_latex": "\\forall z\\in D,\\;((gcd{(n-1 k-1), (n k+1), (n+1 k)}),z)\\\\ne((gcd{(n-1 k), (n k-1), (n+1 k+1)}),z)", "inference_order": 304} {"schema_version": "1.0", "item_id": "rcd2_4c4677256e7219ca", "pair_id": "pair_26e3b4724fe998e0", "statement_latex": "\\neg\\left(\\frac{\\exp(v)}{\\exp(v)+\\exp(b)}\\le\\frac{1}{2}\\right)", "inference_order": 305} {"schema_version": "1.0", "item_id": "rcd2_7817085ed8b24f6e", "pair_id": "pair_1d9fb287ba536efb", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((f(x)))=F((\\sup\\{f(x, \\theta) : x \\in C(\\theta)\\}))", "inference_order": 306} {"schema_version": "1.0", "item_id": "rcd2_b433f9059add20ed", "pair_id": "pair_7f5b2e81002b6a8c", "statement_latex": "\\exists s \\in \\mathbb{R}_{>0}: (K(A) = sup_{|z|) = (1} (|z|-1) ||(z-A)^-1||) + s", "inference_order": 307} {"schema_version": "1.0", "item_id": "rcd2_ec90574f7fe67a71", "pair_id": "pair_9badf2578f18195c", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(max_{1≤i≤m} |B_i|)\\le t\\Rightarrow (ceil(n/m))\\le t", "inference_order": 308} {"schema_version": "1.0", "item_id": "rcd2_666c4a881096a021", "pair_id": "pair_3e53ba3c3c25f411", "statement_latex": "\\max\\{0,(∑_{n=1}^∞ |x^*(x_n)|)-(∞)\\}=0\\;\\land\\;(∑_{n=1}^∞ |x^*(x_n)|)\\ne (∞)", "inference_order": 309} {"schema_version": "1.0", "item_id": "rcd2_ef9683553e8028fe", "pair_id": "pair_4c1e4397893f4b08", "statement_latex": "\\frac{\\exp(sum_{j\\\\ne1}^r d_j)}{\\exp(sum_{j=1}^r d_j)+\\exp(n)}<\\frac{1}{2}", "inference_order": 310} {"schema_version": "1.0", "item_id": "rcd2_3decd7ae24db7aad", "pair_id": "pair_805903dbc220473e", "statement_latex": "\\operatorname{dist}((e*p*(d+1))-(1),(-\\infty,0])=0", "inference_order": 311} {"schema_version": "1.0", "item_id": "rcd2_700a67dfdbd9946d", "pair_id": "pair_ed33e4dbf83d4ba7", "statement_latex": "(\\binom{k-1}{r-1}) = (\\binom{k}{r} \\frac{r}{k})", "inference_order": 312} {"schema_version": "1.0", "item_id": "rcd2_9e0bc7945e4bcbd6", "pair_id": "pair_cd5b18c0bc04a4e3", "statement_latex": "\\max\\{0,(E(R_b))-(E(X^2) / E(X))\\}=0", "inference_order": 313} {"schema_version": "1.0", "item_id": "rcd2_18175dc22bda2904", "pair_id": "pair_36e946694a382b60", "statement_latex": "\\operatorname{dist}((P(|sum_{i=1}^n a_i X_i| ≥ k))-(2 inf_{0 ≤ c ≤ k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz),(-\\infty,0])=0", "inference_order": 314} {"schema_version": "1.0", "item_id": "rcd2_707232dca515533d", "pair_id": "pair_03b535e17b42abf5", "statement_latex": "\\max\\{0,(0)-(D(A) is dense in X, A is dissipative, and A − λ0I is surjective for some λ0)\\}=(0\\;\\land\\;(0)\\ne (D(A) is dense in X, A is dissipative, and A − λ0I is surjective for some λ0)) + \\eta", "inference_order": 315} {"schema_version": "1.0", "item_id": "rcd2_367a0dbc353a32cf", "pair_id": "pair_cdf09e37cd39d59b", "statement_latex": "((1/π) Δ arg f(iy)) =( (p - q)) + \\eta", "inference_order": 316} {"schema_version": "1.0", "item_id": "rcd2_d207ffd3af625083", "pair_id": "pair_1c1af7272fcd3247", "statement_latex": "\\nexists S \\subseteq D: (((SMS^T) \\in S \\land (D ⊕ D) \\notin S) \\lor ((D ⊕ D) \\in S \\land (SMS^T) \\notin S)) + \\eta", "inference_order": 317} {"schema_version": "1.0", "item_id": "rcd2_67abe8faa2906acb", "pair_id": "pair_f55e6d3bd18134eb", "statement_latex": "\\delta_D((\\sum _{k\\\\ne0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}),({\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}))=0", "inference_order": 318} {"schema_version": "1.0", "item_id": "rcd2_4ff69d51c187d2cc", "pair_id": "pair_d88b3777c284deb3", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (prod_{i=1}^n (r_i!)^(1/r_i))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (per(A))\\}", "inference_order": 319} {"schema_version": "1.0", "item_id": "rcd2_a0fc91219455c873", "pair_id": "pair_c8c89c3d8e176519", "statement_latex": "(E(X)_t) \\in {(\\exp(X_t - 1/2 [X]_t))} + \\eta", "inference_order": 320} {"schema_version": "1.0", "item_id": "rcd2_a5142e0a8d891ef2", "pair_id": "pair_8897833468d93ff1", "statement_latex": "\\operatorname{dist}((Pr(S_n >( t))-(exp(-(sigma^2 / a^2) * h(at / sigma^2))),(-\\infty,0])=0) + \\eta", "inference_order": 321} {"schema_version": "1.0", "item_id": "rcd2_a9f7575d7e1b7944", "pair_id": "pair_4c96618158991982", "statement_latex": "\\max\\{0,(sum_{i in I} ||T(e_i)||_H^2)-(+infinity)\\}=0\\;\\land\\;(sum_{i in I} ||T(e_i)||_H^2)\\ne (+infinity)", "inference_order": 322} {"schema_version": "1.0", "item_id": "rcd2_3239dee32a01f89c", "pair_id": "pair_a182580d78d10065", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\ge (max(|A + A|, |A * A|))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (c|A|^{1 + ε})\\}\\right)", "inference_order": 323} {"schema_version": "1.0", "item_id": "rcd2_c06d1c84729e23ee", "pair_id": "pair_f10ded29be6cb1bc", "statement_latex": "\\{((rank(T) + nullity(T)),z):z\\in D\\}=\\{((dim(V)),z):z\\in D\\}", "inference_order": 324} {"schema_version": "1.0", "item_id": "rcd2_cead953d3b015cbf", "pair_id": "pair_a731e1cc162200ed", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;(C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon})\\le t\\Rightarrow (|S_chi(N,H)|)\\le t\\right)", "inference_order": 325} {"schema_version": "1.0", "item_id": "rcd2_c5c38524366fd674", "pair_id": "pair_98ddbcc740b049e4", "statement_latex": "(\\binom{n}{k}) = (\\binom{n-1}{k} + \\binom{n-1}{k-1})", "inference_order": 326} {"schema_version": "1.0", "item_id": "rcd2_8cc03de33c4134cb", "pair_id": "pair_2b2212a7330953a7", "statement_latex": "\\max\\{0,(|F|)-(sum_{i=0}^{d} binom(n,i))\\}=0", "inference_order": 327} {"schema_version": "1.0", "item_id": "rcd2_9ee12cd145de9c24", "pair_id": "pair_f8ad89f7d8b5f072", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(F(z,t)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(e^{g(z)t}))", "inference_order": 328} {"schema_version": "1.0", "item_id": "rcd2_9aa23d5e8070b221", "pair_id": "pair_1a29ce42f38c7db8", "statement_latex": "\\forall z\\in D,\\;(z\\\\ne(lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)])\\Longleftrightarrow z=(sup_{x∈X} (φ(x) - I(x))))", "inference_order": 329} {"schema_version": "1.0", "item_id": "rcd2_64a84fa14fa7b249", "pair_id": "pair_8c8b6e189131e538", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((sum_{k=0}^m c_k 1_{N=k}))=F((sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k))", "inference_order": 330} {"schema_version": "1.0", "item_id": "rcd2_99fb5dd51ce82830", "pair_id": "pair_14413bc1dfa1153b", "statement_latex": "(n)\\mid((sum_{d|n} mu(d)*a_{n/d})-(0))", "inference_order": 331} {"schema_version": "1.0", "item_id": "rcd2_8ea493be0b9aa015", "pair_id": "pair_a1bc22787c2ea82a", "statement_latex": "(0 = 0) \\Rightarrow ((x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k)", "inference_order": 332} {"schema_version": "1.0", "item_id": "rcd2_13e5034c5a067f28", "pair_id": "pair_4b952ec60334dde6", "statement_latex": "\\neg\\left(\\neg(\\neg (\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}={\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}))\\right)", "inference_order": 333} {"schema_version": "1.0", "item_id": "rcd2_7be90b34db9a6f1e", "pair_id": "pair_e5391de45f059134", "statement_latex": "\\neg(\\neg (\\z\\eta (3)=\\sum _{n=1}^{\\infty }{\\frac {1}{n^{3}}}))", "inference_order": 334} {"schema_version": "1.0", "item_id": "rcd2_0f5ce57a91a79c24", "pair_id": "pair_eb0862b965d6951a", "statement_latex": "\\nexists S \\subseteq D: (((H) \\in S \\land (∑_{x,y} ψ^†(x) F(x, y) ψ(y)) \\notin S) \\lor ((∑_{x,y} ψ^†(x) F(x, y) ψ(y)) \\in S \\land (H) \\notin S))", "inference_order": 335} {"schema_version": "1.0", "item_id": "rcd2_d1cfe2d363fc0504", "pair_id": "pair_16ba90f64efc0952", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(φ(+∞) − φ(0)))+\\delta_D(z,(−π(1/2*n0 + nb))))\\\\ne0", "inference_order": 336} {"schema_version": "1.0", "item_id": "rcd2_ac1fe9c976d729db", "pair_id": "pair_4121f4b00f46cbf5", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((w))\\\\neF((xyz))", "inference_order": 337} {"schema_version": "1.0", "item_id": "rcd2_856cf38f1de11279", "pair_id": "pair_a9b53de050384a1c", "statement_latex": "\\forall z\\in D,\\;((\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S}),z)\\\\ne((\\int_{V} (\\nabla \\cdot \\mathbf{F}) dV),z)", "inference_order": 338} {"schema_version": "1.0", "item_id": "rcd2_32b6f7fa49fb54cd", "pair_id": "pair_2d3935f698bb0a98", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)]))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sup_{x∈X} (φ(x) - I(x))))", "inference_order": 339} {"schema_version": "1.0", "item_id": "rcd2_9f780673148e7eb4", "pair_id": "pair_7f04f0fb65db086c", "statement_latex": "\\{((A^{\\mu \\nu}),z):z\\in D\\}=\\{((\\text{conv}\\{w(X) : w \\in W\\}),z):z\\in D\\}", "inference_order": 340} {"schema_version": "1.0", "item_id": "rcd2_2a7953732f9b03a0", "pair_id": "pair_2a51358e4de7a7c8", "statement_latex": "\\sup_{\\varepsilon>(0}\\inf_N\\mathbf{1}_{\\{\\forall n\\ge N:d((lim_{n),(∞} |S_n(β)|^(1/n)))<\\varepsilon\\}}=1) + \\eta", "inference_order": 341} {"schema_version": "1.0", "item_id": "rcd2_0f7211542e9d2983", "pair_id": "pair_c2fe54b2ef5733ba", "statement_latex": "\\nexists S \\subseteq D: (((\\nu(\\chi(1))) \\in S \\land (\\nu(|G:D|)) \\notin S) \\lor ((\\nu(|G:D|)) \\in S \\land (\\nu(\\chi(1))) \\notin S))", "inference_order": 342} {"schema_version": "1.0", "item_id": "rcd2_56c9f9bae87c9944", "pair_id": "pair_7c0997cd728f17a2", "statement_latex": "\\nexists S \\subseteq D: (((a_n) \\in S \\land (a_{n+gcd(h,k)}) \\notin S) \\lor ((a_{n+gcd(h,k)}) \\in S \\land (a_n) \\notin S))", "inference_order": 343} {"schema_version": "1.0", "item_id": "rcd2_ab5db3ff897ee12f", "pair_id": "pair_f1bf1a7725030cd6", "statement_latex": "\\neg\\left(\\forall F:D\\to\\{0,1\\},\\;F((S_T^n))=F((C_1 ∪ ... ∪ C_r))\\right)", "inference_order": 344} {"schema_version": "1.0", "item_id": "rcd2_eec4625b56c1ec8e", "pair_id": "pair_01ca85ce986acdc2", "statement_latex": "\\exists c \\in D: (f(x)) = C(\\th\\eta)\\}) = (f^*(\\th\\eta))", "inference_order": 345} {"schema_version": "1.0", "item_id": "rcd2_85697eddb2d8665d", "pair_id": "pair_132122ca15994f57", "statement_latex": "\\max\\{0,(0)-(g)\\}=0\\;\\land\\;(0)\\ne (g)", "inference_order": 346} {"schema_version": "1.0", "item_id": "rcd2_c9194588b8b63e63", "pair_id": "pair_6a473075088c53e2", "statement_latex": "\\delta_D((ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn)),(2^{n-1}∏_{i=1}^{n}ε(Xi)))=0", "inference_order": 347} {"schema_version": "1.0", "item_id": "rcd2_8566a262f82dd678", "pair_id": "pair_4ee5546adc0c5e7a", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(B))+\\delta_D(z,(SAS^T)))=0", "inference_order": 348} {"schema_version": "1.0", "item_id": "rcd2_24d80d4d36782108", "pair_id": "pair_559d884caa2c51bf", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((Pr[A])-(x(A) * Product_{B in Gamma(A)} (1 - x(B))))=0", "inference_order": 349} {"schema_version": "1.0", "item_id": "rcd2_3c665f7ca2b051ee", "pair_id": "pair_f10b26b568def335", "statement_latex": "(dS) \\in {((dE + PdV) / T)}", "inference_order": 350} {"schema_version": "1.0", "item_id": "rcd2_760e1c005d148fa8", "pair_id": "pair_bc33c3f455d6cce5", "statement_latex": "\\neg\\left((f_3(X ∨ Y)f_4(X ∧ Y)) - (f_1(X)f_2(Y)) \\ge 0\\right)", "inference_order": 351} {"schema_version": "1.0", "item_id": "rcd2_3b00bcfd8e98ddd4", "pair_id": "pair_9bb8830326565899", "statement_latex": "\\exists s\\in\\mathbb{R}_{>0}\\;\\text{such that}\\;(dim_C H^i(X, F))+s=(infinity)", "inference_order": 352} {"schema_version": "1.0", "item_id": "rcd2_7c963bcb56d47806", "pair_id": "pair_2ef4cf0048a1e224", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(|A|)+s\\\\ne(binom(n-1,r-1))", "inference_order": 353} {"schema_version": "1.0", "item_id": "rcd2_5515aee15671d8f4", "pair_id": "pair_3d48063cda278f3f", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((f(s(x))) \\in S \\land (s(f(x))) \\notin S) \\lor ((s(f(x))) \\in S \\land (f(s(x))) \\notin S))\\right)", "inference_order": 354} {"schema_version": "1.0", "item_id": "rcd2_e2351c47a6717814", "pair_id": "pair_b2ced4c51b91761f", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(nullity A)=z\\;\\land\\;z=(nullity H)", "inference_order": 355} {"schema_version": "1.0", "item_id": "rcd2_39238425ed0b1bed", "pair_id": "pair_ea95bf36467431eb", "statement_latex": "\\max\\{0,(p_{n+1}^2)-(p_n#)\\}=0\\;\\land\\;(p_{n+1}^2)\\ne (p_n#)", "inference_order": 356} {"schema_version": "1.0", "item_id": "rcd2_acb5ce45214c45fc", "pair_id": "pair_70cdd0c285c9c5b5", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((f'(c)))=F((f(b)-f(a))/(b-a))", "inference_order": 357} {"schema_version": "1.0", "item_id": "rcd2_5763f647425dd113", "pair_id": "pair_1ac47111a614a2c4", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(\\operatorname{Re} \\langle x, y \\rangle))+\\delta_D(z,(\\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2))))=0", "inference_order": 358} {"schema_version": "1.0", "item_id": "rcd2_08079f15e4468385", "pair_id": "pair_7b6c5180db3270e0", "statement_latex": "\\neg(\\neg (S_βα → S_β1α1 * S_β2α2))", "inference_order": 359} {"schema_version": "1.0", "item_id": "rcd2_11efef3c43870db7", "pair_id": "pair_0bd05325371cd3a1", "statement_latex": "(\\\\varepsilon) - (|Q(x)|) > 0", "inference_order": 360} {"schema_version": "1.0", "item_id": "rcd2_1fe7d8fe8747ac4e", "pair_id": "pair_1c1af7272fcd3247", "statement_latex": "\\nexists S \\subseteq D: (((SMS^T) \\in S \\land (D ⊕ D) \\notin S) \\lor ((D ⊕ D) \\in S \\land (SMS^T) \\notin S))", "inference_order": 361} {"schema_version": "1.0", "item_id": "rcd2_01373d22a1d8976c", "pair_id": "pair_c801bb6cc26fab56", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (|Q(x)|) > x \\land x < (\\\\varepsilon)", "inference_order": 362} {"schema_version": "1.0", "item_id": "rcd2_b11d502bd3c59b9e", "pair_id": "pair_b3b1cd897cceada5", "statement_latex": "\\operatorname{dist}((P(X ≤ a))-(E[X] / a),(-\\infty,0])=0", "inference_order": 363} {"schema_version": "1.0", "item_id": "rcd2_bb8bcdabe524a017", "pair_id": "pair_829611bfc48f59ea", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(rank(T) + nullity(T)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(dim(V)))", "inference_order": 364} {"schema_version": "1.0", "item_id": "rcd2_680356474e4c0615", "pair_id": "pair_f8ca8facbda83359", "statement_latex": "\\forall t\\in\\mathbb{R},\\;((gamma / alpha) * ||u - v||)\\le t\\Rightarrow (||u - u_h||)\\le t", "inference_order": 365} {"schema_version": "1.0", "item_id": "rcd2_8ba386921dddd51b", "pair_id": "pair_2bd72cf3536de671", "statement_latex": "\\frac{\\exp(p_{n+1}^2)}{\\exp(p_{n+1}^2)+\\exp(p_n#)}>\\frac{1}{2}", "inference_order": 366} {"schema_version": "1.0", "item_id": "rcd2_c7efa83bf8bb8aa7", "pair_id": "pair_d7a20dcf9a9e3b07", "statement_latex": "\\{(Σ [k=(0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k)\\}=\\{(w^(-1) * (z + w + m)^m)\\}) + \\eta", "inference_order": 367} {"schema_version": "1.0", "item_id": "rcd2_b2e7f49abda5e83f", "pair_id": "pair_eae7f92f328104f4", "statement_latex": "\\delta_D((lim_{N→inf} (1/(2N+1)) sum_{n=(-N}^{N} |widehat{mu}(n)|^2),(sum_{j} |c_{j}|^2))=0) + \\eta", "inference_order": 368} {"schema_version": "1.0", "item_id": "rcd2_67b50aeed184a445", "pair_id": "pair_445afe00b2b93ffe", "statement_latex": "\\neg\\left(\\forall F:D\\to\\{0,1\\},\\;F((rank(A)))=F((rank([A|b])))\\right)", "inference_order": 369} {"schema_version": "1.0", "item_id": "rcd2_7b11dc25659205be", "pair_id": "pair_4b4f4541df4eb6ae", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(E[X] / a)\\le t\\Rightarrow (P(X ≥ a))\\le t", "inference_order": 370} {"schema_version": "1.0", "item_id": "rcd2_38a8a33e67d1be48", "pair_id": "pair_addc4361b40a860a", "statement_latex": "(d/|S|) - (Pr[P(r1, ..., rn) = 0]) \\ge 0", "inference_order": 371} {"schema_version": "1.0", "item_id": "rcd2_d6b6f61bc5bf1885", "pair_id": "pair_8a8241e1902c5701", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Σ_{v∈V} deg(v)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(2|E|))", "inference_order": 372} {"schema_version": "1.0", "item_id": "rcd2_d773e239e2e4ddd7", "pair_id": "pair_db8433148cedcd9a", "statement_latex": "(\\{(S_T^n)\\}\\times\\{(C_1 ∪ ... ∪ C_r)\\})\\cap\\{(z,z):z\\in D\\}=\\{((S_T^n),(C_1 ∪ ... ∪ C_r))\\}", "inference_order": 373} {"schema_version": "1.0", "item_id": "rcd2_568d4eb829335103", "pair_id": "pair_e0c37a4c1bfbdc09", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (|alpha - b\\eta|) < x \\land x < (|alpha - alpha_i|)", "inference_order": 374} {"schema_version": "1.0", "item_id": "rcd2_e15a9e13f2df41ea", "pair_id": "pair_8580def2e7d38229", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger})))=z\\;\\land\\;z=(\\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha)", "inference_order": 375} {"schema_version": "1.0", "item_id": "rcd2_50369d21c415ea3d", "pair_id": "pair_c211ebf0aa49f29b", "statement_latex": "(f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x)) \\lor (0 = 1)", "inference_order": 376} {"schema_version": "1.0", "item_id": "rcd2_6e70cb42866e735f", "pair_id": "pair_cf1be3c48fa002a8", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((x^{4}+4y^{4}))=F((x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2}))", "inference_order": 377} {"schema_version": "1.0", "item_id": "rcd2_018154856f3daf04", "pair_id": "pair_65e449574dea906b", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(∫[0,∞] x^{s-1} f(x) dx)=z\\;\\land\\;z=(Γ(s) φ(-s))", "inference_order": 378} {"schema_version": "1.0", "item_id": "rcd2_badf58f37c147a02", "pair_id": "pair_b839a04fcfdab816", "statement_latex": "\\forall z\\in D,\\;(z=(([I - A]x)\\Longleftrightarrow z=(d))) + \\eta", "inference_order": 379} {"schema_version": "1.0", "item_id": "rcd2_e516e7ec964bfd22", "pair_id": "pair_146e4530acf79131", "statement_latex": "\\nexists S \\subseteq D: (((f(x)) \\in S \\land (⟨Aφ(x), φ(x)⟩) \\notin S) \\lor ((⟨Aφ(x), φ(x)⟩) \\in S \\land (f(x)) \\notin S))", "inference_order": 380} {"schema_version": "1.0", "item_id": "rcd2_550a29aad94f4e93", "pair_id": "pair_0495db7bad01bf92", "statement_latex": "\\forall z\\in D,\\;((lim_{n→∞} (a_n / n)),z)=(((inf_{n∈N} (a_n / n)),z)) + \\eta", "inference_order": 381} {"schema_version": "1.0", "item_id": "rcd2_089287f9f67ab480", "pair_id": "pair_bf7bb61bb27edde4", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((sum_{p\\\\ne2}^{\\infty} 1/(p-1)))=F((1))", "inference_order": 382} {"schema_version": "1.0", "item_id": "rcd2_9e2f334c5051e805", "pair_id": "pair_3a91b7642636db2e", "statement_latex": "\\neg\\left(\\delta_D((f(-x)),(f(x)))=0\\right)", "inference_order": 383} {"schema_version": "1.0", "item_id": "rcd2_2878c8f968a5c67f", "pair_id": "pair_e95b74888099fc29", "statement_latex": "(T^2) \\in {(-1)}", "inference_order": 384} {"schema_version": "1.0", "item_id": "rcd2_7e77f6b3b43e183e", "pair_id": "pair_7cf348df58b7fdbc", "statement_latex": "\\{((sum_{k=(-a}^{a} (-1)^k * (2a choose k+a)^3),z):z\\in D\\}=\\{(((3a)! / (a!)^3),z):z\\in D\\}) + \\eta", "inference_order": 385} {"schema_version": "1.0", "item_id": "rcd2_74a89e6a4966799e", "pair_id": "pair_e5391de45f059134", "statement_latex": "\\neg(\\neg (\\z\\eta (3)= product_{j=1}^{\\infty} (1/(1-z^j))^{M(alpha, j)}))", "inference_order": 386} {"schema_version": "1.0", "item_id": "rcd2_f0b2bd3a1ec94e45", "pair_id": "pair_6e11c7548fa78961", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(P(A square B))+s= 1 - α))+s=(α)", "inference_order": 387} {"schema_version": "1.0", "item_id": "rcd2_2633491087ae9974", "pair_id": "pair_a16b758a8292c7ed", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((∫_{γ} ∇φ(r) · dr) \\in S \\land (φ(q) - φ(p)) \\notin S) \\lor ((φ(q) - φ(p)) \\in S \\land (∫_{γ} ∇φ(r) · dr) \\notin S))\\right)", "inference_order": 388} {"schema_version": "1.0", "item_id": "rcd2_6b5a8ab35187c6e1", "pair_id": "pair_16fca7c14ac98c81", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(f))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(g ∘ T))", "inference_order": 389} {"schema_version": "1.0", "item_id": "rcd2_ae9644319cbe8474", "pair_id": "pair_ad27237074b33b17", "statement_latex": "(-L(v)) \\\\ne (L(-v))", "inference_order": 390} {"schema_version": "1.0", "item_id": "rcd2_26bd1eec0d2e1325", "pair_id": "pair_0cb3f78da8289508", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (1)\\}", "inference_order": 391} {"schema_version": "1.0", "item_id": "rcd2_723cf757a16646f7", "pair_id": "pair_6da26af4ca3bacf5", "statement_latex": "\\{((P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n)),z):z\\in D\\}=\\{((1),z):z\\in D\\}", "inference_order": 392} {"schema_version": "1.0", "item_id": "rcd2_b51ec5b2eb57a190", "pair_id": "pair_3506c63b499c02d6", "statement_latex": "\\neg\\left(((vol_n(E[X]))^(1/n)) - (E[vol_n(X)^(1/n)]) \\ge 0\\right)", "inference_order": 393} {"schema_version": "1.0", "item_id": "rcd2_c6b7c69e38ce04eb", "pair_id": "pair_eddac251b309ec5a", "statement_latex": "\\forall z\\in D,\\;(([I - A]x),z)=((d),z)", "inference_order": 394} {"schema_version": "1.0", "item_id": "rcd2_7abbcee1a3fcde91", "pair_id": "pair_c920a0de5224242c", "statement_latex": "\\frac{\\exp(P[max_{1≤i≤n} X_i ≥ C])}{\\exp(P[max_{1≤i≤n} X_i ≥ C])+\\exp(E[max(X_n, 0)] / C)}\\le\\frac{1}{2}", "inference_order": 395} {"schema_version": "1.0", "item_id": "rcd2_c6654c3176563cbf", "pair_id": "pair_d7364bd09bd22b23", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(|z|)\\le t\\Rightarrow (|f(z)|)\\le t", "inference_order": 396} {"schema_version": "1.0", "item_id": "rcd2_a3e0e25b81756f83", "pair_id": "pair_f4a0e10a89c19e52", "statement_latex": "(\\infty) - (dim_C H^i(X, F)) < 0", "inference_order": 397} {"schema_version": "1.0", "item_id": "rcd2_57e1f6a7ae470bc3", "pair_id": "pair_7b6dda8e5af25857", "statement_latex": "\\delta_D((H),(\\sum_{i=(1}^{m} H_{i}))=0) + \\eta", "inference_order": 398} {"schema_version": "1.0", "item_id": "rcd2_315f3bc9e7b5f011", "pair_id": "pair_dd53470da43b5872", "statement_latex": "\\neg\\left(\\exists x \\\\in \\mathbb{R}: (1) \\le x \\land x \\le (|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2|)\\right)", "inference_order": 399} {"schema_version": "1.0", "item_id": "rcd2_2066d0cee798eee9", "pair_id": "pair_ed7676e6fb6b895f", "statement_latex": "\\max\\{0,(u(t))-(u(a) * exp(integral from a to t of beta(s) ds))\\}=(0) + \\eta", "inference_order": 400} {"schema_version": "1.0", "item_id": "rcd2_fcdea036fa099398", "pair_id": "pair_e87523e8bb38be46", "statement_latex": "(f'(c)) \\in {((f(b)-f(a))/(b-a))} + \\eta", "inference_order": 401} {"schema_version": "1.0", "item_id": "rcd2_6d945652d01b8348", "pair_id": "pair_15a91b6763f0f21b", "statement_latex": "\\delta_D(([I - A]x),(d))=(0) + \\eta", "inference_order": 402} {"schema_version": "1.0", "item_id": "rcd2_b7a3d27efac4aa4d", "pair_id": "pair_6ee7a4dd20cce0fa", "statement_latex": "(\\prod_{j=1}^{d} ||g_j||_{L^{d-1}}) - (|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1}) \\ge 0", "inference_order": 403} {"schema_version": "1.0", "item_id": "rcd2_84e0b1a9f005abf8", "pair_id": "pair_2cd095f342ccf21f", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(L)=z\\;\\land\\;z=(K(α))", "inference_order": 404} {"schema_version": "1.0", "item_id": "rcd2_72224fce4a69dd00", "pair_id": "pair_0391e512a99b5f30", "statement_latex": "(M^a \\\\ne {x \\in M | f(x)) - (a}) \\le 0", "inference_order": 405} {"schema_version": "1.0", "item_id": "rcd2_77d65605b6131c29", "pair_id": "pair_445afe00b2b93ffe", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((rank(A)))=F((rank([A|b])))", "inference_order": 406} {"schema_version": "1.0", "item_id": "rcd2_6b2f18d6baaacbc1", "pair_id": "pair_155f0177343a183c", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (E(R_b)) \\le x \\land x \\le (E(X^2) / E(X))", "inference_order": 407} {"schema_version": "1.0", "item_id": "rcd2_e76273691be0b259", "pair_id": "pair_8b4a6674e371cf5d", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((Δy))=F((\\sup\\{f(x, \\theta) : x \\in C(\\theta)\\}))", "inference_order": 408} {"schema_version": "1.0", "item_id": "rcd2_f6b71545fdcb7107", "pair_id": "pair_a8cac50bc2a0cb25", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(|∫ Dφ(y)(x) dμ(y)|)+s=(C(x) ‖φ‖_{∞})", "inference_order": 409} {"schema_version": "1.0", "item_id": "rcd2_634364748a90a118", "pair_id": "pair_e84bb06036752d31", "statement_latex": "\\max\\{0,(∫ |∇u*|^p dx)-(∫ |∇u|^p dx)\\}=0", "inference_order": 410} {"schema_version": "1.0", "item_id": "rcd2_948113ba252422c1", "pair_id": "pair_76300ef401478170", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (E[max(X_n, 0)] / C)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (P[max_{1>i≤n} X_i ≥ C])\\}", "inference_order": 411} {"schema_version": "1.0", "item_id": "rcd2_0ebd7cb5404107f0", "pair_id": "pair_642cde1a49451c59", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (∑_{n≥1}P(|S_{n}|>εn))\\Rightarrow t<(∞)", "inference_order": 412} {"schema_version": "1.0", "item_id": "rcd2_39d32922d2917502", "pair_id": "pair_104062319514cb5c", "statement_latex": "\\neg\\left((Av) \\in {(rv)}\\right)", "inference_order": 413} {"schema_version": "1.0", "item_id": "rcd2_21bc8da7af836da8", "pair_id": "pair_6477fb35b6ce3bb5", "statement_latex": "(p - q) \\in {((1/π) Δ arg f(iy))} + \\eta", "inference_order": 414} {"schema_version": "1.0", "item_id": "rcd2_1d460e500f94c730", "pair_id": "pair_be65067772542678", "statement_latex": "\\forall z\\in D,\\;((P(A ∩ B)),z)=((P(B|A)*P(A)),z)", "inference_order": 415} {"schema_version": "1.0", "item_id": "rcd2_761a466d69b2e117", "pair_id": "pair_e986703ca6d30487", "statement_latex": "\\neg\\left(\\exists x \\\\in \\mathbb{R}: ((∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n}) \\le x \\land x \\le ((∑ x_i) / (∑ (1-x_i)))\\right)", "inference_order": 416} {"schema_version": "1.0", "item_id": "rcd2_2a826153a2a8a4eb", "pair_id": "pair_4906cacaf5a529e8", "statement_latex": "\\neg\\left(\\{((sum_{k=m}^{n} f_k Delta g_k),z):z\\in D\\}=\\{(((f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k),z):z\\in D\\}\\right)", "inference_order": 417} {"schema_version": "1.0", "item_id": "rcd2_bf6f6822a7fddd2c", "pair_id": "pair_4e36bd533a1479dd", "statement_latex": "\\neg\\left(\\{((\\lim_{\\theta → \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx),z):z\\in D\\}=\\{((-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)),z):z\\in D\\}\\right)", "inference_order": 418} {"schema_version": "1.0", "item_id": "rcd2_ec9c3d1531b1033a", "pair_id": "pair_07a71df2c6bc5d2b", "statement_latex": "(E(T)) \\in {(n H_n)}", "inference_order": 419} {"schema_version": "1.0", "item_id": "rcd2_7287bdc4e2a4b3c6", "pair_id": "pair_dc0667b982b5a182", "statement_latex": "\\max\\{0,(|f(z)|)-(M)\\}=0\\;\\land\\;(|f(z)|)\\ne (M)", "inference_order": 420} {"schema_version": "1.0", "item_id": "rcd2_9221229f6927e058", "pair_id": "pair_f1a020820d740852", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c))=z\\;\\land\\;z=(Integral transform+Spectral terms)", "inference_order": 421} {"schema_version": "1.0", "item_id": "rcd2_76ce52916005f7d8", "pair_id": "pair_4f9e26b776a14cf7", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(cap(Z)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(H(p/(1+2^s))))", "inference_order": 422} {"schema_version": "1.0", "item_id": "rcd2_5e3b3d503d3fca92", "pair_id": "pair_09840024da1b1478", "statement_latex": "\\delta_D((Pr(Σ_t=A) / Pr(Σ_t=-A)),(e^(At)))=0", "inference_order": 423} {"schema_version": "1.0", "item_id": "rcd2_11908f11076e18e8", "pair_id": "pair_08fd193ce68ff99b", "statement_latex": "\\neg\\left((cr(G)) - (e^3 / (29n^2)) \\ge 0\\right)", "inference_order": 424} {"schema_version": "1.0", "item_id": "rcd2_cad0a9213db8c474", "pair_id": "pair_7bcb28aba798c3bd", "statement_latex": "\\{((integral integral_{Sigma} (curl F) dot dSigma),z):z\\in D\\}=(\\{((integral_{partial Sigma} F dot dr),z):z\\in D\\}) + \\eta", "inference_order": 425} {"schema_version": "1.0", "item_id": "rcd2_476bdef8512d54e1", "pair_id": "pair_6587a84d7e078c85", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\ge (n max_{|z|=1}|P(z)|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (max_{|z|=1}|P'(z)|)\\}\\right)", "inference_order": 426} {"schema_version": "1.0", "item_id": "rcd2_d12f5ba42d0e1c3d", "pair_id": "pair_e3d07e1adcbb1c46", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (T2)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (T1)\\}", "inference_order": 427} {"schema_version": "1.0", "item_id": "rcd2_952aa210a64a182e", "pair_id": "pair_2a3b1c84ee52f49f", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (K|A|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|A+A|)\\}", "inference_order": 428} {"schema_version": "1.0", "item_id": "rcd2_4d8d747a176517b4", "pair_id": "pair_0b390ab6198b4242", "statement_latex": "(f'(x)Δx + εΔx) = (Δy)", "inference_order": 429} {"schema_version": "1.0", "item_id": "rcd2_b45f83cfcaa3046f", "pair_id": "pair_bb5799da15ad86d4", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Ax))\\\\ne\\operatorname{argmin}_{z\\in D}\\delta_D(z,(0))", "inference_order": 430} {"schema_version": "1.0", "item_id": "rcd2_66e56adce49bf3cc", "pair_id": "pair_de9105952dccb41f", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((Σ_{v∈V} deg(v)))\\\\neF((2|E|))", "inference_order": 431} {"schema_version": "1.0", "item_id": "rcd2_f01e9eb16c28f1da", "pair_id": "pair_fc75c626b0423902", "statement_latex": "\\nexists S \\subseteq D: (((p - q) \\in S \\land ((1/π) Δ arg f(iy)) \\notin S) \\lor (((1/π) Δ arg f(iy)) \\in S \\land (p - q) \\notin S)) + \\eta", "inference_order": 432} {"schema_version": "1.0", "item_id": "rcd2_1ede75f36b492471", "pair_id": "pair_00a7c3596833467d", "statement_latex": "\\forall z\\in D,\\;(([x^2 + y^2 + (x+y)^2]^2),z)=((2[x^4 + y^4 + (x+y)^4]),z)", "inference_order": 433} {"schema_version": "1.0", "item_id": "rcd2_ed9b13e997aa522e", "pair_id": "pair_d6ce6b9090f342bf", "statement_latex": "\\neg(\\neg (\\exists prime p such that n < p < 2*n))", "inference_order": 434} {"schema_version": "1.0", "item_id": "rcd2_849631bb16675719", "pair_id": "pair_1f8a821de1dacd7c", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((T^2))=F((-1))", "inference_order": 435} {"schema_version": "1.0", "item_id": "rcd2_1df2771ddc898034", "pair_id": "pair_2b5aa7091a4b7fff", "statement_latex": "\\{((|A+B|),z):z\\in D\\}=\\{((|A+H|+|B+H|-|H|),z):z\\in D\\}", "inference_order": 436} {"schema_version": "1.0", "item_id": "rcd2_756ca1f51135fc19", "pair_id": "pair_e50f798376e786c5", "statement_latex": "\\{(((x_{1}+x_{2}+\\cdots +x_{m})^{n}),z):z\\in D\\}=\\{((\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}),z):z\\in D\\}", "inference_order": 437} {"schema_version": "1.0", "item_id": "rcd2_5895315e0da5a7fa", "pair_id": "pair_655924b18aa2eef2", "statement_latex": "(prod_{n) \\in (1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n \\\\in \\mathbb{Z}} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})) + [0,\\infty)", "inference_order": 438} {"schema_version": "1.0", "item_id": "rcd2_8904a14f6691a743", "pair_id": "pair_9489d141bf687491", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (I(integral f d gamma^n)) \\le x \\land x \\le (integral sqrt(I(f)^2 + |grad f|^2) d gamma^n)) + \\eta_{553}", "inference_order": 439} {"schema_version": "1.0", "item_id": "rcd2_fd85631fedc306af", "pair_id": "pair_4e5df5f9f3cf0c94", "statement_latex": "\\neg(\\neg (Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k = w^(-1) * (z + w + m)^m))", "inference_order": 440} {"schema_version": "1.0", "item_id": "rcd2_b5538e71221e6863", "pair_id": "pair_6a75be454dd202d2", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(lim_{n→∞} (a_n / n)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(inf_{n∈N} (a_n / n)))", "inference_order": 441} {"schema_version": "1.0", "item_id": "rcd2_5f438d6ffdc27107", "pair_id": "pair_1556945bb9b013fc", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (x(A) * Product_{B in Gamma(A)} (1 - x(B)))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (Pr[A])\\}", "inference_order": 442} {"schema_version": "1.0", "item_id": "rcd2_571ea2805cad316a", "pair_id": "pair_d0cb8ab9be7d3bf3", "statement_latex": "\\delta_D((product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2})),(sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}))=0", "inference_order": 443} {"schema_version": "1.0", "item_id": "rcd2_e9a9df7058477054", "pair_id": "pair_50c8ed77b8c448c8", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((P(‖f‖_T > 1 - α))-(α))=0", "inference_order": 444} {"schema_version": "1.0", "item_id": "rcd2_ee877ca0fe3e960a", "pair_id": "pair_8f08cb3b6f39c4a2", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(binom(n,floor(n/2))/2^n)\\le t\\Rightarrow (P(S_n in A))\\le t", "inference_order": 445} {"schema_version": "1.0", "item_id": "rcd2_1b6d7991f94281cc", "pair_id": "pair_f30d6bab79aed469", "statement_latex": "\\frac{\\exp(u(t))}{\\exp(u(t))+\\exp(u(a) * exp(integral from a to t of beta(s) ds))}\\le\\frac{1}{2} + \\eta", "inference_order": 446} {"schema_version": "1.0", "item_id": "rcd2_aa3e8fe89b61fd08", "pair_id": "pair_62cd7cc98da8cbeb", "statement_latex": "\\nexists S \\subseteq D: (((Δy) \\in S \\land (f'(x)Δx + εΔx) \\notin S) \\lor ((f'(x)Δx + εΔx) \\in S \\land (Δy) \\notin S)) + \\eta", "inference_order": 447} {"schema_version": "1.0", "item_id": "rcd2_fc5bf0dffd93e787", "pair_id": "pair_1b49c5c0381c61ca", "statement_latex": "\\{((f'(x)),z):z\\in D\\}=\\{((\\sup\\{f(x, \\theta) : x \\in C(\\theta)\\}),z):z\\in D\\}", "inference_order": 448} {"schema_version": "1.0", "item_id": "rcd2_038d0726487a858a", "pair_id": "pair_f47ee9dba8be9e6f", "statement_latex": "\\frac{\\exp(|F|)}{\\exp(|F|)+\\exp(max_{r: t+2r ≤ n} |F_{n,k,t,r}|)}\\le\\frac{1}{2}", "inference_order": 449} {"schema_version": "1.0", "item_id": "rcd2_25a81df5efdec9b4", "pair_id": "pair_2546bab32833905c", "statement_latex": "\\forall z\\in D,\\;((x_i),z)=((\\det(A_i) / \\det(A)),z)", "inference_order": 450} {"schema_version": "1.0", "item_id": "rcd2_faea42079c3bf9ef", "pair_id": "pair_b9cd6a37747cc338", "statement_latex": "\\forall z\\in D,\\;((p(t)),z)=(((t^6 + c_4t^4 + c_2t^2 + c_1t + c_0),z)) + \\eta", "inference_order": 451} {"schema_version": "1.0", "item_id": "rcd2_f217d2918a27aa75", "pair_id": "pair_8723e74225ca4695", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\sum_{i=r}^{n} \\binom{i}{r}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\binom{n+1}{r+1}))", "inference_order": 452} {"schema_version": "1.0", "item_id": "rcd2_54c3638114213ea6", "pair_id": "pair_7319e7abb0b643e0", "statement_latex": "(j} (a_i b_j - a_j b_i)^2) - ((\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2 = \\sum_{i) > 0", "inference_order": 453} {"schema_version": "1.0", "item_id": "rcd2_3603d890199c57a2", "pair_id": "pair_872c41e33ff1fbee", "statement_latex": "\\max\\{0,(Pr(max_{1≤k≤n} |S_k| ≥ 3alpha))-(3 max_{1≤k≤n} Pr(|S_k| ≥ alpha))\\}=0", "inference_order": 454} {"schema_version": "1.0", "item_id": "rcd2_b707f2d18b1c7603", "pair_id": "pair_5c8a70fc4df807b5", "statement_latex": "\\max\\{0,(\\mu(B))-(\\varepsilon)\\}=0\\;\\land\\;(\\mu(B))\\ne (\\varepsilon)", "inference_order": 455} {"schema_version": "1.0", "item_id": "rcd2_55b2f8b58588ba72", "pair_id": "pair_e1affd579427de18", "statement_latex": "\\{((lim_{n→∞} (a_n / n)),z):z\\in D\\}=\\{((\\sup\\{f(x, \\theta) : x \\in C(\\theta)\\}),z):z\\in D\\}", "inference_order": 456} {"schema_version": "1.0", "item_id": "rcd2_f0495d277cf46a1c", "pair_id": "pair_283bf63d97294a58", "statement_latex": "\\neg\\left(\\frac{\\exp(‖v - Pv‖)}{\\exp(‖v - Pv‖)+\\exp((1 + ‖P‖) inf_{u ∈ U} ‖v - u‖)}\\le\\frac{1}{2}\\right)", "inference_order": 457} {"schema_version": "1.0", "item_id": "rcd2_4ebe666adad051e8", "pair_id": "pair_58561abdaa336b2e", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(z=(|A_d|)\\Longleftrightarrow z=(\\frac{w(d)}{d}X + R_d))\\right)", "inference_order": 458} {"schema_version": "1.0", "item_id": "rcd2_cb9dfe3d07526342", "pair_id": "pair_d5d7a3cab87b8092", "statement_latex": "\\neg\\left(\\delta_D((nu_p(x^n - y^n)),(nu_p(x - y) + nu_p(n)))=0\\right)", "inference_order": 459} {"schema_version": "1.0", "item_id": "rcd2_08abac7bcf556ee1", "pair_id": "pair_ee71fae353ff5df9", "statement_latex": "(E(X^2) / E(X)) - (E(R_b)) \\ge 0 + \\eta", "inference_order": 460} {"schema_version": "1.0", "item_id": "rcd2_b89d82d5fe50eeb9", "pair_id": "pair_4c96618158991982", "statement_latex": "\\max\\{0,(sum_{i in I} ||T(e_i)||_H^2)-(+infinity)\\}\\\\ne0\\;\\land\\;(sum_{i in I} ||T(e_i)||_H^2)\\ne (+infinity)", "inference_order": 461} {"schema_version": "1.0", "item_id": "rcd2_af482c10a640c7f5", "pair_id": "pair_f069c9b2f8a0b471", "statement_latex": "(∑_{x,y} ψ^†(x) F(x, y) ψ(y)) =( (H)) + \\eta", "inference_order": 462} {"schema_version": "1.0", "item_id": "rcd2_7c3981b40a84f176", "pair_id": "pair_2dbab9f71469b22f", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(d/|S|)\\le t\\Rightarrow (Pr[P(r1, ..., rn) =( 0])\\le t) + \\eta", "inference_order": 463} {"schema_version": "1.0", "item_id": "rcd2_d950260e0fe06168", "pair_id": "pair_281988f47944d19e", "statement_latex": "\\delta_D((F),(-V'(x)))=0", "inference_order": 464} {"schema_version": "1.0", "item_id": "rcd2_bc0cd649fd2d1ea9", "pair_id": "pair_14005c0a2df3e066", "statement_latex": "(m)\\mid((ay)-(x)) + \\eta", "inference_order": 465} {"schema_version": "1.0", "item_id": "rcd2_1b264a4e53d2e58e", "pair_id": "pair_359b9eb11918c2c8", "statement_latex": "\\nexists S \\subseteq D: (((\\dim(E^*)) \\in S \\land (\\text{card}(\\mathbb{K}^I)) \\notin S) \\lor ((\\text{card}(\\mathbb{K}^I)) \\in S \\land (\\dim(E^*)) \\notin S))", "inference_order": 466} {"schema_version": "1.0", "item_id": "rcd2_9dda1fb1830fc535", "pair_id": "pair_cf40ec85f5efafdb", "statement_latex": "\\forall z\\in D,\\;((a_n),z)=(((a_{n+gcd(h,k)}),z)) + \\eta", "inference_order": 467} {"schema_version": "1.0", "item_id": "rcd2_7453c7905abd9822", "pair_id": "pair_e7349fa10b476cdb", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (\\prod_{j\\\\ne1}^{d} ||g_j||_{L^{d-1}})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1})\\}", "inference_order": 468} {"schema_version": "1.0", "item_id": "rcd2_bda6f46a48027d50", "pair_id": "pair_5d938b1c1b90a17a", "statement_latex": "\\delta_D((P(A|B)),(P(B|A)*P(A)/P(B)))=0", "inference_order": 469} {"schema_version": "1.0", "item_id": "rcd2_14826c326bd44254", "pair_id": "pair_cfd0bd1bc91a37be", "statement_latex": "(H(t)) \\in {(∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE)}", "inference_order": 470} {"schema_version": "1.0", "item_id": "rcd2_de30a881468d0dc2", "pair_id": "pair_6e3df240b9fbcf9a", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(L))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(lambda*W))", "inference_order": 471} {"schema_version": "1.0", "item_id": "rcd2_c115df5ca69d955e", "pair_id": "pair_82a9034a347d0ac1", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (integral sqrt(I(f)^2 + |grad f|^2) d gamma^n)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (I(integral f d gamma^n))\\}", "inference_order": 472} {"schema_version": "1.0", "item_id": "rcd2_f40743c39794bbb7", "pair_id": "pair_8ba87e9a07410bf5", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((Pr(|X-mu| ≥ k*sigma))-(1/k^2))=0", "inference_order": 473} {"schema_version": "1.0", "item_id": "rcd2_45c0cdcdb1531f6f", "pair_id": "pair_e80ed5f01ae266b6", "statement_latex": "\\operatorname{dist}((|x(S_j)|)-(2t - 1),(-\\infty,0])=0", "inference_order": 474} {"schema_version": "1.0", "item_id": "rcd2_de6d155605e596a6", "pair_id": "pair_9694af5e041e4865", "statement_latex": "\\neg\\left(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(ΔδT))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(0))\\right)", "inference_order": 475} {"schema_version": "1.0", "item_id": "rcd2_137bd49389530c79", "pair_id": "pair_fb13a60d725a9c83", "statement_latex": "\\frac{\\exp(φ(Φ^{-1}(γ^n(A))))}{\\exp(φ(Φ^{-1}(γ^n(A))))+\\exp(liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)})}\\le\\frac{1}{2}", "inference_order": 476} {"schema_version": "1.0", "item_id": "rcd2_e2b6aa7df23ac0ce", "pair_id": "pair_bd2566a0b2c80305", "statement_latex": "\\delta_D((x + y),(z))=(0) + \\eta", "inference_order": 477} {"schema_version": "1.0", "item_id": "rcd2_ebfeea77f541eed5", "pair_id": "pair_d2b6070bb99e7af3", "statement_latex": "(tr W(A, A*)) \\in {(tr W(B, B*))}", "inference_order": 478} {"schema_version": "1.0", "item_id": "rcd2_4f5e0f9776804409", "pair_id": "pair_cb399a8830868f33", "statement_latex": "\\forall z\\in D,\\;(z\\\\ne(p(t))\\Longleftrightarrow z=(t^6 + c_4t^4 + c_2t^2 + c_1t + c_0))", "inference_order": 479} {"schema_version": "1.0", "item_id": "rcd2_dab0a1e32c46cd81", "pair_id": "pair_7eb64502c96c1fde", "statement_latex": "\\forall k\\in\\mathbb{N}_{>(0}\\exists N\\forall n\\ge N:d((lim_{n),(∞} |S_n(β)|^(1/n)))<1/k) + \\eta", "inference_order": 480} {"schema_version": "1.0", "item_id": "rcd2_d98da6e8928a5d71", "pair_id": "pair_f6a0119749023e03", "statement_latex": "\\forall z\\in D,\\;(z=(det(I_m + AB))\\Longleftrightarrow z=(det(I_n + BA)))", "inference_order": 481} {"schema_version": "1.0", "item_id": "rcd2_53f75062555041f8", "pair_id": "pair_e1affd579427de18", "statement_latex": "\\{((lim_{n→∞} (a_n / n)),z):z\\in D\\}=\\{((inf_{n∈N} (a_n / n)),z):z\\in D\\}", "inference_order": 482} {"schema_version": "1.0", "item_id": "rcd2_9f3a2cb183325168", "pair_id": "pair_166d7d4f03deda85", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(z=(A^{\\mu \\nu})\\Longleftrightarrow z=(aG^{\\mu \\nu} + bg^{\\mu \\nu}))\\right)", "inference_order": 483} {"schema_version": "1.0", "item_id": "rcd2_8fc1f7b79ad2b5d8", "pair_id": "pair_926702e331bee626", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(2 inf_{0 ≤ c ≤ k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz)\\le t\\Rightarrow (P(|sum_{i=1}^n a_i X_i| ≥ k))\\le t", "inference_order": 484} {"schema_version": "1.0", "item_id": "rcd2_14c5fe86cf8cd187", "pair_id": "pair_212347c9dfd0b487", "statement_latex": "\\neg\\left(((x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2)) \\in {(z_1^2 + z_2^2 + ... + z_{16}^2)}\\right)", "inference_order": 485} {"schema_version": "1.0", "item_id": "rcd2_cc3a4daa4c10e727", "pair_id": "pair_aa04f1669db18956", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (max_{|z|=1}|P'(z)|) \\le x \\land x \\le (n max_{|z|=1}|P(z)|)", "inference_order": 486} {"schema_version": "1.0", "item_id": "rcd2_2e15a57bf17f8747", "pair_id": "pair_f3a3d104be324795", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((f^{-1})'(b))=F((1 / f'(a)))", "inference_order": 487} {"schema_version": "1.0", "item_id": "rcd2_614f341e01a3dbf4", "pair_id": "pair_e92e652fefccd66a", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (3 max_{1≥k≤n} Pr(|S_k| ≥ alpha))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (Pr(max_{1≤k≤n} |S_k| ≥ 3alpha))\\}", "inference_order": 488} {"schema_version": "1.0", "item_id": "rcd2_d182652f64d15377", "pair_id": "pair_a0a8ca31d2d34e47", "statement_latex": "(A(x)) \\in {((1 / (2*\\pi*i)) * integral_{c-i*\\infty}^{c+i*\\infty} g(z) * (x^z / z) dz)}", "inference_order": 489} {"schema_version": "1.0", "item_id": "rcd2_b52ae67c6578203a", "pair_id": "pair_c5427071f71c1c20", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((P(S_n in A))-(binom(n,floor(n/2))/2^n))\\\\ne0", "inference_order": 490} {"schema_version": "1.0", "item_id": "rcd2_ebe1b10bcf0f7b5a", "pair_id": "pair_61f532c462dec59b", "statement_latex": "\\max\\{0,(P(S_n in A))-(binom(n,floor(n/2))/2^n)\\}=0", "inference_order": 491} {"schema_version": "1.0", "item_id": "rcd2_6aba059cfc48fee4", "pair_id": "pair_eff6336bc71d360d", "statement_latex": "\\neg\\left(\\{(n)\\}=\\{(p1 + p2 + p3)\\}\\right)", "inference_order": 492} {"schema_version": "1.0", "item_id": "rcd2_698b505f15576cf9", "pair_id": "pair_3ff087f01c48eb68", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (-Σ p_i log p_i) \\le x \\land x \\le (-Σ p_i log q_i)) + \\eta", "inference_order": 493} {"schema_version": "1.0", "item_id": "rcd2_7a0ab5268ae91117", "pair_id": "pair_e83f16d4d899ac6b", "statement_latex": "\\neg\\left(((p(z)),(\\omega p^*(z)))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 494} {"schema_version": "1.0", "item_id": "rcd2_632ce6ae2a61542d", "pair_id": "pair_e7ccb17a94b18e14", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;((cap(Z)),z)=((H(p/(1+2^s))),z)\\right)", "inference_order": 495} {"schema_version": "1.0", "item_id": "rcd2_23b6ea248b4d1529", "pair_id": "pair_306bdc1dcf13165e", "statement_latex": "\\{((\\dot{q}),z):z\\in D\\}=\\{((f(q,u)),z):z\\in D\\}", "inference_order": 496} {"schema_version": "1.0", "item_id": "rcd2_2645634f4342ea29", "pair_id": "pair_32a851ae4f8d14e0", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(det(I_m + AB)))=(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(det(I_n + BA)))) + \\eta", "inference_order": 497} {"schema_version": "1.0", "item_id": "rcd2_692ce2fae1d2a99c", "pair_id": "pair_03c6f040b5a258ee", "statement_latex": "\\frac{\\exp(det(M))}{\\exp(det(M))+\\exp(det(A)*det(C))}\\le\\frac{1}{2}", "inference_order": 498} {"schema_version": "1.0", "item_id": "rcd2_5c54e14841a65af5", "pair_id": "pair_18d65636719f07ea", "statement_latex": "\\max\\{0,(P_{n-1}(x)P_{n+1}(x))-(P_n(x)^2)\\}= u_i)])}{\\exp(Pr[Intersection(X_i ≤ u_i)])+\\exp(Pr[Intersection(Y_i ≤ u_i)])}\\le\\frac{1}{2}", "inference_order": 499} {"schema_version": "1.0", "item_id": "rcd2_223aaeb8e2cc29a5", "pair_id": "pair_5c759cd74862e463", "statement_latex": "(F(s) - z\\eta(s)F(s)G(s) - z\\eta'(s)G(s) + (-z\\eta'(s)/z\\eta(s) - F(s))(1 - z\\eta(s)G(s))) = (-z\\eta'(s)/z\\eta(s))", "inference_order": 500} {"schema_version": "1.0", "item_id": "rcd2_0aae7d77a7f28cd1", "pair_id": "pair_7e6abf130473af3e", "statement_latex": "\\neg\\left(((ρ_W(g) ∘ f),(f ∘ ρ_V(g)))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 501} {"schema_version": "1.0", "item_id": "rcd2_9a9f55fef332e85d", "pair_id": "pair_0280c5b83f0aa6bb", "statement_latex": "\\{((\\zeta(s)),z):z\\in D\\}=\\{((\\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}}),z):z\\in D\\}", "inference_order": 502} {"schema_version": "1.0", "item_id": "rcd2_d0268e09a253ad50", "pair_id": "pair_dd0ce9226c3cee7b", "statement_latex": "\\neg(\\neg (product_{m\\\\ne1}^{\\infty} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}) = sum_{n=-\\infty}^{\\infty} x^{n^{2}}y^{2n}))", "inference_order": 503} {"schema_version": "1.0", "item_id": "rcd2_3e1606a8a05dfb32", "pair_id": "pair_9a50f7d859baf64d", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(ζ(s)))+\\delta_D(z,(Σ[n=(1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s))))=0) + \\eta", "inference_order": 504} {"schema_version": "1.0", "item_id": "rcd2_f721475edae8d8a7", "pair_id": "pair_556e68c1fea20514", "statement_latex": "\\forall z\\in D,\\;(z=\\pi_{(n)}((0))", "inference_order": 505} {"schema_version": "1.0", "item_id": "rcd2_538f1b3d64ba3d2e", "pair_id": "pair_ad20f9fc05cf6376", "statement_latex": "\\forall z\\in D,\\;((n),z)=((p1 + p2 + p3),z)", "inference_order": 506} {"schema_version": "1.0", "item_id": "rcd2_cd692272e550967d", "pair_id": "pair_556e68c1fea20514", "statement_latex": "\\forall z\\in D,\\;(z=(\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q})\\Longleftrightarrow z=(\\infty))", "inference_order": 507} {"schema_version": "1.0", "item_id": "rcd2_c38e1a9aa90cb743", "pair_id": "pair_98ddbcc740b049e4", "statement_latex": "\\neg\\left((\\binom{n}{k}) = (\\binom{n-1}{k} + \\binom{n-1}{k-1})\\right)", "inference_order": 508} {"schema_version": "1.0", "item_id": "rcd2_f6837ff58d88c0f4", "pair_id": "pair_f4cb66a7b8057115", "statement_latex": "\\forall z\\in D,\\;(z=(integral integral_{Sigma} (curl F) dot dSigma)\\Longleftrightarrow z=(integral_{partial Sigma} F dot dr))", "inference_order": 509} {"schema_version": "1.0", "item_id": "rcd2_4d7c0c8e56be26ee", "pair_id": "pair_399d605612372801", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(I^nM ∩ N)=z\\;\\land\\;z=(I^{n-k}(I^kM ∩ N))", "inference_order": 510} {"schema_version": "1.0", "item_id": "rcd2_580dfb9f389658fb", "pair_id": "pair_eab506640c937295", "statement_latex": "(\\{(det(A))\\}\\times\\{(det(A_v^u) * det(A_tilde_v^u))\\})\\cap\\{(z,z):z\\in D\\}=\\{((P(\\text{Ad}(K) \\cdot X)),(\\text{conv}\\{w(X) : w \\in W\\}))\\}", "inference_order": 511} {"schema_version": "1.0", "item_id": "rcd2_eb1cdafb46958471", "pair_id": "pair_3328db101c224406", "statement_latex": "\\nexists S \\subseteq D: (((|Irr_{p'}(G)|) \\in S \\land (|Irr_{p'}(N_{G}(P))|) \\notin S) \\lor ((|Irr_{p'}(N_{G}(P))|) \\in S \\land (|Irr_{p'}(G)|) \\notin S))", "inference_order": 512} {"schema_version": "1.0", "item_id": "rcd2_54d0d54ab623cb4d", "pair_id": "pair_5155cc0f2ec17abc", "statement_latex": "\\{((Σ_{v∈V} deg(v)),z):z\\in D\\}=\\{((2|E|),z):z\\in D\\}", "inference_order": 513} {"schema_version": "1.0", "item_id": "rcd2_4237db143118afc2", "pair_id": "pair_f8f86032c5793c42", "statement_latex": "((x-c_{n})y_{n}-d_{n}y_{n-1}) = (y_{n+1})", "inference_order": 514} {"schema_version": "1.0", "item_id": "rcd2_f94a6fd416e7841e", "pair_id": "pair_49ef87d06ac97d3a", "statement_latex": "\\nexists S \\subseteq D: (((mu(t)) \\in S \\land (integral_{-\\infty}^{t} A(s)p(s,t) ds) \\notin S) \\lor ((integral_{-\\infty}^{t} A(s)p(s,t) ds) \\in S \\land (mu(t)) \\notin S))", "inference_order": 515} {"schema_version": "1.0", "item_id": "rcd2_2e9fd5b9286bec59", "pair_id": "pair_eae7f92f328104f4", "statement_latex": "\\delta_D((lim_{N→inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2),(sum_{j} |c_{j}|^2))=0", "inference_order": 516} {"schema_version": "1.0", "item_id": "rcd2_7a5a8e94e83c013b", "pair_id": "pair_1044962ebaf31746", "statement_latex": "\\{(f'(c))\\}=\\{(\\sup\\{f(x, \\theta) : x \\in C(\\theta)\\})\\}", "inference_order": 517} {"schema_version": "1.0", "item_id": "rcd2_fdfd231fc9ccbe65", "pair_id": "pair_d91d8834ce227ffa", "statement_latex": "\\{t\\in\\mathbb{R}:t\\le (c|A|^{1 + ε})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (max(|A + A|, |A * A|))\\} + \\eta", "inference_order": 518} {"schema_version": "1.0", "item_id": "rcd2_8bd7ad40b7309aed", "pair_id": "pair_d0f6afdbc6397485", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (v) \\le x \\land x \\le (b)) + \\eta", "inference_order": 519} {"schema_version": "1.0", "item_id": "rcd2_c65edbe4f6a97d93", "pair_id": "pair_8ccd0f085ba34601", "statement_latex": "\\neg\\left(\\forall F:D\\to\\{0,1\\},\\;F((dS))=F(((dE + PdV) / T))\\right)", "inference_order": 520} {"schema_version": "1.0", "item_id": "rcd2_4e5815f41708e43c", "pair_id": "pair_9151ab5a3672f978", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(c|A|^{1 + ε})+s=((max(|A + A|, |A * A|))) + \\eta", "inference_order": 521} {"schema_version": "1.0", "item_id": "rcd2_d8a6410b8c9d70b8", "pair_id": "pair_b6c715fef411d4a5", "statement_latex": "\\{((E(T)),z):z\\in D\\}\\\\ne\\{((n H_n),z):z\\in D\\}", "inference_order": 522} {"schema_version": "1.0", "item_id": "rcd2_e7c33840cee3d7e2", "pair_id": "pair_ad79b04e942bd29a", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(φ(+∞) − φ(0)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(−π(1/2*n0 + nb)))", "inference_order": 523} {"schema_version": "1.0", "item_id": "rcd2_0347f391dc47e27a", "pair_id": "pair_212aa5f34e7159a5", "statement_latex": "\\forall z\\in D,\\;((Pr(Σ_t=A) / Pr(Σ_t=-A)),z)=((e^(At)),z)", "inference_order": 524} {"schema_version": "1.0", "item_id": "rcd2_eda24c128bb5aa44", "pair_id": "pair_5e9d6d4c3abc1ce3", "statement_latex": "(integral_{partial Sigma} F dot dr) = C(\\th\\eta)\\}) = (f^*(\\th\\eta))", "inference_order": 525} {"schema_version": "1.0", "item_id": "rcd2_1fa68b7e4b7fef94", "pair_id": "pair_dc08afbbdc9791cd", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((∫_{γ} ∇φ(r) · dr))=F((\\sup\\{f(x, \\theta) : x \\in C(\\theta)\\}))", "inference_order": 526} {"schema_version": "1.0", "item_id": "rcd2_6aa99c854981d2e9", "pair_id": "pair_0a8bf98031d19ff2", "statement_latex": "\\neg\\left(\\inf_{z\\in D}(\\delta_D(z,(gcd{(n-1 k-1), (n k+1), (n+1 k)}))+\\delta_D(z,(gcd{(n-1 k), (n k-1), (n+1 k+1)})))=0\\right)", "inference_order": 527} {"schema_version": "1.0", "item_id": "rcd2_9671818087a90f77", "pair_id": "pair_afcc2eb4b5945f11", "statement_latex": "\\exists x \\\\notin \\mathbb{R}: (\\infty} a_n = sup {a_n}) < x \\land x < (lim_{n -)", "inference_order": 528} {"schema_version": "1.0", "item_id": "rcd2_fb61a419b3fcebf7", "pair_id": "pair_9a50f7d859baf64d", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(ζ(s)))+\\delta_D(z,(Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s))))=0", "inference_order": 529} {"schema_version": "1.0", "item_id": "rcd2_6d8133b5fb95fcc3", "pair_id": "pair_a85eb30bedf5dd1e", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(f_3(X ∨ Y)f_4(X ∧ Y))\\le t\\Rightarrow (f_1(X)f_2(Y))\\le t", "inference_order": 530} {"schema_version": "1.0", "item_id": "rcd2_d404b67d9bd66c68", "pair_id": "pair_b4a3d2ab83c383b3", "statement_latex": "(-1) =( (T^2)) + \\eta", "inference_order": 531} {"schema_version": "1.0", "item_id": "rcd2_b1768ebca60d16ee", "pair_id": "pair_d0f6afdbc6397485", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (v) \\le x \\land x \\le (b)", "inference_order": 532} {"schema_version": "1.0", "item_id": "rcd2_400fe0529b08c459", "pair_id": "pair_779cbbe2829c9dda", "statement_latex": "\\neg\\left(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(E))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\bigcup_{k}(a_{k},b_{k})))\\right)", "inference_order": 533} {"schema_version": "1.0", "item_id": "rcd2_f3af443a26ec7f27", "pair_id": "pair_0c6fac2e1ea09bb6", "statement_latex": "\\forall z\\in D,\\;((sum_{k=0}^{n-1} floor(x+k/n)),z)=((floor(n*x)),z)", "inference_order": 534} {"schema_version": "1.0", "item_id": "rcd2_e2a80a031e5a1670", "pair_id": "pair_89c36f5b55aef83f", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((G(k1, ..., km)))=(F(([x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i))) + \\eta", "inference_order": 535} {"schema_version": "1.0", "item_id": "rcd2_40871bd988d22fb8", "pair_id": "pair_8b6a26edadd09743", "statement_latex": "\\neg\\left((\\{(n)\\}\\times\\{((r - 1)(s - 1) + 1)\\})\\cap\\{(z,z):z\\in D\\}=\\{((n),((r - 1)(s - 1) + 1))\\}\\right)", "inference_order": 536} {"schema_version": "1.0", "item_id": "rcd2_0b29094b79c4a298", "pair_id": "pair_c531a60c71a7e456", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(uv))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(xy))", "inference_order": 537} {"schema_version": "1.0", "item_id": "rcd2_1cf7b1b95b50a2b1", "pair_id": "pair_7ce4664145e484c0", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(f(x)))+\\delta_D(z,(0 (mod p^n))))\\\\ne0", "inference_order": 538} {"schema_version": "1.0", "item_id": "rcd2_d885db3a2c67dd12", "pair_id": "pair_6593f453672f9856", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;((E),z)=((F(α)),z)\\right)", "inference_order": 539} {"schema_version": "1.0", "item_id": "rcd2_b4511256257b8f7e", "pair_id": "pair_f8ad89f7d8b5f072", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(F(z,t)))\\\\ne\\operatorname{argmin}_{z\\in D}\\delta_D(z,(e^{g(z)t}))", "inference_order": 540} {"schema_version": "1.0", "item_id": "rcd2_1709eed4c16b936e", "pair_id": "pair_dfa4c12c4d8106e6", "statement_latex": "((⟨j m | T_q^(k) | j' m'⟩),(⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩))\\in\\{(z,z):z\\in D\\}", "inference_order": 541} {"schema_version": "1.0", "item_id": "rcd2_aa5100f1e6dd5124", "pair_id": "pair_f10b26b568def335", "statement_latex": "(dS) \\in {((dE + PdV) / T)} + \\eta", "inference_order": 542} {"schema_version": "1.0", "item_id": "rcd2_58b00fed0fcdea27", "pair_id": "pair_d3a01079b1619e4c", "statement_latex": "(sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}) = (binom(m+n, k)_q)", "inference_order": 543} {"schema_version": "1.0", "item_id": "rcd2_19a38f9ce1a0fef1", "pair_id": "pair_5d0a7c9c202c5c9f", "statement_latex": "\\forall z\\in D,\\;((dim(H_μ)),z)=(((n!),z)) + \\eta", "inference_order": 544} {"schema_version": "1.0", "item_id": "rcd2_aee200fbeb9cb443", "pair_id": "pair_b85ebb089e5e018e", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (1)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (\\sum_{n=0}^{\\infty} n|a_n|^2)\\}", "inference_order": 545} {"schema_version": "1.0", "item_id": "rcd2_116a693e2961b5b0", "pair_id": "pair_efe4c5a3a760f29d", "statement_latex": "((loglogn)^(1/2+ε)) - (|ω(n)−loglogn|) \\\\le 0", "inference_order": 546} {"schema_version": "1.0", "item_id": "rcd2_79ea235f4375ad08", "pair_id": "pair_f8c235f3a597c692", "statement_latex": "(lim_{x -) - (1^-} G(x) = C(\\th\\eta)\\}) = (f^*(\\th\\eta))", "inference_order": 547} {"schema_version": "1.0", "item_id": "rcd2_fa9fe5e91bc44fd4", "pair_id": "pair_36a1be6dbd0b7b10", "statement_latex": "\\frac{\\exp(sqrt(x*y))}{\\exp(sqrt(x*y))+\\exp((x + y)/2)}\\le\\frac{1}{2}", "inference_order": 548} {"schema_version": "1.0", "item_id": "rcd2_78a5e934e382d75b", "pair_id": "pair_0ce1f2b163a78b05", "statement_latex": "\\delta_D((L(E, s)),(sum_{n=1}^∞ a_n / n^s))=0", "inference_order": 549} {"schema_version": "1.0", "item_id": "rcd2_20c8243ebed6a3f2", "pair_id": "pair_e1384a9e4cb6bb24", "statement_latex": "\\max\\{0,(vol(s φ K + t ψ L)^(1/n))-(C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n)))\\}=(0) + \\eta", "inference_order": 550} {"schema_version": "1.0", "item_id": "rcd2_0c4b9ff1d3a7b46c", "pair_id": "pair_2ea2a8b78e5fd705", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(H)=(z\\;\\land\\;z=(∑_{x,y} ψ^†(x) F(x, y) ψ(y))) + \\eta", "inference_order": 551} {"schema_version": "1.0", "item_id": "rcd2_23567af56c84ed47", "pair_id": "pair_c6e582cdfcc9da90", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (exp(lambda^2*(b-a)^2/8))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (E[exp(lambda*(X-E[X]))])\\} + \\eta", "inference_order": 552} {"schema_version": "1.0", "item_id": "rcd2_a089d96e98f97f3c", "pair_id": "pair_b161ffc11ca520e8", "statement_latex": "\\delta_D((A),(k[y_1, ..., y_m]))=(0) + \\eta", "inference_order": 553} {"schema_version": "1.0", "item_id": "rcd2_b762a812f99a27f7", "pair_id": "pair_7e6c93f583fde729", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(L))+\\delta_D(z,(K(α))))=0", "inference_order": 554} {"schema_version": "1.0", "item_id": "rcd2_5b15f72b6a74b35d", "pair_id": "pair_450047a5490e05bc", "statement_latex": "(j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)) - ((sum_i a_i*c_i)*(sum_j b_j*d_j) = W\\}) = (P(\\text{Ad}(K) \\cdot X))", "inference_order": 555} {"schema_version": "1.0", "item_id": "rcd2_760990843ea025a5", "pair_id": "pair_03b178d97836dc4c", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(1))", "inference_order": 556} {"schema_version": "1.0", "item_id": "rcd2_fb64d19b21cf415e", "pair_id": "pair_9914a2ace3684a2c", "statement_latex": "((sum_{n in S} (1/n)),(infinity))\\in\\{(z,z):z\\in D\\} + \\eta", "inference_order": 557} {"schema_version": "1.0", "item_id": "rcd2_2e23884ea05c0362", "pair_id": "pair_fc9cbe6ff290e06b", "statement_latex": "\\nexists S \\subseteq D: (((cap(Z)) \\in S \\land (H(p/(1+2^s))) \\notin S) \\lor ((H(p/(1+2^s))) \\in S \\land (cap(Z)) \\notin S))", "inference_order": 558} {"schema_version": "1.0", "item_id": "rcd2_5ed6d56391973512", "pair_id": "pair_a1bc22787c2ea82a", "statement_latex": "(0 =0\\exists N\\forall n\\ge N:d((X × Y),(X))<\\varepsilon", "inference_order": 559} {"schema_version": "1.0", "item_id": "rcd2_30916e09e2322544", "pair_id": "pair_a8b0d4668ff302dd", "statement_latex": "\\neg\\left(\\operatorname{dist}((P(S_n in A))-(binom(n,floor(n/2))/2^n),(-\\infty,0])=0\\right)", "inference_order": 560} {"schema_version": "1.0", "item_id": "rcd2_98ab3c08773aa3ed", "pair_id": "pair_f406fb7a7a76b61c", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (2 sup_{z in W(A)} |f(z)|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (||f(A)||)\\} + \\eta", "inference_order": 561} {"schema_version": "1.0", "item_id": "rcd2_90a28b5a708a0116", "pair_id": "pair_2317938c849f9486", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(z=(\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S})\\Longleftrightarrow z=(\\int_{V} (\\nabla \\cdot \\mathbf{F}) dV))\\right)", "inference_order": 562} {"schema_version": "1.0", "item_id": "rcd2_a5007a7b18e99098", "pair_id": "pair_52e1cd34f2190ffe", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (d(A,B))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|d(A,X) - d(B,X)|)\\}", "inference_order": 563} {"schema_version": "1.0", "item_id": "rcd2_e52ae3cea7763931", "pair_id": "pair_ca14af12a91fa740", "statement_latex": "\\{t\\in\\mathbb{R}:t\\le (||f(A)||)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (2 sup_{z in W(A)} |f(z)|)\\}", "inference_order": 564} {"schema_version": "1.0", "item_id": "rcd2_f466eb6313390031", "pair_id": "pair_7388d4667755b0d4", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((product_{m=(1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2})))=F((sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}))) + \\eta", "inference_order": 565} {"schema_version": "1.0", "item_id": "rcd2_4de716123c283067", "pair_id": "pair_018e80e713fa9500", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((F_{n-1}*F_{n+1} - F_n^2))=F(((-1)^n))", "inference_order": 566} {"schema_version": "1.0", "item_id": "rcd2_937c470731ce7876", "pair_id": "pair_684cfca00e2cafbe", "statement_latex": "\\{(P[σ(x(t))] / P[σ(x_tilde(t))])\\}=\\{(e^σ[x(t)])\\}", "inference_order": 567} {"schema_version": "1.0", "item_id": "rcd2_53151f02ead082d8", "pair_id": "pair_c5264ec654afc890", "statement_latex": "\\max\\{0,(∑_{n≥1}P(|S_{n}|>εn))-(∞)\\}=0\\;\\land\\;(∑_{n≥1}P(|S_{n}|>εn))\\ne (∞)", "inference_order": 568} {"schema_version": "1.0", "item_id": "rcd2_3f088e54d7e55d7a", "pair_id": "pair_f2da3779253bea8d", "statement_latex": "((lim_{n → infinity} a_n),(sup {a_n}))\\in\\{(z,z):z\\in D\\}", "inference_order": 569} {"schema_version": "1.0", "item_id": "rcd2_22f107839a990c03", "pair_id": "pair_cd5b18c0bc04a4e3", "statement_latex": "\\max\\{0,(E(R_b))-(E(X^2) / E(X))\\}= 1 - α))-(α)\\}=0", "inference_order": 570} {"schema_version": "1.0", "item_id": "rcd2_f9d11a9ef18d3b94", "pair_id": "pair_bd2a70232afd34c6", "statement_latex": "\\forall z\\in D,\\;(z\\\\ne(SO_{\\gamma _{H}}(1_{K_{H}}))\\Longleftrightarrow z=(\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}})))", "inference_order": 571} {"schema_version": "1.0", "item_id": "rcd2_fa26e41b806fc7d0", "pair_id": "pair_1235806a640c1dfa", "statement_latex": "(x + y) \\in {(z)} + \\eta", "inference_order": 572} {"schema_version": "1.0", "item_id": "rcd2_8adce9e68f38ea9f", "pair_id": "pair_650abb376467c729", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(P(|sum_{i=1}^n a_i X_i| ≥ k))+s=(2 inf_{0 ≤ c ≤ k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz)", "inference_order": 573} {"schema_version": "1.0", "item_id": "rcd2_4324ee05114bd7c5", "pair_id": "pair_56083b82314d2107", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (φ(Φ^{-1}(γ^n(A)))) \\le x \\land x \\le (liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)})) + \\eta", "inference_order": 574} {"schema_version": "1.0", "item_id": "rcd2_7a6fdc7fcadacab5", "pair_id": "pair_fdb12ea920679b6a", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(f^{-1})'(b))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(1 / f'(a)))", "inference_order": 575} {"schema_version": "1.0", "item_id": "rcd2_56f605a46add4da0", "pair_id": "pair_9406946a3fa798f0", "statement_latex": "\\frac{\\exp(tr(exp(A+B)))}{\\exp(tr(exp(A+B)))+\\exp(tr(exp(A)*exp(B)))}\\le\\frac{1}{2} + \\eta", "inference_order": 576} {"schema_version": "1.0", "item_id": "rcd2_0d75a202daabcbdd", "pair_id": "pair_a6b2fd605d8ba10e", "statement_latex": "\\forall z\\in D,\\;(z=((U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C)\\Longleftrightarrow z=(|\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C))) + \\eta", "inference_order": 577} {"schema_version": "1.0", "item_id": "rcd2_1558748428af68e5", "pair_id": "pair_bbc41f385ea42f2c", "statement_latex": "(K|A|) - (|A+A|) \\ge 0 + \\eta", "inference_order": 578} {"schema_version": "1.0", "item_id": "rcd2_57d0412689ad4674", "pair_id": "pair_6edf22670f45ca0d", "statement_latex": "\\max\\{0,(v)-(b)\\}=(0) + \\eta", "inference_order": 579} {"schema_version": "1.0", "item_id": "rcd2_cb83f72f83e1e48d", "pair_id": "pair_85b0e22f97be6b9a", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(a+b)^2 / (4ab)\\le t\\Rightarrow (Σ p_i x_i) * (Σ p_i / x_i)\\le t", "inference_order": 580} {"schema_version": "1.0", "item_id": "rcd2_0d3aea59a89de18e", "pair_id": "pair_11dba740c49c2ca0", "statement_latex": "\\forall z\\in D,\\;((gcd{(n-1 k-1), (n k+1), (n+1 k)}),z)=((gcd{(n-1 k), (n k-1), (n+1 k+1)}),z)", "inference_order": 581} {"schema_version": "1.0", "item_id": "rcd2_6d309db7f1ec5026", "pair_id": "pair_9201fc6c681aff23", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(sum_{n in S} (1/n)))+\\delta_D(z,(1)))=0", "inference_order": 582} {"schema_version": "1.0", "item_id": "rcd2_b3196fd766da7961", "pair_id": "pair_d46b6c5f8aeaa4b2", "statement_latex": "\\forall z\\in D,\\;((F),z)\\\\ne((-V'(x)),z)", "inference_order": 583} {"schema_version": "1.0", "item_id": "rcd2_d428512277484fe4", "pair_id": "pair_ccaf4fb48857e447", "statement_latex": "\\max\\{0,(Pr[A])-(x(A) * Product_{B in Gamma(A)} (1 - x(B)))\\}=0", "inference_order": 584} {"schema_version": "1.0", "item_id": "rcd2_9dd923d1c1b44643", "pair_id": "pair_d347f923f1ef78a4", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c)))\\\\ne\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Integral transform+Spectral terms))", "inference_order": 585} {"schema_version": "1.0", "item_id": "rcd2_24003c96d7a2a2dc", "pair_id": "pair_e40a944a01164447", "statement_latex": "\\{t\\in\\mathbb{R}:t\\le (cH log T)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (N(T + H) - N(T))\\}", "inference_order": 586} {"schema_version": "1.0", "item_id": "rcd2_f5e0cd80f537f6d8", "pair_id": "pair_feeced16e8d6886f", "statement_latex": "\\max\\{0,(Pr(|X-mu| ≥ k*sigma))-(1/k^2)\\}=0", "inference_order": 587} {"schema_version": "1.0", "item_id": "rcd2_6544d70d1776ff60", "pair_id": "pair_b3545470fb65329a", "statement_latex": "\\neg\\left(\\neg(\\neg (k e_k = \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i))\\right)", "inference_order": 588} {"schema_version": "1.0", "item_id": "rcd2_2f5327c01596faa1", "pair_id": "pair_1e86c3e59d89f8a2", "statement_latex": "\\frac{\\exp(|A|)}{\\exp(|A|)+\\exp(binom(n-1,r-1))}\\le\\frac{1}{2}", "inference_order": 589} {"schema_version": "1.0", "item_id": "rcd2_2f7f66003e982af3", "pair_id": "pair_70439c43ccdf3fb5", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((\\binom{k}{r} \\frac{r}{k}) \\in S \\land (\\binom{k-1}{r-1}) \\notin S) \\lor ((\\binom{k-1}{r-1}) \\in S \\land (\\binom{k}{r} \\frac{r}{k}) \\notin S))\\right)", "inference_order": 590} {"schema_version": "1.0", "item_id": "rcd2_09166ca3c773844b", "pair_id": "pair_1633d4caed251328", "statement_latex": "(P(A ∩ B)) \\in {(P(B|A)*P(A))} + \\eta", "inference_order": 591} {"schema_version": "1.0", "item_id": "rcd2_ab28979b838e1741", "pair_id": "pair_c90a363eb523867f", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((vol(s φ K + t ψ L)^(1/n))-(C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n))))\\\\ne0", "inference_order": 592} {"schema_version": "1.0", "item_id": "rcd2_c5ab0196ac8e3fc0", "pair_id": "pair_e4628f99e18a1cbb", "statement_latex": "\\frac{\\exp(1)}{\\exp(1)+\\exp(g)}<\\frac{1}{2}", "inference_order": 593} {"schema_version": "1.0", "item_id": "rcd2_0edebdbb3c41ca9c", "pair_id": "pair_dccd828b68e5fd2d", "statement_latex": "\\forall z\\in D,\\;(z=(p_A(A))\\Longleftrightarrow z=(0))", "inference_order": 594} {"schema_version": "1.0", "item_id": "rcd2_9c2d7b83c8d37c21", "pair_id": "pair_4e36bd533a1479dd", "statement_latex": "\\{((\\lim_{\\theta → \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx),z):z\\in D\\}=\\{((-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)),z):z\\in D\\}", "inference_order": 595} {"schema_version": "1.0", "item_id": "rcd2_c681b1405e0173d2", "pair_id": "pair_464dac91dc103256", "statement_latex": "-(j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)) < -((sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i)", "inference_order": 596} {"schema_version": "1.0", "item_id": "rcd2_1dd8360b830c0df6", "pair_id": "pair_d74870837301b2c7", "statement_latex": "\\exists x \\\\in \\mathbb{R}: ((Σ p_i x_i) * (Σ p_i / x_i)) \\le x \\land x \\le ((a+b)^2 / (4ab))", "inference_order": 597} {"schema_version": "1.0", "item_id": "rcd2_6e7fa6189d8c7212", "pair_id": "pair_7e76552834d0e5fb", "statement_latex": "\\operatorname{dist}((|d(X,Y) - d(A,B)|)-(ε),(-\\infty,0])=0", "inference_order": 598} {"schema_version": "1.0", "item_id": "rcd2_b9f7cd5e6e568da7", "pair_id": "pair_bac313ffe4290066", "statement_latex": "\\frac{\\exp(|d(X,Y) - d(A,B)|)}{\\exp(|d(X,Y) - d(A,B)|)+\\exp(ε)}\\le\\frac{1}{2} + \\eta", "inference_order": 599} {"schema_version": "1.0", "item_id": "rcd2_41e12b068a5358d2", "pair_id": "pair_c531a60c71a7e456", "statement_latex": "\\neg\\left(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(uv))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(xy))\\right)", "inference_order": 600} {"schema_version": "1.0", "item_id": "rcd2_9e7f6303a1cabcbb", "pair_id": "pair_258bb255a3b80e81", "statement_latex": "((|B_1|),(|B_2|))\\in\\{(z,z):z\\in D\\} + \\eta", "inference_order": 601} {"schema_version": "1.0", "item_id": "rcd2_97c8427a633aa7b1", "pair_id": "pair_a792d1771917cb18", "statement_latex": "(0 = 0) \\Rightarrow ((∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2)", "inference_order": 602} {"schema_version": "1.0", "item_id": "rcd2_31566370fbd15d59", "pair_id": "pair_700ddbbc3ba5e23c", "statement_latex": "\\forall z\\in D,\\;(z=((uv)\\Longleftrightarrow z=(xy))) + \\eta", "inference_order": 603} {"schema_version": "1.0", "item_id": "rcd2_d2f7bb4a50d88f2b", "pair_id": "pair_6823f5f7cb5fa853", "statement_latex": "\\{(([x^2 + y^2 + (x+y)^2]^2),z):z\\in D\\}\\\\ne\\{((2[x^4 + y^4 + (x+y)^4]),z):z\\in D\\}", "inference_order": 604} {"schema_version": "1.0", "item_id": "rcd2_d1d1e7ae6a119084", "pair_id": "pair_4b952ec60334dde6", "statement_latex": "\\neg(\\neg (\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}={\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}))", "inference_order": 605} {"schema_version": "1.0", "item_id": "rcd2_71db181bd594c0a6", "pair_id": "pair_0e2e358e4ffe1ce4", "statement_latex": "\\{(1/(1-alpha*z))\\}=(\\{(product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)})\\}) + \\eta", "inference_order": 606} {"schema_version": "1.0", "item_id": "rcd2_b6b59c788eace49d", "pair_id": "pair_e0bdcc9e40a780ce", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(z=(α)\\Longleftrightarrow z=(Pr_θ0(X ∈ R)))\\right)", "inference_order": 607} {"schema_version": "1.0", "item_id": "rcd2_cccaa06a55c71a9c", "pair_id": "pair_5b0d932ba96ff32b", "statement_latex": "\\neg\\left(\\delta_D((p),(a^2 + b^2 + c^2 + d^2))=0\\right)", "inference_order": 608} {"schema_version": "1.0", "item_id": "rcd2_06ee32a8f197ecfd", "pair_id": "pair_29e0c41493a7ab1d", "statement_latex": "(\\binom{n-1}{k} + \\binom{n-1}{k-1}) \\in {(\\binom{n}{k})}", "inference_order": 609} {"schema_version": "1.0", "item_id": "rcd2_2b56843269588bdf", "pair_id": "pair_0241fe2c73b9d460", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(integral sqrt(I(f)^2 + |grad f|^2) d gamma^n)\\le t\\Rightarrow (I(integral f d gamma^n))\\le t", "inference_order": 610} {"schema_version": "1.0", "item_id": "rcd2_e3c89872a1ec5fc2", "pair_id": "pair_c4f3cc171e636344", "statement_latex": "\\frac{\\exp(AC * BD)}{\\exp(AC * BD)+\\exp(AB * CD + BC * DA)}\\le\\frac{1}{2} + \\eta", "inference_order": 611} {"schema_version": "1.0", "item_id": "rcd2_85f7214e4c61fdd5", "pair_id": "pair_30eaeef5a02f9db6", "statement_latex": "\\forall z\\in D,\\;((uv),z)=((xy),z)", "inference_order": 612} {"schema_version": "1.0", "item_id": "rcd2_d1ffd58fe65a101e", "pair_id": "pair_bfd8e29c43a8995c", "statement_latex": "\\frac{\\exp([b] for a majorizing b)}{\\exp([b] for a majorizing b)+\\exp([a])}\\le\\frac{1}{2}", "inference_order": 613} {"schema_version": "1.0", "item_id": "rcd2_c4391be1befcc2a7", "pair_id": "pair_4797b17b01882b18", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(L(E, s)))\\\\ne\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{n=1}^∞ a_n / n^s))", "inference_order": 614} {"schema_version": "1.0", "item_id": "rcd2_d2eedb6dbbdf2131", "pair_id": "pair_3e365e30e1404c77", "statement_latex": "\\neg\\left(\\operatorname{argmin}_{z\\in D}\\delta_D(z,([x^2 + y^2 + (x+y)^2]^2))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(2[x^4 + y^4 + (x+y)^4]))\\right)", "inference_order": 615} {"schema_version": "1.0", "item_id": "rcd2_70717d373fc802c4", "pair_id": "pair_7f358154390755c8", "statement_latex": "\\max\\{0,(|g(z)|)-(|f(z)|)\\}=0\\;\\land\\;(|g(z)|)\\ne (|f(z)|)", "inference_order": 616} {"schema_version": "1.0", "item_id": "rcd2_32007156be19f493", "pair_id": "pair_5696551244d2e317", "statement_latex": "\\neg\\left((a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}) = (y^{2})\\right)", "inference_order": 617} {"schema_version": "1.0", "item_id": "rcd2_e0ca28669ff9c5a4", "pair_id": "pair_dc0667b982b5a182", "statement_latex": "\\neg\\left(\\max\\{0,(|f(z)|)-(M)\\}=0\\;\\land\\;(|f(z)|)\\ne (M)\\right)", "inference_order": 618} {"schema_version": "1.0", "item_id": "rcd2_d237fab3e60e88d8", "pair_id": "pair_e0d345690135d662", "statement_latex": "\\exists c \\in D: (A(x)) = c \\land ((1 / (2*\\pi*i)) * integral_{c-i*\\infty}^{c+i*\\infty} g(z) * (x^z / z) dz) = c", "inference_order": 619} {"schema_version": "1.0", "item_id": "rcd2_c0f46c79fa62df68", "pair_id": "pair_9929e779d449e3e4", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;(1)\\le t\\Rightarrow (\\sum_{n=0}^{\\infty} n|a_n|^2)\\le t\\right)", "inference_order": 620} {"schema_version": "1.0", "item_id": "rcd2_1744019a21c9c345", "pair_id": "pair_605dada41b7e23e0", "statement_latex": "((f(b)-f(a))/(b-a)) =( (f'(c))) + \\eta", "inference_order": 621} {"schema_version": "1.0", "item_id": "rcd2_df7262d913d91c3f", "pair_id": "pair_f5a08099e3e6e2cd", "statement_latex": "\\delta_D((-log E[e^{-f(B)}]),(inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]))=(0) + \\eta", "inference_order": 622} {"schema_version": "1.0", "item_id": "rcd2_e107b4d41688e237", "pair_id": "pair_9830fbbd33e911df", "statement_latex": "(Pr(max_{1\\lek\\len}|S_{k}|\\geλ) \\le (1/λ^2) * Var[S_{n}]) \\land (0 = 1 - α) ≤ α))", "inference_order": 623} {"schema_version": "1.0", "item_id": "rcd2_c7ec638fbb4e7bcb", "pair_id": "pair_9929e779d449e3e4", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(1)\\le t\\Rightarrow (\\sum_{n=0}^{\\infty} n|a_n|^2)\\le t", "inference_order": 624} {"schema_version": "1.0", "item_id": "rcd2_f88a3a15473f298a", "pair_id": "pair_28012e96e8dd032b", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(∫[0,∞] x^{s-1} f(x) dx))+\\delta_D(z,(Γ(s) φ(-s))))\\\\ne0", "inference_order": 625} {"schema_version": "1.0", "item_id": "rcd2_f342e836b39f7f70", "pair_id": "pair_d8361ccc1d5ada75", "statement_latex": "-(limsup_{n-) < -(\\infty} E_n = intersection_{n=1 to \\infty} union_{k=n to \\infty} E_k)", "inference_order": 626} {"schema_version": "1.0", "item_id": "rcd2_a3f24e69be744f81", "pair_id": "pair_3cda7a7134b7ed5d", "statement_latex": "\\frac{\\exp(p(a,d))}{\\exp(p(a,d))+\\exp(c * d^L)}<\\frac{1}{2}", "inference_order": 627} {"schema_version": "1.0", "item_id": "rcd2_dacff1bab44377c7", "pair_id": "pair_0a813312aa42184e", "statement_latex": "\\delta_D((1/(1-alpha*z)),(product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}))=0", "inference_order": 628} {"schema_version": "1.0", "item_id": "rcd2_942007af534b59e6", "pair_id": "pair_c3675bb5c0ee01f9", "statement_latex": "\\neg(\\neg (P[max_{1\\lei\\len} X_i \\ge C] \\le E[\\max(X_n, 0)] / C)) + \\eta", "inference_order": 629} {"schema_version": "1.0", "item_id": "rcd2_a6bda8c5256b3428", "pair_id": "pair_a19ee483627cccf3", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((P(X ≥ a))-(inf_{t>0} M(t)e^{-ta}))=0", "inference_order": 630} {"schema_version": "1.0", "item_id": "rcd2_f1d67cf89a02f4e2", "pair_id": "pair_b2ced4c51b91761f", "statement_latex": "\\neg\\left(\\exists z\\in D\\;\\text{such that}\\;(nullity A)=z\\;\\land\\;z=(nullity H)\\right)", "inference_order": 631} {"schema_version": "1.0", "item_id": "rcd2_284c4e36069f740d", "pair_id": "pair_59753ac1886d9034", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (sum_{k=0}^n a_k/binom(n,k)) \\le x \\land x \\le (1)", "inference_order": 632} {"schema_version": "1.0", "item_id": "rcd2_dc572f545240c21a", "pair_id": "pair_8a4d879c421bc411", "statement_latex": "\\neg\\left(\\operatorname{dist}((E[vol_n(X)^(1/n)])-(vol_n(E[X]))^(1/n),(-\\infty,0])=0\\right)", "inference_order": 633} {"schema_version": "1.0", "item_id": "rcd2_fbf68290c9316d3d", "pair_id": "pair_8225c2b4cc75a1b9", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((chi'_l(K_{n,n})))=F((n))", "inference_order": 634} {"schema_version": "1.0", "item_id": "rcd2_d140174e28ee1376", "pair_id": "pair_b37e9d154ec9ff23", "statement_latex": "\\{(F(X))\\}=\\{(inf_S sup_I (|I|/|S|))\\}", "inference_order": 635} {"schema_version": "1.0", "item_id": "rcd2_2725f5f9f2c3ad29", "pair_id": "pair_93fe24a690c51338", "statement_latex": "\\neg\\left((\\{(F(X))\\}\\times\\{(inf_S sup_I (|I|/|S|))\\})\\cap\\{(z,z):z\\in D\\}=\\{((F(X)),(inf_S sup_I (|I|/|S|)))\\}\\right)", "inference_order": 636} {"schema_version": "1.0", "item_id": "rcd2_b194512e976689ec", "pair_id": "pair_acd5fa1b90c9234c", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((x + y))\\\\neF((z))", "inference_order": 637} {"schema_version": "1.0", "item_id": "rcd2_7cca1f9d94753683", "pair_id": "pair_c4b87097eb905603", "statement_latex": "\\neg\\left(\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(γ)+s=(ε^h / 4h)\\right)", "inference_order": 638} {"schema_version": "1.0", "item_id": "rcd2_e5e1f8f74ab435f5", "pair_id": "pair_c4b87097eb905603", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(γ)+s=(ε^h / 4h)", "inference_order": 639} {"schema_version": "1.0", "item_id": "rcd2_7d8931cab48db1ed", "pair_id": "pair_264da9a64fd9b593", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (W(r,k)) \\le x \\land x \\le (2^{2^{r^{2^{2^{k+9}}}}})", "inference_order": 640} {"schema_version": "1.0", "item_id": "rcd2_c523bdcb28fec188", "pair_id": "pair_ec1c8a1691508201", "statement_latex": "\\neg(\\neg (Pr(max_{1\\lek\\len}|S_{k}|\\geλ) \\le (1/λ^2) * Var[S_{n}]))", "inference_order": 641} {"schema_version": "1.0", "item_id": "rcd2_e9b5912af126f4da", "pair_id": "pair_babc97b6c4670ef5", "statement_latex": "\\operatorname{dist}((-Σ p_i log p_i)-(-Σ p_i log q_i),(-\\infty,0])=(0) + \\eta", "inference_order": 642} {"schema_version": "1.0", "item_id": "rcd2_a477cacd1cf391fb", "pair_id": "pair_13376ffee15f1c4a", "statement_latex": "\\nexists S \\subseteq D: (((L) \\in S \\land (lambda*W) \\notin S) \\lor ((lambda*W) \\in S \\land (L) \\notin S))", "inference_order": 643} {"schema_version": "1.0", "item_id": "rcd2_7fc6ac76d4bec155", "pair_id": "pair_8dad7963cae83fb6", "statement_latex": "\\neg\\left((C |u|_{W^{m,p}(\\Omega)}) - (\\inf_{v \\in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)}) \\ge 0\\right)", "inference_order": 644} {"schema_version": "1.0", "item_id": "rcd2_56f76965713227a5", "pair_id": "pair_7a8bdf1b0d487b92", "statement_latex": "\\neg\\left((per(A)) - (prod_{i=1}^n (r_i!)^(1/r_i)) \\le 0\\right)", "inference_order": 645} {"schema_version": "1.0", "item_id": "rcd2_fe29b83f296013a2", "pair_id": "pair_4a0c38e00c95f935", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((sum_{n in S} (1/n)))=F((1))", "inference_order": 646} {"schema_version": "1.0", "item_id": "rcd2_d644a5555e82c43b", "pair_id": "pair_268e80b80b0b9e32", "statement_latex": "\\neg\\left((b) - (v) \\ge 0\\right)", "inference_order": 647} {"schema_version": "1.0", "item_id": "rcd2_863144497be80cef", "pair_id": "pair_c314f9126c32d148", "statement_latex": "\\neg\\left(\\frac{\\exp(prod_{i} d_{i})}{\\exp(prod_{i} d_{i})+\\exp(Vol(M))}\\le\\frac{1}{2}\\right)", "inference_order": 648} {"schema_version": "1.0", "item_id": "rcd2_e9cfb5b4eec9cdb0", "pair_id": "pair_96c5f0d0d68f63d4", "statement_latex": "\\delta_D((T^2),(-1))=0", "inference_order": 649} {"schema_version": "1.0", "item_id": "rcd2_34d2a927ddd1129f", "pair_id": "pair_30f0ad301fd0b73f", "statement_latex": "(liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)}) - (φ(Φ^{-1}(γ^n(A)))) \\ge 0 + \\eta", "inference_order": 650} {"schema_version": "1.0", "item_id": "rcd2_515d7ced002a9b70", "pair_id": "pair_52bb252c4aef512b", "statement_latex": "\\{((\\sum _{k=\\{((xy),z):z\\in D\\}", "inference_order": 651} {"schema_version": "1.0", "item_id": "rcd2_449045aecf10e217", "pair_id": "pair_0d0774d1842318f7", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((H))=(F((∑_{x,y} ψ^†(x) F(x, y) ψ(y)))) + \\eta", "inference_order": 652} {"schema_version": "1.0", "item_id": "rcd2_371b9cea27feee89", "pair_id": "pair_7fece1a2d46268e3", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;(∑_{i=1}^∞ P(A_i))\\le t\\Rightarrow (P(∪_{i=1}^∞ A_i))\\le t\\right)", "inference_order": 653} {"schema_version": "1.0", "item_id": "rcd2_9bf0832f8f94f777", "pair_id": "pair_fab273b3ae0b5862", "statement_latex": "\\neg(\\neg (\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty))", "inference_order": 654} {"schema_version": "1.0", "item_id": "rcd2_4d4aab83bd9b79ff", "pair_id": "pair_53d590a4a10745ac", "statement_latex": "\\{(ζ(s))\\}=\\{(Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s))\\}", "inference_order": 655} {"schema_version": "1.0", "item_id": "rcd2_bd60644b867e61a1", "pair_id": "pair_7b03ecbcf8eac7c2", "statement_latex": "\\forall z\\in D,\\;(z\\\\ne(\\binom{n-1}{k} + \\binom{n-1}{k-1})\\Longleftrightarrow z=(\\binom{n}{k}))", "inference_order": 656} {"schema_version": "1.0", "item_id": "rcd2_140a0a59275c058e", "pair_id": "pair_01ca85ce986acdc2", "statement_latex": "\\exists c \\in D: (f(x)) = c \\land (⟨Aφ(x), φ(x)⟩) = c", "inference_order": 657} {"schema_version": "1.0", "item_id": "rcd2_8138304064ceb27c", "pair_id": "pair_8869a246a5096af4", "statement_latex": "\\neg\\left((N}c^{-r}K(m) \\mid (∑_{c) - (0 ,n,c)g(4π√mn/c)=Integral transform+Spectral terms)\\right)", "inference_order": 658} {"schema_version": "1.0", "item_id": "rcd2_cfc82b4caabb831f", "pair_id": "pair_b3545470fb65329a", "statement_latex": "\\neg(\\neg (k e_k = \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i))", "inference_order": 659} {"schema_version": "1.0", "item_id": "rcd2_41b05dde5d1f34e5", "pair_id": "pair_7f98da523688f020", "statement_latex": "\\max\\{0,(e*p*(d+1))-(1)\\}= 1 - α))-(α)\\}=0", "inference_order": 660} {"schema_version": "1.0", "item_id": "rcd2_3027fcf7c8dc6b33", "pair_id": "pair_896e46d259b33077", "statement_latex": "\\neg(\\neg (P(S_n - E[S_n] \\ge t) \\le \\exp(-2*t^2 / sum_i (b_i-a_i)^2))) + \\eta", "inference_order": 661} {"schema_version": "1.0", "item_id": "rcd2_47263e81fcbc8780", "pair_id": "pair_c8f703d16a021cf7", "statement_latex": "\\max\\{0,(|x(S_j)|)-(2t - 1)\\}\\\\ne0", "inference_order": 662} {"schema_version": "1.0", "item_id": "rcd2_64e6c72608846363", "pair_id": "pair_2080c04a558ced51", "statement_latex": "\\nexists S \\subseteq D: (((y^{2}) \\in S \\land (a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}) \\notin S) \\lor ((a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}) \\in S \\land (y^{2}) \\notin S))", "inference_order": 663} {"schema_version": "1.0", "item_id": "rcd2_8efb3046b53d7291", "pair_id": "pair_7a56872f6b2b1e3b", "statement_latex": "(\\{(x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2})\\}\\times\\{(0)\\})\\cap\\{(z,z):z\\in D\\}=\\{((uv),(xy))\\}", "inference_order": 664} {"schema_version": "1.0", "item_id": "rcd2_0d8c0670d4f03b48", "pair_id": "pair_ff37a8a291b33407", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((max_matching_size))=F((1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)))", "inference_order": 665} {"schema_version": "1.0", "item_id": "rcd2_8a9deda1b688b01e", "pair_id": "pair_954641ed0018434a", "statement_latex": "{(binom(m+n, k)_q)} =( {(sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)})}) + \\eta", "inference_order": 666} {"schema_version": "1.0", "item_id": "rcd2_04cdded7da9d69cc", "pair_id": "pair_48b02de8a3f2db85", "statement_latex": "\\operatorname{dist}((P(∪_{i=(1}^∞ A_i))-(∑_{i=1}^∞ P(A_i)),(-\\infty,0])=0) + \\eta", "inference_order": 667} {"schema_version": "1.0", "item_id": "rcd2_b24ebda69d5748ab", "pair_id": "pair_68123f5a28616058", "statement_latex": "(ω(t)) \\\\ne (ω(-t))", "inference_order": 668} {"schema_version": "1.0", "item_id": "rcd2_9168cf7e4ec22bfe", "pair_id": "pair_183c91ad7131218b", "statement_latex": "\\forall z\\in D,\\;(z=(\\operatorname{Re} \\langle x, y \\rangle)\\Longleftrightarrow z=(\\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2)))", "inference_order": 669} {"schema_version": "1.0", "item_id": "rcd2_1aa51d4357229ff2", "pair_id": "pair_7f9142810d9f056d", "statement_latex": "\\{(dim(H_μ))\\}\\\\ne\\{(n!)\\}", "inference_order": 670} {"schema_version": "1.0", "item_id": "rcd2_8a926f857dae1e3f", "pair_id": "pair_8c3bfd8751a0760a", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;((\\nu(\\chi(1))),z)=((\\nu(|G:D|)),z)\\right)", "inference_order": 671} {"schema_version": "1.0", "item_id": "rcd2_7675c5800ff26d04", "pair_id": "pair_287f5f2260d68082", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(-log E[e^{-f(B)}]))\\\\ne\\operatorname{argmin}_{z\\in D}\\delta_D(z,(inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]))", "inference_order": 672} {"schema_version": "1.0", "item_id": "rcd2_4c656adde11cce84", "pair_id": "pair_31da46759658f091", "statement_latex": "\\forall z\\in D,\\;((p_{n,N}),z)=((2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}),z)", "inference_order": 673} {"schema_version": "1.0", "item_id": "rcd2_73382c6671d7a7e4", "pair_id": "pair_42741fbca0eb3b94", "statement_latex": "\\{t\\in\\mathbb{R}:t\\le (sqrt(x*y))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le ((x + y)/2)\\}", "inference_order": 674} {"schema_version": "1.0", "item_id": "rcd2_7eedc912c750029b", "pair_id": "pair_0cac4d95c341c96b", "statement_latex": "(rv) = (Av)", "inference_order": 675} {"schema_version": "1.0", "item_id": "rcd2_b774d9f88a081ec4", "pair_id": "pair_860f0feb3ef7da3b", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (‖v - Pv‖) \\le x \\land x \\le ((1 + ‖P‖) \\inf_{u ∈ U} ‖v - u‖)", "inference_order": 676} {"schema_version": "1.0", "item_id": "rcd2_11e20257a0343a39", "pair_id": "pair_3429f36ab1133a7d", "statement_latex": "(d(A,B)) - (|d(A,X) - d(B,X)|) \\ge 0", "inference_order": 677} {"schema_version": "1.0", "item_id": "rcd2_2c0c590dc38e713c", "pair_id": "pair_46cdbfbda51a15a4", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2)))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (P(X_N - X_0 < epsilon))\\}", "inference_order": 678} {"schema_version": "1.0", "item_id": "rcd2_42acb3e20cdb35e2", "pair_id": "pair_b85ebb089e5e018e", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\ge (1)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (\\sum_{n=0}^{\\infty} n|a_n|^2)\\}\\right)", "inference_order": 679} {"schema_version": "1.0", "item_id": "rcd2_f476df9c0195810f", "pair_id": "pair_29a0df0cd0992b24", "statement_latex": "\\forall z\\in D,\\;(z=(a_n)\\Longleftrightarrow z=(a_{n+gcd(h,k)}))", "inference_order": 680} {"schema_version": "1.0", "item_id": "rcd2_d36b73d914376f82", "pair_id": "pair_ad4d673b2c49f71f", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((P(S_n - E[S_n] ≥( t))-(exp(-2*t^2 / sum_i (b_i-a_i)^2)))=0) + \\eta", "inference_order": 681} {"schema_version": "1.0", "item_id": "rcd2_d4919de59b28c6b8", "pair_id": "pair_5175bf60f1e27ef7", "statement_latex": "((T^2),(-1))\\in\\{(z,z):z\\in D\\}", "inference_order": 682} {"schema_version": "1.0", "item_id": "rcd2_c34a3749e5249dea", "pair_id": "pair_03a574c6fb4079df", "statement_latex": "((∑_{n≥1}P(|S_{n}|>εn)),(∞))\\in\\{(u,v)\\in\\mathbb{R}^2:u\\le v\\}\\setminus\\{(u,u):u\\in\\mathbb{R}\\}", "inference_order": 683} {"schema_version": "1.0", "item_id": "rcd2_cd798d267a9f7dce", "pair_id": "pair_c90b61360c4a2644", "statement_latex": "\\neg\\left(((|Irr_{p'}(G)|),(|Irr_{p'}(N_{G}(P))|))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 684} {"schema_version": "1.0", "item_id": "rcd2_1522c1604ce63b02", "pair_id": "pair_1591c96ff8365829", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (sum_{i=(0}^{d} binom(n,i))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|F|)\\}) + \\eta", "inference_order": 685} {"schema_version": "1.0", "item_id": "rcd2_15d99cc07a09b062", "pair_id": "pair_cff375bb595ba29a", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(∮_{C}(L dx + M dy)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(∬_{D} (∂M/∂x - ∂L/∂y) dA))", "inference_order": 686} {"schema_version": "1.0", "item_id": "rcd2_a81eaa15bf94d335", "pair_id": "pair_7e76552834d0e5fb", "statement_latex": "\\operatorname{dist}((|d(X,Y) - d(A,B)|)-(ε),(-\\infty,0])=(0) + \\eta", "inference_order": 687} {"schema_version": "1.0", "item_id": "rcd2_7525d4038685ff84", "pair_id": "pair_0cb9aad03553e4ec", "statement_latex": "(SAS^T) \\\\ne (B)", "inference_order": 688} {"schema_version": "1.0", "item_id": "rcd2_2db97c67f9117372", "pair_id": "pair_a9c5c2accbf799c9", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(f'(x_0)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(0))", "inference_order": 689} {"schema_version": "1.0", "item_id": "rcd2_aa82f501ce247ee4", "pair_id": "pair_c48b8100122b5f9d", "statement_latex": "\\forall z\\in D,\\;((α),z)=((Pr_θ0(X ∈ R)),z)", "inference_order": 690} {"schema_version": "1.0", "item_id": "rcd2_1121d7f9c170e80f", "pair_id": "pair_323720428879542f", "statement_latex": "(0 = 0) \\Rightarrow (f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x))", "inference_order": 691} {"schema_version": "1.0", "item_id": "rcd2_36cad5745e697150", "pair_id": "pair_da45955f6783e7b6", "statement_latex": "(a}) - (M^a = W\\}) = (P(\\text{Ad}(K) \\cdot X))", "inference_order": 692} {"schema_version": "1.0", "item_id": "rcd2_a0db399d494d6cc3", "pair_id": "pair_9c77a2d919371475", "statement_latex": "\\nexists S \\subseteq D: (((ΔδT) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (ΔδT) \\notin S))", "inference_order": 693} {"schema_version": "1.0", "item_id": "rcd2_d103ecd58360ce43", "pair_id": "pair_fbebe7f63e764297", "statement_latex": "\\neg(\\neg ((x+m+1)\\sum _{i=(0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}=(x-m){\\dbinom {x}{m}}))) + \\eta", "inference_order": 694} {"schema_version": "1.0", "item_id": "rcd2_36f3c24413d9865a", "pair_id": "pair_e7ccb17a94b18e14", "statement_latex": "\\forall z\\in D,\\;((cap(Z)),z)=((H(p/(1+2^s))),z)", "inference_order": 695} {"schema_version": "1.0", "item_id": "rcd2_bcb6b491db2cf325", "pair_id": "pair_80f38ae50f25b80a", "statement_latex": "\\arctan(AC * BD)\\le\\arctan(AB * CD + BC * DA)", "inference_order": 696} {"schema_version": "1.0", "item_id": "rcd2_fa4ff5680134fac4", "pair_id": "pair_6391668d69d59dd4", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge ((1/λ^2) * Var[S_{n}])\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (Pr(max_{1≤k≤n}|S_{k}|≥λ))\\}", "inference_order": 697} {"schema_version": "1.0", "item_id": "rcd2_329d353a6eb4c91b", "pair_id": "pair_e40ae504348a33a1", "statement_latex": "\\{((det(I_m + AB)),z):z\\in D\\}=\\{((det(I_n + BA)),z):z\\in D\\}", "inference_order": 698} {"schema_version": "1.0", "item_id": "rcd2_995e5cd574e03e9e", "pair_id": "pair_970b6e0da7498d68", "statement_latex": "\\exists x \\\\notin \\mathbb{R}: (tr(\\exp(A+B))) \\le x \\land x \\le (tr(\\exp(A)*\\exp(B)))", "inference_order": 699} {"schema_version": "1.0", "item_id": "rcd2_7b83d4c4ea8558cc", "pair_id": "pair_258bb255a3b80e81", "statement_latex": "((|B_1|),(|B_2|))\\in\\{(z,z):z\\in D\\}", "inference_order": 700} {"schema_version": "1.0", "item_id": "rcd2_0cf6fcecae3434be", "pair_id": "pair_5889d634a00a262f", "statement_latex": "(f(x)) \\in {(0 (mod p^n))}", "inference_order": 701} {"schema_version": "1.0", "item_id": "rcd2_be4ff09239c0b1f5", "pair_id": "pair_38bef7e39776b407", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\ge (C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|S_chi(N,H)|)\\}\\right)", "inference_order": 702} {"schema_version": "1.0", "item_id": "rcd2_9b8f21562034ceac", "pair_id": "pair_b108537d3b3f2322", "statement_latex": "\\{((\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger}))),z):z\\in D\\}=\\{((\\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha),z):z\\in D\\}", "inference_order": 703} {"schema_version": "1.0", "item_id": "rcd2_233e1c22ab71f448", "pair_id": "pair_dd97fce64c526fc4", "statement_latex": "\\{((n),z):z\\in D\\}=\\{((p1 + p2 + p3),z):z\\in D\\}", "inference_order": 704} {"schema_version": "1.0", "item_id": "rcd2_84c24e4318c53e53", "pair_id": "pair_f333d3cc43b2dace", "statement_latex": "\\neg\\left((\\{(L)\\}\\times\\{(K(α))\\})\\cap\\{(z,z):z\\in D\\}=\\{((L),(K(α)))\\}\\right)", "inference_order": 705} {"schema_version": "1.0", "item_id": "rcd2_2412d6b170787f03", "pair_id": "pair_d9cb91f14d45422c", "statement_latex": "\\delta_D((sum_{n in S} (1/n)),(infinity))=0", "inference_order": 706} {"schema_version": "1.0", "item_id": "rcd2_fc624f6c6eea273a", "pair_id": "pair_4b4f4541df4eb6ae", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(E[X] / a)\\le t\\Rightarrow (P(X ≤ a))\\le t", "inference_order": 707} {"schema_version": "1.0", "item_id": "rcd2_97d36d88d3b74c5d", "pair_id": "pair_50badbdc0b8507e4", "statement_latex": "\\forall z\\in D,\\;((-log E[e^{-f(B)}]),z)=((inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]),z)", "inference_order": 708} {"schema_version": "1.0", "item_id": "rcd2_f95d5f9eda29de92", "pair_id": "pair_bffaa425232ec001", "statement_latex": "\\operatorname{dist}((V(K)^((n-1)/n) V(L)^(1/n))-(V_1(K, L)),(-\\infty,0])=0", "inference_order": 709} {"schema_version": "1.0", "item_id": "rcd2_18e6e4b6ef5a385c", "pair_id": "pair_e7972ff39e4aa6e3", "statement_latex": "\\max\\{0,(|| \\prod_{j=(1}^{d} g_j \\circ \\pi_j ||_{L^1})-(\\prod_{j=1}^{d} ||g_j||_{L^{d-1}})\\}=0) + \\eta", "inference_order": 710} {"schema_version": "1.0", "item_id": "rcd2_9c3eb0cff1973c09", "pair_id": "pair_702dbfcff4139f12", "statement_latex": "\\neg\\left(\\exists x \\\\in \\mathbb{R}: (cH log T) \\le x \\land x \\le (N(T + H) - N(T))\\right)", "inference_order": 711} {"schema_version": "1.0", "item_id": "rcd2_0323683fa7cdb075", "pair_id": "pair_6684444833625b35", "statement_latex": "\\forall z\\in D,\\;(z=(n)\\Longleftrightarrow z=(p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}))", "inference_order": 712} {"schema_version": "1.0", "item_id": "rcd2_f24d674cd9bb9c51", "pair_id": "pair_792ccbd07ca1182e", "statement_latex": "\\exists s \\in \\mathbb{R}_{>0}: (limsup_{n-) = (\\infty} E_n = intersection_{n=1 to \\infty} union_{k=n to \\infty} E_k) + s", "inference_order": 713} {"schema_version": "1.0", "item_id": "rcd2_7b5af430a16d572f", "pair_id": "pair_163d943e8a6144a7", "statement_latex": "\\neg(\\neg (|Y^X/G| =-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3))", "inference_order": 714} {"schema_version": "1.0", "item_id": "rcd2_1e7e767a751b8627", "pair_id": "pair_4480985b4fafc15e", "statement_latex": "\\{((sum_{p=\\{(((-1)^n),z):z\\in D\\}", "inference_order": 715} {"schema_version": "1.0", "item_id": "rcd2_4c1c1dbd72e8b721", "pair_id": "pair_5f847c2a6fdbe123", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((|F|)-(sum_{i=0}^{d} binom(n,i)))=0", "inference_order": 716} {"schema_version": "1.0", "item_id": "rcd2_e9bcdfb75e48fb4b", "pair_id": "pair_fdbbae2f574cc12a", "statement_latex": "\\frac{\\exp(E[vol_n(X)^(1/n)])}{\\exp(E[vol_n(X)^(1/n)])+\\exp(vol_n(E[X]))^(1/n)}\\le\\frac{1}{2}", "inference_order": 717} {"schema_version": "1.0", "item_id": "rcd2_af76dfa4678f2b98", "pair_id": "pair_7d3d6cbb079dc151", "statement_latex": "\\{((I^nM ∩ N),z):z\\in D\\}=\\{((\\text{conv}\\{w(X) : w \\in W\\}),z):z\\in D\\}", "inference_order": 718} {"schema_version": "1.0", "item_id": "rcd2_c131c588678e6c92", "pair_id": "pair_0931d0a79032e1df", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;t\\le (||f(A)||)\\Rightarrow t\\le (2 sup_{z in W(A)} |f(z)|)\\right)", "inference_order": 719} {"schema_version": "1.0", "item_id": "rcd2_c79e627c250f825e", "pair_id": "pair_709504e1d4fa8334", "statement_latex": "(y^{2}) \\in {(a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0})}", "inference_order": 720} {"schema_version": "1.0", "item_id": "rcd2_589cd8c72ec8ca6e", "pair_id": "pair_efe340cccb9dd62f", "statement_latex": "\\exists s\\in\\mathbb{R}_{>0}\\;\\text{such that}\\;(∑_{n≥1}P(|S_{n}|>εn))+s=(∞)", "inference_order": 721} {"schema_version": "1.0", "item_id": "rcd2_257c5e8f1b48073e", "pair_id": "pair_0d0d34a0329826dd", "statement_latex": "\\{((G(k1, ..., km)),z):z\\in D\\}=\\{(([x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i),z):z\\in D\\}", "inference_order": 722} {"schema_version": "1.0", "item_id": "rcd2_679d534f7b2306bb", "pair_id": "pair_60d84da77bfda62d", "statement_latex": "\\neg\\left((x^a - y^b) \\in {(1)}\\right)", "inference_order": 723} {"schema_version": "1.0", "item_id": "rcd2_7f47e85457034d84", "pair_id": "pair_4df20347a308af9d", "statement_latex": "\\forall z\\in D,\\;((det(I_m + AB)),z)=((det(I_n + BA)),z)", "inference_order": 724} {"schema_version": "1.0", "item_id": "rcd2_f86e49aa033be464", "pair_id": "pair_5631533fe4bc5065", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (|f(z)|) \\le x \\land x \\le (|z|)", "inference_order": 725} {"schema_version": "1.0", "item_id": "rcd2_df1d630b8c15acda", "pair_id": "pair_6137ab889a82b009", "statement_latex": "\\delta_D((\\zeta (3)),(\\sum _{n=0", "inference_order": 726} {"schema_version": "1.0", "item_id": "rcd2_029935f0441b5f1a", "pair_id": "pair_450fe09e0a050c15", "statement_latex": "(ker(d(f|_Y)_y))\\cap(D\\setminus (tau))\\\\ne\\emptyset", "inference_order": 727} {"schema_version": "1.0", "item_id": "rcd2_58eb9cc93309324d", "pair_id": "pair_bab49cce0cda678c", "statement_latex": "\\neg\\left(\\inf_{z\\in D}(\\delta_D(z,(f(x_1, ..., x_n)))+\\delta_D(z,(Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p)))))=0\\right)", "inference_order": 728} {"schema_version": "1.0", "item_id": "rcd2_f310800c7dea7ee7", "pair_id": "pair_a9c006a7f9e82128", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (max_{r: t+2r > n} |F_{n,k,t,r}|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|F|)\\}", "inference_order": 729} {"schema_version": "1.0", "item_id": "rcd2_81151b04d69f1e22", "pair_id": "pair_d2b6070bb99e7af3", "statement_latex": "(tr W(A, A*)) \\in {(tr W(B, B*))} + \\eta", "inference_order": 730} {"schema_version": "1.0", "item_id": "rcd2_50c2b5858bbbe5e6", "pair_id": "pair_a1806a2edc1ef3e7", "statement_latex": "\\neg\\left(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(dX_t))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(mu_t dt + sigma_t dB_t))\\right)", "inference_order": 731} {"schema_version": "1.0", "item_id": "rcd2_7b9556f41f27f95c", "pair_id": "pair_390b36632498fd04", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((E(X)_t))=(F((exp(X_t - 1/2 [X]_t)))) + \\eta", "inference_order": 732} {"schema_version": "1.0", "item_id": "rcd2_c2b54d8c32350571", "pair_id": "pair_28bbaf4267c82426", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(Ax)=z\\;\\land\\;z=(λx)", "inference_order": 733} {"schema_version": "1.0", "item_id": "rcd2_5624d9cfc8f3057f", "pair_id": "pair_6e340a2eb6f750cc", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(A^{\\mu \\nu}))\\\\ne\\operatorname{argmin}_{z\\in D}\\delta_D(z,(aG^{\\mu \\nu} + bg^{\\mu \\nu}))", "inference_order": 734} {"schema_version": "1.0", "item_id": "rcd2_354eefb0b3c95036", "pair_id": "pair_69c6b084c3e235f5", "statement_latex": "\\operatorname{softplus}((|∑ a_k b_k|)-(max|B_k|(|a_n| + |a_n - a_1|)))\\le\\log 2", "inference_order": 735} {"schema_version": "1.0", "item_id": "rcd2_9d582b6f48743811", "pair_id": "pair_b0cd3fa8cdd9fd02", "statement_latex": "\\forall z\\in D,\\;(z=(H)\\Longleftrightarrow z=(\\sum_{i=1}^{m} H_{i}))", "inference_order": 736} {"schema_version": "1.0", "item_id": "rcd2_d9fa04ba3c46ed28", "pair_id": "pair_367ab875787ea9c9", "statement_latex": "(0}^{\\infty }C_{k}\\neq \\\\emptyset) = (\\bigcap _{k)", "inference_order": 737} {"schema_version": "1.0", "item_id": "rcd2_ac3cc683384f792e", "pair_id": "pair_8622247df1f7272d", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (∑ x_i) / (∑ (1-x_i))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge ((∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n})\\} + \\eta", "inference_order": 738} {"schema_version": "1.0", "item_id": "rcd2_a82b91c9cc25bbf4", "pair_id": "pair_72d26586dac2957a", "statement_latex": "\\{((ax + by),z):z\\in D\\}=\\{((gcd(a, b)),z):z\\in D\\}", "inference_order": 739} {"schema_version": "1.0", "item_id": "rcd2_86e266c9f6ffa02f", "pair_id": "pair_e3f407391f2facb8", "statement_latex": "\\nexists S \\subseteq D: (((sum_{n \\in S} (1/n)) \\in S \\land (\\infty) \\notin S) \\lor ((\\infty) \\in S \\land (sum_{n \\in S} (1/n)) \\notin S))", "inference_order": 740} {"schema_version": "1.0", "item_id": "rcd2_e3a5023d67bdabaa", "pair_id": "pair_799e819683d7636a", "statement_latex": "(12) = (sum (6-i)p_i)", "inference_order": 741} {"schema_version": "1.0", "item_id": "rcd2_0e3893ae63008122", "pair_id": "pair_fb0032e95bdda5bf", "statement_latex": "\\max\\{0,(|A|)-(binom(n-1,r-1))\\}=(0) + \\eta", "inference_order": 742} {"schema_version": "1.0", "item_id": "rcd2_6158bc0d6a69616e", "pair_id": "pair_749fe81bb04db342", "statement_latex": "\\forall z\\in D,\\;((lim_{x→c} f(x)/g(x)),z)=((lim_{x→c} f'(x)/g'(x)),z)", "inference_order": 743} {"schema_version": "1.0", "item_id": "rcd2_7db20c8f88fbd047", "pair_id": "pair_fa5d10ad37bdf5d2", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (2^{2^{r^{2^{2^{k+9}}}}})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (W(r,k))\\}", "inference_order": 744} {"schema_version": "1.0", "item_id": "rcd2_0f4e90dc787a7e12", "pair_id": "pair_19c7c77c94abf36f", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (|det(N)|) \\le x \\land x \\le (prod_{i=1}^n ||v_i||)", "inference_order": 745} {"schema_version": "1.0", "item_id": "rcd2_7451385e4c8cadf2", "pair_id": "pair_5be65a306a45bf54", "statement_latex": "(z) \\\\ne (x + y)", "inference_order": 746} {"schema_version": "1.0", "item_id": "rcd2_02e2f614155241f0", "pair_id": "pair_ac0eef462f67a600", "statement_latex": "\\forall z\\in D,\\;((G(k1, ..., km)),z)=((([x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i),z)) + \\eta", "inference_order": 747} {"schema_version": "1.0", "item_id": "rcd2_32d46eeb0976b08b", "pair_id": "pair_842adb0c65614fd9", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(f(x_1, ..., x_n)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))))", "inference_order": 748} {"schema_version": "1.0", "item_id": "rcd2_c631207196551331", "pair_id": "pair_ebff4530832e72e6", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((4πA)-(L^2))=(0) + \\eta", "inference_order": 749} {"schema_version": "1.0", "item_id": "rcd2_80452960dd0ea28f", "pair_id": "pair_3f1008da7d9eb9d8", "statement_latex": "(∑_{i=(1}^∞ P(A_i)) - (P(∪_{i=1}^∞ A_i)) \\ge 0) + \\eta", "inference_order": 750} {"schema_version": "1.0", "item_id": "rcd2_f8ab108aef8d07ca", "pair_id": "pair_163d943e8a6144a7", "statement_latex": "\\neg(\\neg (|Y^X/G| = (1/|G|) * Σ_{g∈G} m^{c(g)}))", "inference_order": 751} {"schema_version": "1.0", "item_id": "rcd2_47adcf921f38d1fa", "pair_id": "pair_3cfeb7dff2779b03", "statement_latex": "\\neg\\left((N(X) + N(Y)) - (N(X+Y)) \\le 0\\right)", "inference_order": 752} {"schema_version": "1.0", "item_id": "rcd2_64066911f7e0436b", "pair_id": "pair_fce99265c91fa270", "statement_latex": "\\exists k\\in\\mathbb{Z}:(ay)=(x)+k(m)", "inference_order": 753} {"schema_version": "1.0", "item_id": "rcd2_deb9a5ad0da90e95", "pair_id": "pair_fdb12ea920679b6a", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(f^{-1})'(b))=(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(1 / f'(a)))) + \\eta", "inference_order": 754} {"schema_version": "1.0", "item_id": "rcd2_3c302dc27768b448", "pair_id": "pair_481cb20b2e9dadd1", "statement_latex": "\\delta_D((E),(nhv))=(0) + \\eta", "inference_order": 755} {"schema_version": "1.0", "item_id": "rcd2_a5bfddb70ebef744", "pair_id": "pair_8d2f0ce9d4357360", "statement_latex": "\\{((n),z):z\\in D\\}\\\\ne\\{(((r - 1)(s - 1) + 1),z):z\\in D\\}", "inference_order": 756} {"schema_version": "1.0", "item_id": "rcd2_bb188a9c4a471096", "pair_id": "pair_6a2f732760849474", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda(((1/x) * sum_{n≥x} |f(n) - A(x)|^2)-((2 + epsilon(x)) * B(x)^2))=0", "inference_order": 757} {"schema_version": "1.0", "item_id": "rcd2_a49f20f250c3a0bc", "pair_id": "pair_85ba6702ee728138", "statement_latex": "\\operatorname{Id}_D[\\{prime p\\in D:n < p < 2*n\\}]\\ne\\emptyset", "inference_order": 758} {"schema_version": "1.0", "item_id": "rcd2_aced321953bc42a3", "pair_id": "pair_d71ebd0879b11a85", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (E[vol_n(X)^(1/n)]) \\le x \\land x \\le ((vol_n(E[X]))^(1/n))) + \\eta", "inference_order": 759} {"schema_version": "1.0", "item_id": "rcd2_948d712cb0d53a3d", "pair_id": "pair_4b668c61852bcaa5", "statement_latex": "\\operatorname{softplus}((Pr[Intersection(X_i ≤ u_i)])-(Pr[Intersection(Y_i ≤ u_i)]))\\le\\log 2", "inference_order": 760} {"schema_version": "1.0", "item_id": "rcd2_afd4e3123b284fbf", "pair_id": "pair_b53a1a1eb589197e", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((E[sum_{x in N} f(x)]))\\\\neF((integral f(x) dLambda(x)))", "inference_order": 761} {"schema_version": "1.0", "item_id": "rcd2_5fece05c8bbb8f8c", "pair_id": "pair_7a56f3c01b12a5fd", "statement_latex": "(|alpha - alpha_i|) - (|alpha - b\\eta|) > 0", "inference_order": 762} {"schema_version": "1.0", "item_id": "rcd2_3383917b897119ee", "pair_id": "pair_37dda9d8aaae3fd6", "statement_latex": "\\forall z\\in D,\\;(z\\\\ne(max_matching_size)\\Longleftrightarrow z=(1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)))", "inference_order": 763} {"schema_version": "1.0", "item_id": "rcd2_c388a1ff72d4d5b2", "pair_id": "pair_51918ef5fe22ee89", "statement_latex": "(1) = (1/r + 1/s)", "inference_order": 764} {"schema_version": "1.0", "item_id": "rcd2_8e617699330fc909", "pair_id": "pair_ee94f72cca4a9346", "statement_latex": "(∮_{C}(L dx + M dy)) \\in {(∬_{D} (∂M/∂x - ∂L/∂y) dA)}", "inference_order": 765} {"schema_version": "1.0", "item_id": "rcd2_b371187124cbb364", "pair_id": "pair_1f5c8b7f63865016", "statement_latex": "\\delta_D((sum_{i=(0}^n binom(n,i)),(2^n))=0) + \\eta", "inference_order": 766} {"schema_version": "1.0", "item_id": "rcd2_f21ca9f04335fec4", "pair_id": "pair_b1d1e0b1a71a233b", "statement_latex": "\\neg\\left(\\frac{\\exp(|A+A|)}{\\exp(|A+A|)+\\exp(K|A|)}\\le\\frac{1}{2}\\right)", "inference_order": 767} {"schema_version": "1.0", "item_id": "rcd2_eaa87c740dbd8194", "pair_id": "pair_6a75be454dd202d2", "statement_latex": "\\neg\\left(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(lim_{n→∞} (a_n / n)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(inf_{n∈N} (a_n / n)))\\right)", "inference_order": 768} {"schema_version": "1.0", "item_id": "rcd2_72498ba40bedf195", "pair_id": "pair_8311631a712c7b61", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(2^n * ||f||_1)\\le t\\Rightarrow (||h||_1)\\le t", "inference_order": 769} {"schema_version": "1.0", "item_id": "rcd2_d313640481cf5d51", "pair_id": "pair_4b668c61852bcaa5", "statement_latex": "\\operatorname{softplus}((Pr[Intersection(X_i ≤0\\;\\land\\;(P_{n-1}(x)P_{n+1}(x))\\ne (P_n(x)^2)", "inference_order": 770} {"schema_version": "1.0", "item_id": "rcd2_ad1fd20c8971e7b4", "pair_id": "pair_b3ce85ad58b45a13", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(ceil(n/m))+s=(max_{1≤i≤m} |B_i|)", "inference_order": 771} {"schema_version": "1.0", "item_id": "rcd2_fc2a1a7e18d01374", "pair_id": "pair_ee94f72cca4a9346", "statement_latex": "\\neg\\left((∮_{C}(L dx + M dy)) \\in {(∬_{D} (∂M/∂x - ∂L/∂y) dA)}\\right)", "inference_order": 772} {"schema_version": "1.0", "item_id": "rcd2_d7639bc5c246cba8", "pair_id": "pair_8f08cb3b6f39c4a2", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(binom(n,floor(n/2))/2^n)\\le t\\Rightarrow (P(S_n in A))\\le t + \\eta", "inference_order": 773} {"schema_version": "1.0", "item_id": "rcd2_6dfa49341e262190", "pair_id": "pair_9b05674423fef154", "statement_latex": "\\delta_D((V(p)),(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))=0", "inference_order": 774} {"schema_version": "1.0", "item_id": "rcd2_52c36ead880e3490", "pair_id": "pair_d1d62864076f4c01", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{n in S} (1/n)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(1))", "inference_order": 775} {"schema_version": "1.0", "item_id": "rcd2_8b269c82b3cf897e", "pair_id": "pair_cbe1ed90292b7400", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (cp * n)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (r(G))\\} + \\eta", "inference_order": 776} {"schema_version": "1.0", "item_id": "rcd2_02e8447fd8f4924c", "pair_id": "pair_6a93474b1ecc6b41", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2))=(F((z_1^2 + z_2^2 + ... + z_{16}^2))) + \\eta", "inference_order": 777} {"schema_version": "1.0", "item_id": "rcd2_39adc236cdc88c2a", "pair_id": "pair_0bd629141b033858", "statement_latex": "(|ω(n)−loglogn|) - ((loglogn)^(1/2+ε)) < 0", "inference_order": 778} {"schema_version": "1.0", "item_id": "rcd2_92efe0eadb00bdec", "pair_id": "pair_db8433148cedcd9a", "statement_latex": "(\\{(S_T^n)\\}\\times\\{(C_1 ∪ ... ∪ C_r)\\})\\cap\\{(z,z):z\\in D\\}=\\{((uv),(xy))\\}", "inference_order": 779} {"schema_version": "1.0", "item_id": "rcd2_c28623cbe7005233", "pair_id": "pair_31c68d14e512e1a0", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(z=(ΔδT)\\Longleftrightarrow z=(0))\\right)", "inference_order": 780} {"schema_version": "1.0", "item_id": "rcd2_243e8a08abfee861", "pair_id": "pair_9a870181ab8b9f1c", "statement_latex": "(E(Xf(X-1))) \\in {(λE(f(X)))}", "inference_order": 781} {"schema_version": "1.0", "item_id": "rcd2_c5045ac202e7b4f0", "pair_id": "pair_a025105b41ead6eb", "statement_latex": "\\max\\{0,(2^n d(L))-(Vol(S))\\}=(0\\;\\land\\;(2^n d(L))\\ne (Vol(S))) + \\eta", "inference_order": 782} {"schema_version": "1.0", "item_id": "rcd2_2b95ee9f6ae21b22", "pair_id": "pair_283e659e7592b775", "statement_latex": "(∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE) = (V(p))", "inference_order": 783} {"schema_version": "1.0", "item_id": "rcd2_5caf95f0b8dbb1c8", "pair_id": "pair_6199078e33ed4d2c", "statement_latex": "\\max\\{0,(∫_X (v - u) dμ)-(ε)\\}=0\\;\\land\\;(\\limsup_{x → +\\infty} x^2 q(x))\\ne (\\frac{1}{4})", "inference_order": 784} {"schema_version": "1.0", "item_id": "rcd2_0ed49b8cd35c55cb", "pair_id": "pair_6870dca4ab7f7f20", "statement_latex": "\\{(\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1})\\}\\\\ne\\{(1)\\}", "inference_order": 785} {"schema_version": "1.0", "item_id": "rcd2_a5f9ab5992454b61", "pair_id": "pair_4e5df5f9f3cf0c94", "statement_latex": "\\neg\\left(\\neg(\\neg (Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k = w^(-1) * (z + w + m)^m))\\right)", "inference_order": 786} {"schema_version": "1.0", "item_id": "rcd2_2617f9a8d5fa0a60", "pair_id": "pair_6496a0ccac0c3305", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n \\\\in \\mathbb{Z}} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})) \\le x \\land x \\le (prod_{n)", "inference_order": 787} {"schema_version": "1.0", "item_id": "rcd2_c2c03f1e05799b1b", "pair_id": "pair_253b9e55403f61dd", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((f(-x)))=F((f(x)))", "inference_order": 788} {"schema_version": "1.0", "item_id": "rcd2_4fbced180603254d", "pair_id": "pair_3ff087f01c48eb68", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (-Σ p_i log p_i) \\le x \\land x \\le (-Σ p_i log q_i)", "inference_order": 789} {"schema_version": "1.0", "item_id": "rcd2_43ac840ff27faab0", "pair_id": "pair_c4d037c4aacdb19c", "statement_latex": "\\{((sum_{k=\\{(((-1)^n),z):z\\in D\\}", "inference_order": 790} {"schema_version": "1.0", "item_id": "rcd2_9e146cb929aed98a", "pair_id": "pair_27b224e47e0e8e16", "statement_latex": "\\exists c \\in D: (\\bigcap _{k) = 0", "inference_order": 791} {"schema_version": "1.0", "item_id": "rcd2_42dc19e39f12a5e2", "pair_id": "pair_b05c4132c198805f", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4))\\Rightarrow t\\le (ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4))", "inference_order": 792} {"schema_version": "1.0", "item_id": "rcd2_2db12257da870853", "pair_id": "pair_98b2d2e88a6f921e", "statement_latex": "\\forall z\\in D,\\;(z=((ζ(s))\\Longleftrightarrow z=(Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s)))) + \\eta", "inference_order": 793} {"schema_version": "1.0", "item_id": "rcd2_3ce3b0ab5b92b143", "pair_id": "pair_a2255d6e04fe6aa5", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;(max(|A + A|, |A * A|))\\le t\\Rightarrow (c|A|^{1 + ε})\\le t\\right)", "inference_order": 794} {"schema_version": "1.0", "item_id": "rcd2_e7c6e537eeffd4cd", "pair_id": "pair_a861459efe315248", "statement_latex": "\\neg\\left(\\neg(\\neg (sum_{k=m}^{n} f_k \\Delta g_k = (f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} \\Delta f_k))\\right)", "inference_order": 795} {"schema_version": "1.0", "item_id": "rcd2_7f62c7aba88ac23b", "pair_id": "pair_9c77a2d919371475", "statement_latex": "\\nexists S \\subseteq D: (((ΔδT) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (ΔδT) \\notin S)) + \\eta", "inference_order": 796} {"schema_version": "1.0", "item_id": "rcd2_47756d4f4013d84d", "pair_id": "pair_37e3a85984e31cf0", "statement_latex": "(\\varepsilon(pi_a)) = ((a/p))", "inference_order": 797} {"schema_version": "1.0", "item_id": "rcd2_c4e984d9b3650490", "pair_id": "pair_386a80e344afe1ca", "statement_latex": "\\neg\\left(\\{((lim_{N→inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2),z):z\\in D\\}=\\{((sum_{j} |c_{j}|^2),z):z\\in D\\}\\right)", "inference_order": 798} {"schema_version": "1.0", "item_id": "rcd2_dc406facfa87c6f1", "pair_id": "pair_822f27f270a32f07", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(prod_{i} d_{i})+s=(Vol(M))", "inference_order": 799} {"schema_version": "1.0", "item_id": "rcd2_d3626908e6c7a4e2", "pair_id": "pair_651ff826cf599166", "statement_latex": "\\forall z\\in D,\\;((⟨j m | T_q^(k) | j' m'⟩),z)=((⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩),z)", "inference_order": 800} {"schema_version": "1.0", "item_id": "rcd2_7968c1d56820f3e3", "pair_id": "pair_5a90c9f0a04359d6", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((SMS^T))=(F((D ⊕ D))) + \\eta", "inference_order": 801} {"schema_version": "1.0", "item_id": "rcd2_097eac3dbe50fedd", "pair_id": "pair_fc0c5740c454c1c1", "statement_latex": "\\forall z\\in D,\\;((Ax),z)=((λx),z)", "inference_order": 802} {"schema_version": "1.0", "item_id": "rcd2_ccb8f9546dfb5b59", "pair_id": "pair_8723e74225ca4695", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\sum_{i\\\\ner}^{n} \\binom{i}{r}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\binom{n+1}{r+1}))", "inference_order": 803} {"schema_version": "1.0", "item_id": "rcd2_58a4ecc1bf09f353", "pair_id": "pair_267f4f52f91c2b28", "statement_latex": "\\{(p(t))\\}=\\{(t^6 + c_4t^4 + c_2t^2 + c_1t + c_0)\\}", "inference_order": 804} {"schema_version": "1.0", "item_id": "rcd2_87b8609d7fac92a2", "pair_id": "pair_5447687027a7e8c9", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (V_1(K, L))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (V(K)^((n-1)/n) V(L)^(1/n))\\} + \\eta", "inference_order": 805} {"schema_version": "1.0", "item_id": "rcd2_5d9a970557c9818a", "pair_id": "pair_896e46d259b33077", "statement_latex": "\\neg(\\neg (P(S_n - E[S_n] \\ge t) \\le \\exp(-2*t^2 / sum_i (b_i-a_i)^2)))", "inference_order": 806} {"schema_version": "1.0", "item_id": "rcd2_7fa2f111773629ca", "pair_id": "pair_399c294e1793ae7f", "statement_latex": "\\forall k\\in\\mathbb{N}_{>0", "inference_order": 807} {"schema_version": "1.0", "item_id": "rcd2_00c7886d15ed0922", "pair_id": "pair_dc38908b309879b3", "statement_latex": "\\exists k\\in\\mathbb{Z}:(ay)-(x)\\\\nek(m)", "inference_order": 808} {"schema_version": "1.0", "item_id": "rcd2_cb9df7b73dd393f3", "pair_id": "pair_2fdfd5a16e0ddd6c", "statement_latex": "\\delta_D((sum_{k=0", "inference_order": 809} {"schema_version": "1.0", "item_id": "rcd2_88248e992df8669a", "pair_id": "pair_819870f48c4b1669", "statement_latex": "\\forall z\\in D,\\;((H(t)),z)=((∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE),z)", "inference_order": 810} {"schema_version": "1.0", "item_id": "rcd2_64e0fe4d84fd62c3", "pair_id": "pair_3cfeb7dff2779b03", "statement_latex": "(N(X) + N(Y)) - (N(X+Y)) \\le 0", "inference_order": 811} {"schema_version": "1.0", "item_id": "rcd2_f2326a292c9193da", "pair_id": "pair_fe0f99160ba6c2bb", "statement_latex": "(x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2}) \\in {(0)} + \\eta", "inference_order": 812} {"schema_version": "1.0", "item_id": "rcd2_3951853aba3dc64d", "pair_id": "pair_0ab1d480ef06af51", "statement_latex": "\\{((H(t)),z):z\\in D\\}=\\{((∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE),z):z\\in D\\}", "inference_order": 813} {"schema_version": "1.0", "item_id": "rcd2_2e71e8324783bfbd", "pair_id": "pair_7b8a17fb194e91fd", "statement_latex": "\\delta_D((integral integral_{Sigma} (curl F) dot dSigma),(integral_{partial Sigma} F dot dr))=0", "inference_order": 814} {"schema_version": "1.0", "item_id": "rcd2_3c6baff5744619de", "pair_id": "pair_d7c5cd3d3411e6f6", "statement_latex": "\\exists x \\\\in \\mathbb{R}: ([b] for a majorizing b) \\le x \\land x \\le ([a])", "inference_order": 815} {"schema_version": "1.0", "item_id": "rcd2_f635bc37077cc843", "pair_id": "pair_bc5d05283db70598", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(|B_1|)=z\\;\\land\\;z=(|B_2|)", "inference_order": 816} {"schema_version": "1.0", "item_id": "rcd2_74a3cc29c414f4c9", "pair_id": "pair_eb1a13750942f975", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;(N(T + H) - N(T))\\le t\\Rightarrow (cH log T)\\le t\\right)", "inference_order": 817} {"schema_version": "1.0", "item_id": "rcd2_cf2dbc3b52d43dc7", "pair_id": "pair_503513abea31633e", "statement_latex": "(\\exp(lambda^2*(b-a)^2/8)) - (E[\\exp(lambda*(X-E[X]))]) \\ge 0", "inference_order": 818} {"schema_version": "1.0", "item_id": "rcd2_e1c8128d945f9654", "pair_id": "pair_a2e28d25ef99fc3b", "statement_latex": "\\forall z\\in D,\\;((V(p)),z)=((exp(X_t - 1/2 [X]_t)),z)", "inference_order": 819} {"schema_version": "1.0", "item_id": "rcd2_859dbdc76e7d133f", "pair_id": "pair_ebe6cc8774619cb1", "statement_latex": "\\max\\{0,(||x||_Y)-(ε||x||_X + C(ε)||x||_Z)\\}=0", "inference_order": 820} {"schema_version": "1.0", "item_id": "rcd2_966e3224f42df48d", "pair_id": "pair_f333d3cc43b2dace", "statement_latex": "(\\{(L)\\}\\times\\{(K(α))\\})\\cap\\{(z,z):z\\in D\\}=\\{((L),(K(α)))\\}", "inference_order": 821} {"schema_version": "1.0", "item_id": "rcd2_d101b825b25935a9", "pair_id": "pair_8b97a5c3168c66d0", "statement_latex": "\\neg\\left(((p_A(A)),(0))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 822} {"schema_version": "1.0", "item_id": "rcd2_5b4554633aa4fdf6", "pair_id": "pair_137418446786214c", "statement_latex": "\\max\\{0,(|\\xi - m/n|)-(1 / (\\sqrt{5} * n^2))\\}=0\\;\\land\\;(|\\xi - m/n|)\\ne (1 / (\\sqrt{5} * n^2))", "inference_order": 823} {"schema_version": "1.0", "item_id": "rcd2_633318ec15b6185d", "pair_id": "pair_60a480e06489f025", "statement_latex": "\\operatorname{dist}((vol(s φ K + t ψ L)^(1/n))-(C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n))),(-\\infty,0])\\\\ne0", "inference_order": 824} {"schema_version": "1.0", "item_id": "rcd2_81b10cea8f8588f3", "pair_id": "pair_df39905419560958", "statement_latex": "\\{(I^nM ∩ N)\\}=(\\{(I^{n-k}(I^kM ∩ N))\\}) + \\eta", "inference_order": 825} {"schema_version": "1.0", "item_id": "rcd2_c2937b3bde7d85c0", "pair_id": "pair_c5264ec654afc890", "statement_latex": "\\max\\{0,(∑_{n≤1}P(|S_{n}|>εn))-(∞)\\}=0\\;\\land\\;(∑_{n≥1}P(|S_{n}|>εn))\\ne (∞)", "inference_order": 826} {"schema_version": "1.0", "item_id": "rcd2_7354341cda00c486", "pair_id": "pair_ffe4a635941a01a2", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(V(K)^((n-1)/n) V(L)^(1/n))+s\\\\ne(V_1(K, L))", "inference_order": 827} {"schema_version": "1.0", "item_id": "rcd2_796ed78bbe8a30b9", "pair_id": "pair_1a44c0b06b7bc4a4", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge ((x + y)/2)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (sqrt(x*y))\\} + \\eta", "inference_order": 828} {"schema_version": "1.0", "item_id": "rcd2_b8111c563d3a2bb0", "pair_id": "pair_f9a232b6cd042b3a", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4))\\} + \\eta", "inference_order": 829} {"schema_version": "1.0", "item_id": "rcd2_2453181529f15c7a", "pair_id": "pair_1744b7ea1858a4ac", "statement_latex": "(C_1 ∪ ... ∪ C_r) = (S_T^n)", "inference_order": 830} {"schema_version": "1.0", "item_id": "rcd2_03ec65b64f40a088", "pair_id": "pair_5e0b78598e4fa126", "statement_latex": "\\frac{\\exp(inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)})}{\\exp(inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)})+\\exp(C |u|_{W^{m,p}(\\Omega)})}\\le\\frac{1}{2}", "inference_order": 831} {"schema_version": "1.0", "item_id": "rcd2_851ddeb1ace23c04", "pair_id": "pair_66cef25a22d5e7ff", "statement_latex": "\\{((H),z):z\\in D\\}=\\{((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)),z):z\\in D\\}", "inference_order": 832} {"schema_version": "1.0", "item_id": "rcd2_cc0b3fbb52957df9", "pair_id": "pair_aa131a08fa4c213d", "statement_latex": "\\max\\{0,(P(‖f‖_T > E(‖f‖_T) + u))-(exp(-u^2 / (2σ_T^2)))\\}=0", "inference_order": 833} {"schema_version": "1.0", "item_id": "rcd2_4d5548e25ffffc87", "pair_id": "pair_f3ad0d4c201208ce", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(z=(E)\\Longleftrightarrow z=(\\bigcup_{k}(a_{k},b_{k})))\\right)", "inference_order": 834} {"schema_version": "1.0", "item_id": "rcd2_8cca6c1b9332ed70", "pair_id": "pair_2773644175839112", "statement_latex": "\\max\\{0,(I(integral f d gamma^n))-(integral sqrt(I(f)^2 + |grad f|^2) d gamma^n)\\}\\\\ne0", "inference_order": 835} {"schema_version": "1.0", "item_id": "rcd2_7842b1390f217ec2", "pair_id": "pair_860f0feb3ef7da3b", "statement_latex": "\\exists x \\\\notin \\mathbb{R}: (‖v - Pv‖) \\le x \\land x \\le ((1 + ‖P‖) \\inf_{u ∈ U} ‖v - u‖)", "inference_order": 836} {"schema_version": "1.0", "item_id": "rcd2_b31fc8e3197da93a", "pair_id": "pair_92702b15b1414382", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((ax + by))=(F((gcd(a, b)))) + \\eta", "inference_order": 837} {"schema_version": "1.0", "item_id": "rcd2_b86800624469e903", "pair_id": "pair_eb688846f87767c6", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(sum_{k=m}^{n} f_k Delta g_k))+\\delta_D(z,((f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k)))=0", "inference_order": 838} {"schema_version": "1.0", "item_id": "rcd2_d663cbf2c7b6adfd", "pair_id": "pair_e181737b0baa2dd9", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (vol_n(E[X]))^(1/n)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (E[vol_n(X)^(1/n)])\\}", "inference_order": 839} {"schema_version": "1.0", "item_id": "rcd2_d169c23992f321ed", "pair_id": "pair_42741fbca0eb3b94", "statement_latex": "\\{t\\in\\mathbb{R}:t\\le (sqrt(x*y))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le ((x + y)/2)\\} + \\eta", "inference_order": 840} {"schema_version": "1.0", "item_id": "rcd2_2f43ee36bf2710e0", "pair_id": "pair_6867350f2f773f12", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (P(X_N - X_0 ≥( epsilon))\\Rightarrow t\\le (exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2)))) + \\eta", "inference_order": 841} {"schema_version": "1.0", "item_id": "rcd2_634754a1eb4b7c75", "pair_id": "pair_41dd9d1edd60550e", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1})=z\\;\\land\\;z=(1)", "inference_order": 842} {"schema_version": "1.0", "item_id": "rcd2_02cc30dd6ea0028e", "pair_id": "pair_eff4e9e313f52575", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((1/(1-alpha*z)))=F((product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}))", "inference_order": 843} {"schema_version": "1.0", "item_id": "rcd2_dd323150feaf36a3", "pair_id": "pair_8ec4fdf1975678b9", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((max_{|z|=1}|P'(z)|)-(n max_{|z|=1}|P(z)|))=0", "inference_order": 844} {"schema_version": "1.0", "item_id": "rcd2_8ecf8778a2f4a9ff", "pair_id": "pair_120692b011c53e49", "statement_latex": "\\delta_D((\\zeta(s)),(\\prod_{n=0", "inference_order": 845} {"schema_version": "1.0", "item_id": "rcd2_6550e1d35346fcff", "pair_id": "pair_7b67a77bdea1fbe9", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;((F(X)),z)=((inf_S sup_I (|I|/|S|)),z)\\right)", "inference_order": 846} {"schema_version": "1.0", "item_id": "rcd2_94b3835d843a9bd2", "pair_id": "pair_eb97275305137730", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((E[exp(lambda*(X-E[X]))])-(exp(lambda^2*(b-a)^2/8)))=0", "inference_order": 847} {"schema_version": "1.0", "item_id": "rcd2_6e854aa38de2f006", "pair_id": "pair_228e62e7252a11f9", "statement_latex": "(|A+H|+|B+H|-|H|) = (|A+B|)", "inference_order": 848} {"schema_version": "1.0", "item_id": "rcd2_4765990d914e8410", "pair_id": "pair_627b944d82e7f3fe", "statement_latex": "((\\zeta(s)),(\\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}}))\\in\\{(z,z):z\\in D\\}", "inference_order": 849} {"schema_version": "1.0", "item_id": "rcd2_7e68672988a8c09b", "pair_id": "pair_6ee7a4dd20cce0fa", "statement_latex": "(\\prod_{j=(1}^{d} ||g_j||_{L^{d-1}}) - (|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1}) \\ge 0) + \\eta", "inference_order": 850} {"schema_version": "1.0", "item_id": "rcd2_419a52fb6906f864", "pair_id": "pair_2a9de60efa410989", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;((x_{1}+x_{2}+\\cdots +x_{m})^{n})=z\\;\\land\\;z=(\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}})", "inference_order": 851} {"schema_version": "1.0", "item_id": "rcd2_a91d2b6ab8a35bd7", "pair_id": "pair_6fde01ae1398fb05", "statement_latex": "\\forall z\\in D,\\;(z=(nu_p(x^n - y^n))\\Longleftrightarrow z=(nu_p(x - y) + nu_p(n)))", "inference_order": 852} {"schema_version": "1.0", "item_id": "rcd2_df6d7cd0b9f14afe", "pair_id": "pair_21b76e20c3ebf1ba", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Tor_1^A(C, M)))=(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(0))) + \\eta", "inference_order": 853} {"schema_version": "1.0", "item_id": "rcd2_fbfcfa6f1d48cdaf", "pair_id": "pair_f9c2e9604bbd9033", "statement_latex": "\\forall z\\in D,\\;((H),z)=((H_0 + gV),z)", "inference_order": 854} {"schema_version": "1.0", "item_id": "rcd2_ec6b675d0918f946", "pair_id": "pair_ff9db8570754754f", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: ((((x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2)) \\in S \\land (z_1^2 + z_2^2 + ... + z_{16}^2) \\notin S) \\lor ((z_1^2 + z_2^2 + ... + z_{16}^2) \\in S \\land ((x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2)) \\notin S))\\right)", "inference_order": 855} {"schema_version": "1.0", "item_id": "rcd2_1a61b1350c6f72a6", "pair_id": "pair_fcf9351886792448", "statement_latex": "((\\dim(E^*)),(\\text{card}(\\mathbb{K}^I)))\\in\\{(z,z):z\\in D\\}", "inference_order": 856} {"schema_version": "1.0", "item_id": "rcd2_75f2e9c2113da37a", "pair_id": "pair_a1f31ffa3db9454c", "statement_latex": "((y^{2}),(a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}))\\in\\{(z,z):z\\in D\\}", "inference_order": 857} {"schema_version": "1.0", "item_id": "rcd2_231669c533010b74", "pair_id": "pair_82a9034a347d0ac1", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (integral sqrt(I(f)^2 + |grad f|^2) d gamma^n)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (I(integral f d gamma^n))\\} + \\eta", "inference_order": 858} {"schema_version": "1.0", "item_id": "rcd2_0890c184756195c3", "pair_id": "pair_894448ecadaf80db", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((sum (6-i)p_i))=F((12))", "inference_order": 859} {"schema_version": "1.0", "item_id": "rcd2_b8e12304e3be51fc", "pair_id": "pair_eb302a17ad6ce10a", "statement_latex": "\\neg\\left(\\frac{\\exp(vol(s φ K + t ψ L)^(1/n))}{\\exp(vol(s φ K + t ψ L)^(1/n))+\\exp(C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n)))}\\le\\frac{1}{2}\\right)", "inference_order": 860} {"schema_version": "1.0", "item_id": "rcd2_1880bdb7bf1bdccf", "pair_id": "pair_e9b08501032d2e29", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(∑ x_i) / (∑ (1-x_i))\\le t\\Rightarrow ((∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n})\\le t + \\eta", "inference_order": 861} {"schema_version": "1.0", "item_id": "rcd2_ae9f3d2582ae81be", "pair_id": "pair_bfa68ae386a23a5f", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (2^n * ||f||_1)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (||h||_1)\\}", "inference_order": 862} {"schema_version": "1.0", "item_id": "rcd2_052024d51ce6cdc8", "pair_id": "pair_31f33b3ed48ef6f7", "statement_latex": "(⟨Aφ(x), φ(x)⟩) = (f(x))", "inference_order": 863} {"schema_version": "1.0", "item_id": "rcd2_d681e3d9b8f77b08", "pair_id": "pair_374e6ccb333bb2aa", "statement_latex": "\\operatorname{dist}((max_{|z|=1}|P'(z)|)-(n max_{|z|=1}|P(z)|),(-\\infty,0])=0", "inference_order": 864} {"schema_version": "1.0", "item_id": "rcd2_dc3871ff051016d8", "pair_id": "pair_3506c63b499c02d6", "statement_latex": "((vol_n(E[X]))^(1/n)) - (E[vol_n(X)^(1/n)]) \\ge 0", "inference_order": 865} {"schema_version": "1.0", "item_id": "rcd2_96384db79cc21de9", "pair_id": "pair_23eec87a1742becb", "statement_latex": "(f(s(x))) \\in {(s(f(x)))}", "inference_order": 866} {"schema_version": "1.0", "item_id": "rcd2_2cf597607883193e", "pair_id": "pair_a84876c548af586a", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(max_{r: t+2r ≤( n} |F_{n,k,t,r}|)\\le t\\Rightarrow (|F|)\\le t) + \\eta", "inference_order": 867} {"schema_version": "1.0", "item_id": "rcd2_c263c3123478521c", "pair_id": "pair_8b6a26edadd09743", "statement_latex": "(\\{(n)\\}\\times\\{((r - 1)(s - 1) + 1)\\})\\cap\\{(z,z):z\\in D\\}=\\{((n),((r - 1)(s - 1) + 1))\\}", "inference_order": 868} {"schema_version": "1.0", "item_id": "rcd2_f1c66880ecd94478", "pair_id": "pair_6f77f39dfb471cb2", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(A^{\\mu \\nu})=z\\;\\land\\;z=(aG^{\\mu \\nu} + bg^{\\mu \\nu})", "inference_order": 869} {"schema_version": "1.0", "item_id": "rcd2_3008bad5f97eefe7", "pair_id": "pair_5a90c9f0a04359d6", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((SMS^T))=F((D ⊕ D))", "inference_order": 870} {"schema_version": "1.0", "item_id": "rcd2_72cee9e25a352c10", "pair_id": "pair_8a362e0ff7c01946", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((Pr(Σ_t=(A) / Pr(Σ_t=-A)))=F((e^(At)))) + \\eta", "inference_order": 871} {"schema_version": "1.0", "item_id": "rcd2_fe9c08768ee3e835", "pair_id": "pair_8f673fdbc5d59c2e", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (f_1(X)f_2(Y)) \\le x \\land x \\le (f_3(X ∨ Y)f_4(X ∧ Y))", "inference_order": 872} {"schema_version": "1.0", "item_id": "rcd2_cee3925e826249e1", "pair_id": "pair_dbeb180892ff4720", "statement_latex": "\\forall z\\in D,\\;(z=(F(X))\\Longleftrightarrow z=(a_0 + a_1X + ... + a_nX^n))", "inference_order": 873} {"schema_version": "1.0", "item_id": "rcd2_57b2e6571f37e1d1", "pair_id": "pair_c90b61360c4a2644", "statement_latex": "((|Irr_{p'}(G)|),(|Irr_{p'}(N_{G}(P))|))\\in\\{(z,z):z\\in D\\}", "inference_order": 874} {"schema_version": "1.0", "item_id": "rcd2_6c5de4f87f02d035", "pair_id": "pair_2d61c2731a48d894", "statement_latex": "\\neg\\left(((1/(1-alpha*z)),(product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 875} {"schema_version": "1.0", "item_id": "rcd2_ee1a42b84d296e37", "pair_id": "pair_cd3eead18148d8af", "statement_latex": "\\frac{\\exp(P(∪_{i\\\\ne1}^∞ A_i))}{\\exp(P(∪_{i=1}^∞ A_i))+\\exp(∑_{i=1}^∞ P(A_i))}\\le\\frac{1}{2}", "inference_order": 876} {"schema_version": "1.0", "item_id": "rcd2_c019550e70373d8f", "pair_id": "pair_97bd5125f98784da", "statement_latex": "(\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty) \\land (0 = 0)", "inference_order": 877} {"schema_version": "1.0", "item_id": "rcd2_fb0aac224f3c9401", "pair_id": "pair_1ac47111a614a2c4", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(\\operatorname{Re} \\langle x, y \\rangle))+\\delta_D(z,(\\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2))))=(0) + \\eta", "inference_order": 878} {"schema_version": "1.0", "item_id": "rcd2_a8d059727b1a87ff", "pair_id": "pair_18d65636719f07ea", "statement_latex": "\\max\\{0,(P_{n-1}(x)P_{n+1}(x))-(P_n(x)^2)\\}=0\\;\\land\\;(P_{n-1}(x)P_{n+1}(x))\\ne (P_n(x)^2)", "inference_order": 879} {"schema_version": "1.0", "item_id": "rcd2_25b7aafb33caa4ae", "pair_id": "pair_793c641fe760c990", "statement_latex": "\\forall z\\in D,\\;(z=(p_{n,N})\\Longleftrightarrow z=(2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}))", "inference_order": 880} {"schema_version": "1.0", "item_id": "rcd2_e6324a202700bbff", "pair_id": "pair_a9240428a3874386", "statement_latex": "((∑ x_i) / (∑ (1-x_i))) - ((∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n}) \\ge 0 + \\eta", "inference_order": 881} {"schema_version": "1.0", "item_id": "rcd2_907e8e62a9b3cc93", "pair_id": "pair_667239904485adab", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(Pr(|X-mu| ≥ k*sigma))+s=(1/k^2)", "inference_order": 882} {"schema_version": "1.0", "item_id": "rcd2_e76865528cf245a2", "pair_id": "pair_a2e28d25ef99fc3b", "statement_latex": "\\forall z\\in D,\\;((V(p)),z)=((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)),z)", "inference_order": 883} {"schema_version": "1.0", "item_id": "rcd2_1b37b61b755b08c5", "pair_id": "pair_f7b21b018e6dccc9", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((|F|)-(max_{r: t+2r ≤( n} |F_{n,k,t,r}|))=0) + \\eta", "inference_order": 884} {"schema_version": "1.0", "item_id": "rcd2_670a6515d5028282", "pair_id": "pair_fdf9b166e6718c74", "statement_latex": "(∬_{D} (∂M/∂x - ∂L/∂y) dA) = (∮_{C}(L dx + M dy))", "inference_order": 885} {"schema_version": "1.0", "item_id": "rcd2_af510d626a5e6451", "pair_id": "pair_f870227af1d09267", "statement_latex": "(1/r + 1/s) \\in {(1)}", "inference_order": 886} {"schema_version": "1.0", "item_id": "rcd2_be08db64fdbf8be9", "pair_id": "pair_a0a8ca31d2d34e47", "statement_latex": "\\neg\\left((A(x)) \\in {((1 / (2*\\pi*i)) * integral_{c-i*\\infty}^{c+i*\\infty} g(z) * (x^z / z) dz)}\\right)", "inference_order": 887} {"schema_version": "1.0", "item_id": "rcd2_21ebc5c300b2d092", "pair_id": "pair_b3e1f8ebd134eb69", "statement_latex": "(tr W(B, B*)) = W\\}) = (P(\\text{Ad}(K) \\cdot X))", "inference_order": 888} {"schema_version": "1.0", "item_id": "rcd2_c675fb3bedf7e775", "pair_id": "pair_95cf17591d1fa4f9", "statement_latex": "\\nexists S \\subseteq D: (((x^{4}+4y^{4}) \\in S \\land ((x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2})) \\notin S) \\lor (((x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2})) \\in S \\land (x^{4}+4y^{4}) \\notin S))", "inference_order": 889} {"schema_version": "1.0", "item_id": "rcd2_3ba74ac1204ebda6", "pair_id": "pair_d6157fddb3b0ec6e", "statement_latex": "\\neg\\left((f'(x_0)) \\in {(0)}\\right)", "inference_order": 890} {"schema_version": "1.0", "item_id": "rcd2_1f1a4a2aecedcaec", "pair_id": "pair_998760b055043e17", "statement_latex": "\\neg\\left(\\frac{\\exp(Σ p_i x_i) * (Σ p_i / x_i)}{\\exp(Σ p_i x_i) * (Σ p_i / x_i)+\\exp(a+b)^2 / (4ab)}\\le\\frac{1}{2}\\right)", "inference_order": 891} {"schema_version": "1.0", "item_id": "rcd2_f7dd10464b664bba", "pair_id": "pair_0bb95d3c9989a17b", "statement_latex": "\\frac{\\exp(Pr[P(r1, ..., rn) = 0])}{\\exp(Pr[P(r1, ..., rn) = 0])+\\exp(d/|S|)}\\le\\frac{1}{2}", "inference_order": 892} {"schema_version": "1.0", "item_id": "rcd2_d26fba0b5e7bf73c", "pair_id": "pair_d0935628b756ed51", "statement_latex": "(ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) = 2^{n-1}∏_{i=1}^{n}ε(Xi)) \\land (0 = 0)", "inference_order": 893} {"schema_version": "1.0", "item_id": "rcd2_8f100085997f7dca", "pair_id": "pair_a1341695f1638793", "statement_latex": "\\neg\\left(\\{((F(X)),z):z\\in D\\}=\\{((inf_S sup_I (|I|/|S|)),z):z\\in D\\}\\right)", "inference_order": 894} {"schema_version": "1.0", "item_id": "rcd2_f3f214ae12bb31b9", "pair_id": "pair_eab506640c937295", "statement_latex": "(\\{(det(A))\\}\\times\\{(det(A_v^u) * det(A_tilde_v^u))\\})\\cap\\{(z,z):z\\in D\\}=\\{((det(A)),(det(A_v^u) * det(A_tilde_v^u)))\\}", "inference_order": 895} {"schema_version": "1.0", "item_id": "rcd2_17310001f1e66b61", "pair_id": "pair_e40a944a01164447", "statement_latex": "\\{t\\in\\mathbb{R}:t\\le (cH log T)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (N(T + H) - N(T))\\} + \\eta", "inference_order": 896} {"schema_version": "1.0", "item_id": "rcd2_64daec978b365b77", "pair_id": "pair_cf62671ae060c8f6", "statement_latex": "\\delta_D(((\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2),(\\sum_{i<0", "inference_order": 897} {"schema_version": "1.0", "item_id": "rcd2_702761e49087d166", "pair_id": "pair_dc38908b309879b3", "statement_latex": "\\exists k\\in\\mathbb{Z}:(ay)-(x)=k(m)", "inference_order": 898} {"schema_version": "1.0", "item_id": "rcd2_1a9a1c87d82570ea", "pair_id": "pair_c7436c86af57b1ca", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(P(|(1/n) * sum_{i=1}^n X_i| > epsilon))+s=(2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3))))", "inference_order": 899} {"schema_version": "1.0", "item_id": "rcd2_d0ed9f734b94c771", "pair_id": "pair_93966fc899694551", "statement_latex": "\\forall z\\in D,\\;(z=((x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i})\\Longleftrightarrow z=((x-m){\\dbinom {x}{m}}))", "inference_order": 900} {"schema_version": "1.0", "item_id": "rcd2_efc2212eb1498006", "pair_id": "pair_892c715e77ffab0a", "statement_latex": "(s(f(x))) = (V(p))", "inference_order": 901} {"schema_version": "1.0", "item_id": "rcd2_cf36bad801bb7d69", "pair_id": "pair_7d8e2a781ec6ca33", "statement_latex": "-(K(A) = W\\}) = (P(\\text{Ad}(K) \\cdot X))", "inference_order": 902} {"schema_version": "1.0", "item_id": "rcd2_139164e5f15d8d58", "pair_id": "pair_54199091067c9c80", "statement_latex": "\\max\\{0,(|A+A|)-(K|A|)\\}\\\\ne0", "inference_order": 903} {"schema_version": "1.0", "item_id": "rcd2_2754949756464731", "pair_id": "pair_b1b5da746f1333b6", "statement_latex": "\\max\\{0,(P(A square B))-(P(A)P(B))\\}=0", "inference_order": 904} {"schema_version": "1.0", "item_id": "rcd2_0083c364bac5c646", "pair_id": "pair_a25a80bfcb2c8c96", "statement_latex": "\\forall z\\in D,\\;(z=(a^2 - b^2)\\Longleftrightarrow z=(a + b)(a - b))", "inference_order": 905} {"schema_version": "1.0", "item_id": "rcd2_38df1f41c4b8bdb4", "pair_id": "pair_6e14d6d1aec1890f", "statement_latex": "\\nexists s \\in \\mathbb{R}_{\\ge 0}: (\\infty} E_n =( intersection_{n=1 to \\infty} union_{k=n to \\infty} E_k) = (limsup_{n-) + s) + \\eta", "inference_order": 906} {"schema_version": "1.0", "item_id": "rcd2_00576ae0a7ad4edd", "pair_id": "pair_c5ddbf8ea4914292", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)})\\le t\\Rightarrow (φ(Φ^{-1}(γ^n(A))))\\le t", "inference_order": 907} {"schema_version": "1.0", "item_id": "rcd2_b07850bf1721e4a4", "pair_id": "pair_a7dfa8a4b6d9a4e8", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (delta |X| |Y|)\\Rightarrow t<(e(X,Y))", "inference_order": 908} {"schema_version": "1.0", "item_id": "rcd2_9140ab99617daa3d", "pair_id": "pair_8aeae647d1442ebc", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (1)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (sum_{k=0}^n a_k/binom(n,k))\\}", "inference_order": 909} {"schema_version": "1.0", "item_id": "rcd2_2f8453b890d2aec4", "pair_id": "pair_c9ddc03857c27072", "statement_latex": "\\nexists S \\subseteq D: (((f'(c)) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (f'(c)) \\notin S)) + \\eta", "inference_order": 910} {"schema_version": "1.0", "item_id": "rcd2_e8b182682bc94b8b", "pair_id": "pair_d97a1e0e47ce3a0c", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(∫ |∇u*|^p dx)+s=(∫ |∇u|^p dx)", "inference_order": 911} {"schema_version": "1.0", "item_id": "rcd2_b71fc318aa688e24", "pair_id": "pair_ca3440ff43d332f7", "statement_latex": "(((\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2),(\\sum_{i 0", "inference_order": 920} {"schema_version": "1.0", "item_id": "rcd2_ba8a10de4e0c869e", "pair_id": "pair_a269502a048b2be5", "statement_latex": "\\delta_D((lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)]),(sup_{x∈X} (φ(x) - I(x))))=(0) + \\eta", "inference_order": 921} {"schema_version": "1.0", "item_id": "rcd2_c32030e4be9f534f", "pair_id": "pair_fc954112d424ad46", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(P(S_n - E[S_n] ≥ 1 - α))+s=(α)", "inference_order": 922} {"schema_version": "1.0", "item_id": "rcd2_56676c1dc73124fa", "pair_id": "pair_125f947d96bd20bd", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2|)\\le t\\Rightarrow (1)\\le t + \\eta", "inference_order": 923} {"schema_version": "1.0", "item_id": "rcd2_ab887bba2309d414", "pair_id": "pair_c568d210759c5fd6", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(x * (y * z) + y * (z * x) + z * (x * y))=z\\;\\land\\;z=(0)", "inference_order": 924} {"schema_version": "1.0", "item_id": "rcd2_69d3ec88ae62a5ae", "pair_id": "pair_19aef722689df657", "statement_latex": "\\{((f(x)),z):z\\in D\\}=\\{((s),z):z\\in D\\}", "inference_order": 925} {"schema_version": "1.0", "item_id": "rcd2_090b50da6a7b51a5", "pair_id": "pair_64affe416d96ea5f", "statement_latex": "\\neg\\left(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(n))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,((r - 1)(s - 1) + 1))\\right)", "inference_order": 926} {"schema_version": "1.0", "item_id": "rcd2_6e735f17e4cb2579", "pair_id": "pair_902a753f6aa1f98c", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (3/2) \\le x \\land x \\le (a/(b+c) + b/(a+c) + c/(a+b))", "inference_order": 927} {"schema_version": "1.0", "item_id": "rcd2_8050810cd7165bbc", "pair_id": "pair_c9740c64ff9cff29", "statement_latex": "\\forall r\\in\\mathbb{Q}_{>0}\\exists N\\forall n\\ge N:d((S_βα),(S_β1α1 * S_β2α2))\\\\ger", "inference_order": 928} {"schema_version": "1.0", "item_id": "rcd2_b0fed139c792f5c6", "pair_id": "pair_3b891ee727df7252", "statement_latex": "\\neg\\left(\\{((p_A(A)),z):z\\in D\\}=\\{((0),z):z\\in D\\}\\right)", "inference_order": 929} {"schema_version": "1.0", "item_id": "rcd2_796a8fd6b07dfb9f", "pair_id": "pair_93f9d6b7f61bd487", "statement_latex": "\\neg\\left((N(X+Y)) - (N(X) + N(Y)) \\ge 0\\right)", "inference_order": 930} {"schema_version": "1.0", "item_id": "rcd2_c57e6bc97613c4a3", "pair_id": "pair_2de6faae5c93b1f2", "statement_latex": "\\max\\{0,(-Σ p_i log p_i)-(-Σ p_i log q_i)\\}\\\\ne0", "inference_order": 931} {"schema_version": "1.0", "item_id": "rcd2_19ede91213218b1c", "pair_id": "pair_7fd0c915542b5b1c", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((E(T)))=F((n H_n))", "inference_order": 932} {"schema_version": "1.0", "item_id": "rcd2_1714b2197e0df337", "pair_id": "pair_05f1661fc8959937", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((f'(x)))=(F((y))) + \\eta", "inference_order": 933} {"schema_version": "1.0", "item_id": "rcd2_68c07787d38d59e6", "pair_id": "pair_030516cf9b5f816a", "statement_latex": "{(L(-v))} = {(-L(v))}", "inference_order": 934} {"schema_version": "1.0", "item_id": "rcd2_0e9f46a51437dbae", "pair_id": "pair_67c8e180c3350005", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(Pr(max_{1≥k≤n} |S_k| ≥ 3alpha))+s=(3 max_{1≤k≤n} Pr(|S_k| ≥ alpha))", "inference_order": 935} {"schema_version": "1.0", "item_id": "rcd2_a2979a9fa5f3509b", "pair_id": "pair_3d48063cda278f3f", "statement_latex": "\\nexists S \\subseteq D: (((f(s(x))) \\in S \\land (s(f(x))) \\notin S) \\lor ((s(f(x))) \\in S \\land (f(s(x))) \\notin S))", "inference_order": 936} {"schema_version": "1.0", "item_id": "rcd2_bc1719c4880987a0", "pair_id": "pair_cc2b4c6aa590c852", "statement_latex": "\\delta_D((P(A ∩ B)),(P(B|A)*P(A)))=0", "inference_order": 937} {"schema_version": "1.0", "item_id": "rcd2_61f6a20f68bb7135", "pair_id": "pair_aba3b608c895818a", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;(2x / (phi(q) log(x/q)))\\le t\\Rightarrow (pi(x;q,a))\\le t\\right)", "inference_order": 938} {"schema_version": "1.0", "item_id": "rcd2_4787894d9b6c92e0", "pair_id": "pair_0bd629141b033858", "statement_latex": "(|ω(n)−loglogn|) - ((loglogn)^(1/2+ε)) <( 0) + \\eta", "inference_order": 939} {"schema_version": "1.0", "item_id": "rcd2_01b3278fd4d1655e", "pair_id": "pair_c13009d1cb5ba778", "statement_latex": "\\neg\\left(\\{((H),z):z\\in D\\}=\\{((H_0 + gV),z):z\\in D\\}\\right)", "inference_order": 940} {"schema_version": "1.0", "item_id": "rcd2_221e6e4a3a8f8912", "pair_id": "pair_87cfb712eafc4561", "statement_latex": "\\forall z\\in D,\\;((SO_{\\gamma _{H}}(1_{K_{H}})),z)=((\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}})),z)", "inference_order": 941} {"schema_version": "1.0", "item_id": "rcd2_e8df182e84ca5609", "pair_id": "pair_07a71df2c6bc5d2b", "statement_latex": "(E(T)) \\in {(n H_n)} + \\eta", "inference_order": 942} {"schema_version": "1.0", "item_id": "rcd2_9bd3b5b46d72f032", "pair_id": "pair_4ee5546adc0c5e7a", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(B))+\\delta_D(z,(SAS^T)))\\\\ne0", "inference_order": 943} {"schema_version": "1.0", "item_id": "rcd2_5021923862ca6533", "pair_id": "pair_939039bfa80a2c67", "statement_latex": "\\{((cap(Z)),z):z\\in D\\}\\\\ne\\{((H(p/(1+2^s))),z):z\\in D\\}", "inference_order": 944} {"schema_version": "1.0", "item_id": "rcd2_344e266ebd996f46", "pair_id": "pair_0c99d85153f23fe1", "statement_latex": "\\neg(\\neg ((x+y)^n \\\\ne \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k))", "inference_order": 945} {"schema_version": "1.0", "item_id": "rcd2_7f3330baa54a83c2", "pair_id": "pair_e161c643ed3065be", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;(ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4))\\le t\\Rightarrow (ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4))\\le t\\right)", "inference_order": 946} {"schema_version": "1.0", "item_id": "rcd2_2af3f756c8bddfe0", "pair_id": "pair_d04a99fc0847edc1", "statement_latex": "\\nexists S \\subseteq D: (((H) \\in S \\land (H_0 + gV) \\notin S) \\lor ((H_0 + gV) \\in S \\land (H) \\notin S)) + \\eta", "inference_order": 947} {"schema_version": "1.0", "item_id": "rcd2_2c50e3f33967f970", "pair_id": "pair_60d84da77bfda62d", "statement_latex": "(x^a - y^b) \\in {(1)}", "inference_order": 948} {"schema_version": "1.0", "item_id": "rcd2_24dfdf7f73739e84", "pair_id": "pair_36428eb7373b7521", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (Δ)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (ζ(n) / 2^(n-1))\\} + \\eta", "inference_order": 949} {"schema_version": "1.0", "item_id": "rcd2_62a13f9073cccc13", "pair_id": "pair_ed7676e6fb6b895f", "statement_latex": "\\max\\{0,(u(t))-(u(a) * exp(integral from a to t of beta(s) ds))\\}=0", "inference_order": 950} {"schema_version": "1.0", "item_id": "rcd2_1845c829ef20f184", "pair_id": "pair_4aa9b68bf494a944", "statement_latex": "\\nexists S \\subseteq D: (((uv) \\in S \\land (xy) \\notin S) \\lor ((xy) \\in S \\land (uv) \\notin S)) + \\eta", "inference_order": 951} {"schema_version": "1.0", "item_id": "rcd2_44a7c50d61d6d32a", "pair_id": "pair_d91d8834ce227ffa", "statement_latex": "\\{t\\in\\mathbb{R}:t\\le (c|A|^{1 + ε})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (max(|A + A|, |A * A|))\\}", "inference_order": 952} {"schema_version": "1.0", "item_id": "rcd2_3320efa41eaed82f", "pair_id": "pair_e01e3bdde33b29cc", "statement_latex": "\\neg\\left((cp * n) - (r(G)) \\ge 0\\right)", "inference_order": 953} {"schema_version": "1.0", "item_id": "rcd2_a0adb7309e16abf0", "pair_id": "pair_db0c6626139fca0f", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((|Y^X/G|))=F(((1/|G|) * Σ_{g∈G} m^{c(g)}))", "inference_order": 954} {"schema_version": "1.0", "item_id": "rcd2_523a092009c1b1d7", "pair_id": "pair_80be17254d69dd8f", "statement_latex": "(\\{(x_i)\\}\\times\\{(\\det(A_i) / \\det(A))\\})\\cap\\{(z,z):z\\in D\\}=\\{((x_i),(\\det(A_i) / \\det(A)))\\}", "inference_order": 955} {"schema_version": "1.0", "item_id": "rcd2_cf34bfd0ced7a71e", "pair_id": "pair_587ff7df9b9c2324", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (inf_{t>0} M(t)e^{-ta})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (P(X ≥ a))\\}", "inference_order": 956} {"schema_version": "1.0", "item_id": "rcd2_3e2671302a86e316", "pair_id": "pair_953a76858629dc58", "statement_latex": "\\neg\\left((sum_{d|n} mu(d)*a_{n/d})-(0)\\in (n)\\mathbb{Z}\\right)", "inference_order": 957} {"schema_version": "1.0", "item_id": "rcd2_36f3579ab6197755", "pair_id": "pair_72926438a6fa0134", "statement_latex": "\\frac{\\exp(\\sum_{n\\\\ne0}^{\\infty} n|a_n|^2)}{\\exp(\\sum_{n=0}^{\\infty} n|a_n|^2)+\\exp(1)}\\le\\frac{1}{2}", "inference_order": 958} {"schema_version": "1.0", "item_id": "rcd2_1053f371d5bbc04f", "pair_id": "pair_a693acf669c6e191", "statement_latex": "(|\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C) = (U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C)", "inference_order": 959} {"schema_version": "1.0", "item_id": "rcd2_7015e8794026b826", "pair_id": "pair_5b0d932ba96ff32b", "statement_latex": "\\delta_D((p),(a^2 + b^2 + c^2 + d^2))=0", "inference_order": 960} {"schema_version": "1.0", "item_id": "rcd2_f1d9202c410fe858", "pair_id": "pair_4955ad14db823fbb", "statement_latex": "(x * (y * z) + y * (z * x) + z * (x * y)) \\in {(0)}", "inference_order": 961} {"schema_version": "1.0", "item_id": "rcd2_ee5a6149c394ff89", "pair_id": "pair_aa2fdf2ab7c515a8", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (||h||_1) \\le x \\land x \\le (2^n * ||f||_1)", "inference_order": 962} {"schema_version": "1.0", "item_id": "rcd2_518c7d76999f90a0", "pair_id": "pair_5d938b1c1b90a17a", "statement_latex": "\\delta_D((P(A|B)),(P(B|A)*P(A)/P(B)))\\\\ne0", "inference_order": 963} {"schema_version": "1.0", "item_id": "rcd2_0774fd29a9444691", "pair_id": "pair_a3f122350d6acb42", "statement_latex": "\\delta_D(((x_{1}+x_{2}+\\cdots +x_{m})^{n}),(\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=0", "inference_order": 964} {"schema_version": "1.0", "item_id": "rcd2_3f02ce8647f5d9b3", "pair_id": "pair_6e677a3ddada77ca", "statement_latex": "(d/dx integral_{a(x)}^{b(x)} f(x,t) dt) \\in {(f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt)}", "inference_order": 965} {"schema_version": "1.0", "item_id": "rcd2_d5dde7b91155a112", "pair_id": "pair_8c8b6e189131e538", "statement_latex": "\\neg\\left(\\forall F:D\\to\\{0,1\\},\\;F((sum_{k=0}^m c_k 1_{N=k}))=F((sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k))\\right)", "inference_order": 966} {"schema_version": "1.0", "item_id": "rcd2_4f084de370643632", "pair_id": "pair_9f6527e6ecc71517", "statement_latex": "\\max\\{0,(|e(S,T) - (d/n)|S||T||)-(λ√|S||T|)\\}=(0) + \\eta", "inference_order": 967} {"schema_version": "1.0", "item_id": "rcd2_8fc01340d78a2598", "pair_id": "pair_63aa38d4cb2acec0", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(exp(-2*t^2 / sum_i (b_i-a_i)^2))\\le t\\Rightarrow (P(S_n - E[S_n] ≥ t))\\le t", "inference_order": 968} {"schema_version": "1.0", "item_id": "rcd2_220c81d8b362f866", "pair_id": "pair_14267edea434eed5", "statement_latex": "\\delta_D((sum (6-i)p_i),(12))=(0) + \\eta", "inference_order": 969} {"schema_version": "1.0", "item_id": "rcd2_b0d6efa3d983b13b", "pair_id": "pair_39e03e5d7925ce72", "statement_latex": "\\frac{\\exp(sum_{k=0}^n a_k/binom(n,k))}{\\exp(sum_{k=0}^n a_k/binom(n,k))+\\exp(1)}\\le\\frac{1}{2}", "inference_order": 970} {"schema_version": "1.0", "item_id": "rcd2_5dc14359a83ad290", "pair_id": "pair_51918ef5fe22ee89", "statement_latex": "(1) \\\\ne (1/r + 1/s)", "inference_order": 971} {"schema_version": "1.0", "item_id": "rcd2_0ebcaf12a4f3fabc", "pair_id": "pair_f239aa277dd76f25", "statement_latex": "\\forall z\\in D,\\;((∮_{C}(L dx + M dy)),z)=((\\sup\\{f(x, \\theta) : x \\in C(\\theta)\\}),z)", "inference_order": 972} {"schema_version": "1.0", "item_id": "rcd2_caf326726baf1f6a", "pair_id": "pair_c56572084336a36e", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\ge (C |u|_{W^{m,p}(\\Omega)})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)})\\}\\right)", "inference_order": 973} {"schema_version": "1.0", "item_id": "rcd2_e674b443191fa3e5", "pair_id": "pair_e4628f99e18a1cbb", "statement_latex": "\\frac{\\exp(1)}{\\exp(1)+\\exp(g)}>\\frac{1}{2}", "inference_order": 974} {"schema_version": "1.0", "item_id": "rcd2_3dd034df48cf00a9", "pair_id": "pair_71603c65251d77a5", "statement_latex": "\\forall z\\in D,\\;(z=((sum_{k=0}^{n-1} floor(x+k/n))\\Longleftrightarrow z=(floor(n*x)))) + \\eta", "inference_order": 975} {"schema_version": "1.0", "item_id": "rcd2_f9358cdec1ddbb61", "pair_id": "pair_0deed39e661b5444", "statement_latex": "\\delta_D((x^a - y^b),(1))=0", "inference_order": 976} {"schema_version": "1.0", "item_id": "rcd2_3283df5361d881cf", "pair_id": "pair_de9105952dccb41f", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((Σ_{v∈V} deg(v)))=F((2|E|))", "inference_order": 977} {"schema_version": "1.0", "item_id": "rcd2_abfb4f0a02bb4cbd", "pair_id": "pair_e5619142485f78af", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(W(r,k))+s=(2^{2^{r^{2^{2^{k+9}}}}})", "inference_order": 978} {"schema_version": "1.0", "item_id": "rcd2_312c7c894ebb0841", "pair_id": "pair_6d2a7e3d67e8bfc5", "statement_latex": "\\forall z\\in D,\\;(z=(E)\\Longleftrightarrow z=(nhv))", "inference_order": 979} {"schema_version": "1.0", "item_id": "rcd2_c0a66776b99a5cda", "pair_id": "pair_ad7da540fe04b892", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (prod_{i=(1}^n ||v_i||)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|det(N)|)\\}) + \\eta", "inference_order": 980} {"schema_version": "1.0", "item_id": "rcd2_e63c83991d967091", "pair_id": "pair_aa10ecd99f42dac6", "statement_latex": "\\{(∫[0,∞] x^{s-1} f(x) dx)\\}=(\\{(Γ(s) φ(-s))\\}) + \\eta", "inference_order": 981} {"schema_version": "1.0", "item_id": "rcd2_4e257edef3763734", "pair_id": "pair_32c294284fc38a6f", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (exp(-u^2 / (2σ_T^2)))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (P(‖f‖_T \\\\le E(‖f‖_T) + u))\\}", "inference_order": 982} {"schema_version": "1.0", "item_id": "rcd2_ce572f6689579f9e", "pair_id": "pair_9e80b4903d06c18d", "statement_latex": "\\neg\\left((n H_n) = (E(T))\\right)", "inference_order": 983} {"schema_version": "1.0", "item_id": "rcd2_4896aaaef9cf8431", "pair_id": "pair_4a1c8057eb955840", "statement_latex": "(\\neg (f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x))) \\Rightarrow (0 = 1)", "inference_order": 984} {"schema_version": "1.0", "item_id": "rcd2_d3796675a03efc33", "pair_id": "pair_0492dd723f6d2eca", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(sum_{i=(0}^{d} binom(n,i))\\le t\\Rightarrow (|F|)\\le t) + \\eta", "inference_order": 985} {"schema_version": "1.0", "item_id": "rcd2_c34474d3e5fcaf3b", "pair_id": "pair_86e4e3efef6d92ba", "statement_latex": "\\forall z\\in D,\\;(z=((rank(A))\\Longleftrightarrow z=(rank([A|b])))) + \\eta", "inference_order": 986} {"schema_version": "1.0", "item_id": "rcd2_6940414e1caf93e8", "pair_id": "pair_f837b9d0966556ac", "statement_latex": "\\delta_D((p(z)),(\\omega p^*(z)))=(0) + \\eta", "inference_order": 987} {"schema_version": "1.0", "item_id": "rcd2_e07e8a525668b9a1", "pair_id": "pair_4f47dc32a7f986ec", "statement_latex": "\\neg\\left(\\max\\{0,(\\limsup_{x → +\\infty} x^2 q(x))-(\\frac{1}{4})\\}=0\\;\\land\\;(\\limsup_{x → +\\infty} x^2 q(x))\\ne (\\frac{1}{4})\\right)", "inference_order": 988} {"schema_version": "1.0", "item_id": "rcd2_bb6c8e4b6fe5be28", "pair_id": "pair_6e14d6d1aec1890f", "statement_latex": "\\nexists s \\in \\mathbb{R}_{\\ge 0}: (\\infty} E_n = intersection_{n=1 to \\infty} union_{k=n to \\infty} E_k) = (limsup_{n-) + s", "inference_order": 989} {"schema_version": "1.0", "item_id": "rcd2_5a950f010f75e986", "pair_id": "pair_c2ca52c4e476ceed", "statement_latex": "\\nexists S \\subseteq D: (((ω(-t)) \\in S \\land (ω(t)) \\notin S) \\lor ((ω(t)) \\in S \\land (ω(-t)) \\notin S)) + \\eta", "inference_order": 990} {"schema_version": "1.0", "item_id": "rcd2_5f78a6caf5fb7b0b", "pair_id": "pair_2953f21c0ffd15d1", "statement_latex": "\\max\\{0,(ceil(n/m))-(max_{1≤i≤m} |B_i|)\\}=0", "inference_order": 991} {"schema_version": "1.0", "item_id": "rcd2_07f026e3826722d9", "pair_id": "pair_b83b9cbd4e911f3c", "statement_latex": "\\neg\\left(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(d/dt integral_{Omega(t)} f dV))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA))\\right)", "inference_order": 992} {"schema_version": "1.0", "item_id": "rcd2_afe45b299f26f5db", "pair_id": "pair_b92550da758cb933", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((y^{2}))\\\\neF((a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}))", "inference_order": 993} {"schema_version": "1.0", "item_id": "rcd2_9245ac2d68fde542", "pair_id": "pair_2e4e1c9254e8cd66", "statement_latex": "\\neg\\left((E) \\in {(F(α))}\\right)", "inference_order": 994} {"schema_version": "1.0", "item_id": "rcd2_a597f88e03b8f3f6", "pair_id": "pair_454cf526b96d8ff2", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;(max|B_k|(|a_n| + |a_n - a_1|))\\le t\\Rightarrow (|∑ a_k b_k|)\\le t\\right)", "inference_order": 995} {"schema_version": "1.0", "item_id": "rcd2_bbea76c382b5f9a6", "pair_id": "pair_359b9eb11918c2c8", "statement_latex": "\\nexists S \\subseteq D: (((\\dim(E^*)) \\in S \\land (\\text{card}(\\mathbb{K}^I)) \\notin S) \\lor ((\\text{card}(\\mathbb{K}^I)) \\in S \\land (\\dim(E^*)) \\notin S)) + \\eta", "inference_order": 996} {"schema_version": "1.0", "item_id": "rcd2_67b20158f5f3c59d", "pair_id": "pair_040362726de7e564", "statement_latex": "((constant_term(prod_{1≤i≠j≤n} (1-t_i/t_j)^{a_i})),(sum a_i)! / prod(a_i!))\\in\\{(z,z):z\\in D\\}", "inference_order": 997} {"schema_version": "1.0", "item_id": "rcd2_43498570b9f5ed2b", "pair_id": "pair_34730235e4c579fb", "statement_latex": "(inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]) = (-log E[e^{-f(B)}])", "inference_order": 998} {"schema_version": "1.0", "item_id": "rcd2_ac6fee486de13b11", "pair_id": "pair_e61b90b39de947fa", "statement_latex": "(e(X,Y)) - (delta |X| |Y|) >( 0) + \\eta", "inference_order": 999} {"schema_version": "1.0", "item_id": "rcd2_2491b51ecb7add07", "pair_id": "pair_5e9a6b46177aa61f", "statement_latex": "\\delta_D((x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2),(z_1^2 + z_2^2 + ... + z_{16}^2))\\\\ne0", "inference_order": 1000} {"schema_version": "1.0", "item_id": "rcd2_9ef836058c9852b2", "pair_id": "pair_6255afb17f28bab9", "statement_latex": "\\nexists S \\subseteq D: (((E) \\in S \\land (\\bigcup_{k}(a_{k},b_{k})) \\notin S) \\lor ((\\bigcup_{k}(a_{k},b_{k})) \\in S \\land (E) \\notin S)) + \\eta", "inference_order": 1001} {"schema_version": "1.0", "item_id": "rcd2_d39d5db4e351d914", "pair_id": "pair_72d26586dac2957a", "statement_latex": "\\{((ax + by),z):z\\in D\\}\\\\ne\\{((gcd(a, b)),z):z\\in D\\}", "inference_order": 1002} {"schema_version": "1.0", "item_id": "rcd2_7f60b4d7bd538dcc", "pair_id": "pair_41c3ac1ea9c66336", "statement_latex": "\\operatorname{dist}((Pr(max_{1≤(k≤n} |S_k| ≥ 3alpha))-(3 max_{1≤k≤n} Pr(|S_k| ≥ alpha)),(-\\infty,0])=0) + \\eta", "inference_order": 1003} {"schema_version": "1.0", "item_id": "rcd2_b4f39cbdbd92cac6", "pair_id": "pair_ac61fe8bb58499d3", "statement_latex": "\\neg\\left(\\exists x \\\\in \\mathbb{R}: (4πA) \\le x \\land x \\le (L^2)\\right)", "inference_order": 1004} {"schema_version": "1.0", "item_id": "rcd2_b3e09b09c8e7a128", "pair_id": "pair_8df812c685e65f0c", "statement_latex": "\\{((uv),z):z\\in D\\}=\\{((-L(v)),z):z\\in D\\}", "inference_order": 1005} {"schema_version": "1.0", "item_id": "rcd2_4e4388fce0fc3626", "pair_id": "pair_f5b0bdc923472cec", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((P(A square B))-(P(A)P(B)))\\\\ne0", "inference_order": 1006} {"schema_version": "1.0", "item_id": "rcd2_7b30c47e62cd70c3", "pair_id": "pair_8a8241e1902c5701", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Σ_{v∈V} deg(v)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(xy))", "inference_order": 1007} {"schema_version": "1.0", "item_id": "rcd2_f23638ce87b5236e", "pair_id": "pair_e509ffee2b4ca75c", "statement_latex": "\\{((E[sum_{x in N} f(x)]),z):z\\in D\\}=(\\{((integral f(x) dLambda(x)),z):z\\in D\\}) + \\eta", "inference_order": 1008} {"schema_version": "1.0", "item_id": "rcd2_871e60af784b6cfc", "pair_id": "pair_d0cb8ab9be7d3bf3", "statement_latex": "\\neg\\left(\\delta_D((product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2})),(sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}))=0\\right)", "inference_order": 1009} {"schema_version": "1.0", "item_id": "rcd2_244848d2006756c6", "pair_id": "pair_b2d6360d4da2fc30", "statement_latex": "(a^2 + b^2 + c^2 + d^2) = (p)", "inference_order": 1010} {"schema_version": "1.0", "item_id": "rcd2_a8591160976e2bbc", "pair_id": "pair_d2c51e84eca5d8bb", "statement_latex": "\\forall t\\in\\mathbb{R},\\;((x + y)/2)\\le t\\Rightarrow (sqrt(x*y))\\le t", "inference_order": 1011} {"schema_version": "1.0", "item_id": "rcd2_17b28f42d0daff3e", "pair_id": "pair_18c902475bdd0794", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(sum_{i=0}^n binom(n,i))=z\\;\\land\\;z=(2^n)", "inference_order": 1012} {"schema_version": "1.0", "item_id": "rcd2_cf6131a88c5702dd", "pair_id": "pair_0bd39448aa802271", "statement_latex": "\\pi_{(n)}((sum_{d|n} mu(d)*a_{n/d}))=(\\pi_{(n)}((0))) + \\eta", "inference_order": 1013} {"schema_version": "1.0", "item_id": "rcd2_0e258b6bb675d4a0", "pair_id": "pair_c13009d1cb5ba778", "statement_latex": "\\{((H),z):z\\in D\\}=\\{((H_0 + gV),z):z\\in D\\}", "inference_order": 1014} {"schema_version": "1.0", "item_id": "rcd2_5d519a6e86800976", "pair_id": "pair_779853b36e5441ba", "statement_latex": "\\neg\\left(\\exists z\\in D\\;\\text{such that}\\;(\\operatorname{Re} \\langle x, y \\rangle)=z\\;\\land\\;z=(\\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2))\\right)", "inference_order": 1015} {"schema_version": "1.0", "item_id": "rcd2_d6256fe8ae9c0301", "pair_id": "pair_f250270489ac1d01", "statement_latex": "\\max\\{0,(0)-(d_)\\}=0\\;\\land\\;(0)\\ne (d_)", "inference_order": 1016} {"schema_version": "1.0", "item_id": "rcd2_fcfaab3ef248947a", "pair_id": "pair_65d918bd8c72ddfc", "statement_latex": "\\forall z\\in D,\\;(z=((ω(-t))\\Longleftrightarrow z=(ω(t)))) + \\eta", "inference_order": 1017} {"schema_version": "1.0", "item_id": "rcd2_9dae063dc56fd3ba", "pair_id": "pair_1744b7ea1858a4ac", "statement_latex": "\\neg\\left((C_1 ∪ ... ∪ C_r) = (S_T^n)\\right)", "inference_order": 1018} {"schema_version": "1.0", "item_id": "rcd2_546d95a02dfd916b", "pair_id": "pair_799797f130b57cbb", "statement_latex": "\\neg(\\neg (L(E, s) = sum_{n=1}^∞ a_n / n^s))", "inference_order": 1019} {"schema_version": "1.0", "item_id": "rcd2_3f79c4fbaea8ee80", "pair_id": "pair_8622247df1f7272d", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (∑ x_i) / (∑ (1-x_i))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge ((∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n})\\}", "inference_order": 1020} {"schema_version": "1.0", "item_id": "rcd2_6f34f8d0cf0d2b15", "pair_id": "pair_771305494d7e3cbe", "statement_latex": "\\sup_{\\lambda\\in[0,\\infty)}\\lambda((infinity)-(dim_C H^i(X, F)))=+\\infty", "inference_order": 1021} {"schema_version": "1.0", "item_id": "rcd2_8314e468a62279d2", "pair_id": "pair_432f76a3be5ea262", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(lim_{n→∞} σ_n(f,x))=(z\\;\\land\\;z=(f(x))) + \\eta", "inference_order": 1022} {"schema_version": "1.0", "item_id": "rcd2_3eb88713bc29fc13", "pair_id": "pair_3b45f54bb8c17863", "statement_latex": "\\exists x \\\\in \\mathbb{R}: ((sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i) > x \\land x < (j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i))", "inference_order": 1023} {"schema_version": "1.0", "item_id": "rcd2_4e5a54f9578a1003", "pair_id": "pair_086b86fca81af642", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(A)=z\\;\\land\\;z=(\\text{conv}\\{w(X) : w \\in W\\})", "inference_order": 1024} {"schema_version": "1.0", "item_id": "rcd2_a16554ed1c7d80f6", "pair_id": "pair_53af381ae41c746a", "statement_latex": "\\forall z\\in D,\\;((max_matching_size),z)=(((1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)),z)) + \\eta", "inference_order": 1025} {"schema_version": "1.0", "item_id": "rcd2_d3403565b00e5b14", "pair_id": "pair_6870dca4ab7f7f20", "statement_latex": "\\{(\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1})\\}=\\{(1)\\}", "inference_order": 1026} {"schema_version": "1.0", "item_id": "rcd2_68bc150e83fe54ad", "pair_id": "pair_77418fcbb4a25140", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(prod_{n≥1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2})))+\\delta_D(z,(sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1}))))=0", "inference_order": 1027} {"schema_version": "1.0", "item_id": "rcd2_3efeebf990d21724", "pair_id": "pair_8897833468d93ff1", "statement_latex": "\\operatorname{dist}((Pr(S_n > t))-(exp(-(sigma^2 / a^2) * h(at / sigma^2))),(-\\infty,0])=0", "inference_order": 1028} {"schema_version": "1.0", "item_id": "rcd2_23596090158edf6c", "pair_id": "pair_e338851eae0d68b0", "statement_latex": "\\{((V(p)),z):z\\in D\\}=\\{((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)),z):z\\in D\\}", "inference_order": 1029} {"schema_version": "1.0", "item_id": "rcd2_693fbd3b8e1b67ae", "pair_id": "pair_19aef722689df657", "statement_latex": "\\{((f(x)),z):z\\in D\\}=\\{((\\sup\\{f(x, \\theta) : x \\in C(\\theta)\\}),z):z\\in D\\}", "inference_order": 1030} {"schema_version": "1.0", "item_id": "rcd2_39638a357deb4ba3", "pair_id": "pair_033e9a3e45c3852f", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(F))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(-V'(x)))", "inference_order": 1031} {"schema_version": "1.0", "item_id": "rcd2_f7a8e363ba445803", "pair_id": "pair_9d0fbc57dbfd60a9", "statement_latex": "\\delta_D((w(AB|X)),(w(A|X)w(B|AX)))=0", "inference_order": 1032} {"schema_version": "1.0", "item_id": "rcd2_a2b19b2a9a3d9ee1", "pair_id": "pair_957d318d7d663fd5", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(w(S)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{x in S} w(x)))", "inference_order": 1033} {"schema_version": "1.0", "item_id": "rcd2_31643e59601b7728", "pair_id": "pair_200ee8ecdac87cf8", "statement_latex": "\\nexists S \\subseteq D: (((Tor_1^A(C, M)) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (Tor_1^A(C, M)) \\notin S)) + \\eta", "inference_order": 1034} {"schema_version": "1.0", "item_id": "rcd2_a88b04ab08c0568b", "pair_id": "pair_0a8bf98031d19ff2", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(gcd{(n-1 k-1), (n k+1), (n+1 k)}))+\\delta_D(z,(gcd{(n-1 k), (n k-1), (n+1 k+1)})))=0", "inference_order": 1035} {"schema_version": "1.0", "item_id": "rcd2_b51fee6fe29de607", "pair_id": "pair_d8c587f203bea9d3", "statement_latex": "((sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i) - (j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)) < 0", "inference_order": 1036} {"schema_version": "1.0", "item_id": "rcd2_3929fec4ad3eb4ad", "pair_id": "pair_b9c888410517f10e", "statement_latex": "\\sup_{\\lambda\\in[0,\\infty)}\\lambda((floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)))-(1/2))\\\\ne+\\infty", "inference_order": 1037} {"schema_version": "1.0", "item_id": "rcd2_92d22c7368982b1d", "pair_id": "pair_13a9589daa8ad1ae", "statement_latex": "\\neg\\left(((y_{n+1}),((x-c_{n})y_{n}-d_{n}y_{n-1}))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 1038} {"schema_version": "1.0", "item_id": "rcd2_6d765398ff87bf0f", "pair_id": "pair_78d48fd996b18846", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(P(∪_{i= 1 - α))+s=(α)", "inference_order": 1039} {"schema_version": "1.0", "item_id": "rcd2_6e9fc170b9b780ef", "pair_id": "pair_283bf63d97294a58", "statement_latex": "\\frac{\\exp(‖v - Pv‖)}{\\exp(‖v - Pv‖)+\\exp((1 + ‖P‖) inf_{u ∈ U} ‖v - u‖)}\\le\\frac{1}{2}", "inference_order": 1040} {"schema_version": "1.0", "item_id": "rcd2_fd8e09e4be656699", "pair_id": "pair_fd14740d9f3d7997", "statement_latex": "\\neg\\left(\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1})+s=(\\prod_{j=1}^{d} ||g_j||_{L^{d-1}})\\right)", "inference_order": 1041} {"schema_version": "1.0", "item_id": "rcd2_cec1e2ba840ba517", "pair_id": "pair_5a6ae949eb4abd45", "statement_latex": "(\\exp(-u^2 / (2σ_T^2))) - (P(‖f‖_T > E(‖f‖_T) + u)) \\ge 0", "inference_order": 1042} {"schema_version": "1.0", "item_id": "rcd2_95985190ecca4d66", "pair_id": "pair_8129b00b78a45cc6", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (d/|S|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (Pr[P(r1, ..., rn) = 0])\\}", "inference_order": 1043} {"schema_version": "1.0", "item_id": "rcd2_bcee159fc31934c2", "pair_id": "pair_258c4aaccb957157", "statement_latex": "\\nexists S \\subseteq D: (((\\binom{n-1}{k} + \\binom{n-1}{k-1}) \\in S \\land (\\binom{n}{k}) \\notin S) \\lor ((\\binom{n}{k}) \\in S \\land (\\binom{n-1}{k} + \\binom{n-1}{k-1}) \\notin S)) + \\eta", "inference_order": 1044} {"schema_version": "1.0", "item_id": "rcd2_cfc80f04c5aa4555", "pair_id": "pair_97f93e6320295f9f", "statement_latex": "\\neg\\left((f(x)) \\in {(⟨Aφ(x), φ(x)⟩)}\\right)", "inference_order": 1045} {"schema_version": "1.0", "item_id": "rcd2_190ca1d899cdf4ae", "pair_id": "pair_2033f60eb4132d67", "statement_latex": "\\nexists S \\subseteq D: (((nullity A) \\in S \\land (nullity H) \\notin S) \\lor ((nullity H) \\in S \\land (nullity A) \\notin S))", "inference_order": 1046} {"schema_version": "1.0", "item_id": "rcd2_5a8d8a8c0dcdad95", "pair_id": "pair_4807a28ce3c3b23d", "statement_latex": "\\max\\{0,(Pr[Intersection(X_i > u_i)])-(Pr[Intersection(Y_i ≤ u_i)])\\}=0", "inference_order": 1047} {"schema_version": "1.0", "item_id": "rcd2_793aceeafe82e861", "pair_id": "pair_908290996865a0ea", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;((lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)]),z)=((sup_{x∈X} (φ(x) - I(x))),z)\\right)", "inference_order": 1048} {"schema_version": "1.0", "item_id": "rcd2_944433ba6c355b91", "pair_id": "pair_22d1f33cf284f830", "statement_latex": "(\\{({m+n \\choose r})\\}\\times\\{(\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k})\\})\\cap\\{(z,z):z\\in D\\}=\\{(({m+n \\choose r}),(\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}))\\}", "inference_order": 1049} {"schema_version": "1.0", "item_id": "rcd2_af9a177f5cbd6c05", "pair_id": "pair_8660fca3b1e05ab9", "statement_latex": "(f(x)) \\in {(s)}", "inference_order": 1050} {"schema_version": "1.0", "item_id": "rcd2_341c71eedf0090c9", "pair_id": "pair_926702e331bee626", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(2 inf_{0 ≤( c ≤ k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz)\\le t\\Rightarrow (P(|sum_{i=1}^n a_i X_i| ≥ k))\\le t) + \\eta", "inference_order": 1051} {"schema_version": "1.0", "item_id": "rcd2_dbc514a2f24ac528", "pair_id": "pair_72bb2a233c9f4b9b", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(1/(1-alpha*z)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}))", "inference_order": 1052} {"schema_version": "1.0", "item_id": "rcd2_60030e5b63e3af9f", "pair_id": "pair_fae6b7a1422f76f2", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((ω(-t)))=F((ω(t)))", "inference_order": 1053} {"schema_version": "1.0", "item_id": "rcd2_b650746152392497", "pair_id": "pair_81282f23015ae50f", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge ((2 + epsilon(x)) * B(x)^2)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge ((1/x) * sum_{n≤x} |f(n) - A(x)|^2)\\}", "inference_order": 1054} {"schema_version": "1.0", "item_id": "rcd2_767846360d8c4333", "pair_id": "pair_ed21bdf93f8cdaa6", "statement_latex": "\\max\\{0,(0)-(prod_{n=1}^\\infty \\int \\sqrt{d\\mu_n d\\nu_n})\\}=0\\;\\land\\;(0)\\ne (prod_{n=1}^\\infty \\int \\sqrt{d\\mu_n d\\nu_n})", "inference_order": 1055} {"schema_version": "1.0", "item_id": "rcd2_941f2cdbe5148439", "pair_id": "pair_a19ee483627cccf3", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((P(X < a))-(inf_{t>0} M(t)e^{-ta}))=0", "inference_order": 1056} {"schema_version": "1.0", "item_id": "rcd2_fd222614e7fb28cc", "pair_id": "pair_9259a14a33b4e402", "statement_latex": "\\operatorname{dist}((v)-(b),(-\\infty,0])=0", "inference_order": 1057} {"schema_version": "1.0", "item_id": "rcd2_eebd6b7162958bd9", "pair_id": "pair_6f93eab1652df6f0", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (\\inf_{v \\in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)}) \\le x \\land x \\le (C |u|_{W^{m,p}(\\Omega)})", "inference_order": 1058} {"schema_version": "1.0", "item_id": "rcd2_d2cfca90c5462668", "pair_id": "pair_e986703ca6d30487", "statement_latex": "\\exists x \\\\in \\mathbb{R}: ((∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n}) \\le x \\land x \\le ((∑ x_i) / (∑ (1-x_i)))", "inference_order": 1059} {"schema_version": "1.0", "item_id": "rcd2_2bd2a97a35a567a4", "pair_id": "pair_d7364bd09bd22b23", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;(|z|)\\le t\\Rightarrow (|f(z)|)\\le t\\right)", "inference_order": 1060} {"schema_version": "1.0", "item_id": "rcd2_48286380f84af5f6", "pair_id": "pair_9bb8830326565899", "statement_latex": "\\exists s\\in\\mathbb{R}_{>(8/9)", "inference_order": 1061} {"schema_version": "1.0", "item_id": "rcd2_cfbb1aa385189c64", "pair_id": "pair_8809ccb51bf62476", "statement_latex": "\\forall z\\in D,\\;(z=({m+n \\choose r})\\Longleftrightarrow z=(\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}))", "inference_order": 1062} {"schema_version": "1.0", "item_id": "rcd2_a35351e4016be346", "pair_id": "pair_15a91b6763f0f21b", "statement_latex": "\\delta_D(([I - A]x),(d))=0", "inference_order": 1063} {"schema_version": "1.0", "item_id": "rcd2_e107fe3b6fed2cdd", "pair_id": "pair_d0d44e5f91ddf73a", "statement_latex": "\\delta_D((∮_{C}(L dx + M dy)),(∬_{D} (∂M/∂x - ∂L/∂y) dA))=(0) + \\eta", "inference_order": 1064} {"schema_version": "1.0", "item_id": "rcd2_8d88e26efcbfd644", "pair_id": "pair_a0bb3ad12d18d2e7", "statement_latex": "\\{(Ax)\\}=\\{(λx)\\}", "inference_order": 1065} {"schema_version": "1.0", "item_id": "rcd2_88d072b751a637f3", "pair_id": "pair_23eec87a1742becb", "statement_latex": "\\neg\\left((f(s(x))) \\in {(s(f(x)))}\\right)", "inference_order": 1066} {"schema_version": "1.0", "item_id": "rcd2_647ef1a497eb0ec3", "pair_id": "pair_ff420571c42152ad", "statement_latex": "\\neg\\left((ΔδT) \\in {(0)}\\right)", "inference_order": 1067} {"schema_version": "1.0", "item_id": "rcd2_7bf8e183d5ebd752", "pair_id": "pair_21574c4f0404d295", "statement_latex": "\\forall z\\in D,\\;((Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k),z)=((w^(-1) * (z + w + m)^m),z)", "inference_order": 1068} {"schema_version": "1.0", "item_id": "rcd2_33fd191b26a5e871", "pair_id": "pair_627b944d82e7f3fe", "statement_latex": "((\\zeta(s)),(\\prod_{n\\\\ne1}^{\\infty} \\frac{1}{1-p_n^{-s}}))\\in\\{(z,z):z\\in D\\}", "inference_order": 1069} {"schema_version": "1.0", "item_id": "rcd2_fb7c634e8d3fc268", "pair_id": "pair_09840024da1b1478", "statement_latex": "\\delta_D((Pr(Σ_t\\\\neA) / Pr(Σ_t=-A)),(e^(At)))=0", "inference_order": 1070} {"schema_version": "1.0", "item_id": "rcd2_bb73da85addec97f", "pair_id": "pair_109dfe40e8ddc10e", "statement_latex": "((∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2) \\lor (0 = 1)", "inference_order": 1071} {"schema_version": "1.0", "item_id": "rcd2_90f78859c14cbbbe", "pair_id": "pair_c9740c64ff9cff29", "statement_latex": "\\forall r\\in\\mathbb{Q}_{>0}\\exists N\\forall n\\ge N:d((S_βα),(S_β1α1 * S_β2α2)) epsilon))}{\\exp(P(|(1/n) * sum_{i=1}^n X_i| > epsilon))+\\exp(2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3))))}\\le\\frac{1}{2}", "inference_order": 1102} {"schema_version": "1.0", "item_id": "rcd2_2a197999b9591aa6", "pair_id": "pair_09100147d4f94c39", "statement_latex": "\\neg\\left(\\max\\{0,(max_{|z|=1}|P'(z)|)-(n max_{|z|=1}|P(z)|)\\}=0\\right)", "inference_order": 1103} {"schema_version": "1.0", "item_id": "rcd2_2c8461ee86eddb03", "pair_id": "pair_e1384a9e4cb6bb24", "statement_latex": "\\max\\{0,(vol(s φ K + t ψ L)^(1/n))-(C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n)))\\}=0", "inference_order": 1104} {"schema_version": "1.0", "item_id": "rcd2_517c42360016a53d", "pair_id": "pair_56083b82314d2107", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (φ(Φ^{-1}(γ^n(A)))) \\le x \\land x \\le (liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)})", "inference_order": 1105} {"schema_version": "1.0", "item_id": "rcd2_d69b6862a9a89c12", "pair_id": "pair_2c929668e411a67e", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((P[σ(x(t))] / P[σ(x_tilde(t))]) \\in S \\land (e^σ[x(t)]) \\notin S) \\lor ((e^σ[x(t)]) \\in S \\land (P[σ(x(t))] / P[σ(x_tilde(t))]) \\notin S))\\right)", "inference_order": 1106} {"schema_version": "1.0", "item_id": "rcd2_7587757d52c29b6c", "pair_id": "pair_884e51f3bea2d75d", "statement_latex": "\\delta_D((sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3),((3a)! / (a!)^3))=0", "inference_order": 1107} {"schema_version": "1.0", "item_id": "rcd2_55a44ddb5f6fabde", "pair_id": "pair_30c813cf58194699", "statement_latex": "\\{((A(x)),z):z\\in D\\}=\\{(((1 / (2*pi*i)) * integral_{c-i*infinity}^{c+i*infinity} g(z) * (x^z / z) dz),z):z\\in D\\}", "inference_order": 1108} {"schema_version": "1.0", "item_id": "rcd2_cfbdaa71e99301c5", "pair_id": "pair_dbeb180892ff4720", "statement_latex": "\\forall z\\in D,\\;(z=((F(X))\\Longleftrightarrow z=(a_0 + a_1X + ... + a_nX^n))) + \\eta", "inference_order": 1109} {"schema_version": "1.0", "item_id": "rcd2_a6311bbbe49ce73c", "pair_id": "pair_6eb58c98c883891a", "statement_latex": "\\neg\\forall x\\in D,x\\notin \\{prime p\\in D:n > p < 2*n\\}", "inference_order": 1110} {"schema_version": "1.0", "item_id": "rcd2_283c5c30644963e8", "pair_id": "pair_6bcbd22b959d46d2", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\ge ([a])\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge ([b] for a majorizing b)\\}\\right)", "inference_order": 1111} {"schema_version": "1.0", "item_id": "rcd2_39d13f2433329ea2", "pair_id": "pair_6d2a7e3d67e8bfc5", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(z=(E)\\Longleftrightarrow z=(nhv))\\right)", "inference_order": 1112} {"schema_version": "1.0", "item_id": "rcd2_31327a568592e299", "pair_id": "pair_151551d6d4e89a2d", "statement_latex": "\\neg\\left(\\forall F:D\\to\\{0,1\\},\\;F((sum_{k=m}^{n} f_k Delta g_k))=F(((f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k))\\right)", "inference_order": 1113} {"schema_version": "1.0", "item_id": "rcd2_60301da20e6f1763", "pair_id": "pair_132122ca15994f57", "statement_latex": "\\max\\{0,(0)-(g)\\}\\\\ne0\\;\\land\\;(0)\\ne (g)", "inference_order": 1114} {"schema_version": "1.0", "item_id": "rcd2_ee3e90cff8968545", "pair_id": "pair_367ffeec7237a91c", "statement_latex": "\\{(F(z,t))\\}=\\{(e^{g(z)t})\\}", "inference_order": 1115} {"schema_version": "1.0", "item_id": "rcd2_52ac27e141145b01", "pair_id": "pair_b0a6d3f6ab775e50", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((mu(t)))=F((integral_{-infinity}^{t} A(s)p(s,t) ds))", "inference_order": 1116} {"schema_version": "1.0", "item_id": "rcd2_d84bb6be2f864d83", "pair_id": "pair_03b178d97836dc4c", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1}))\\\\ne\\operatorname{argmin}_{z\\in D}\\delta_D(z,(1))", "inference_order": 1117} {"schema_version": "1.0", "item_id": "rcd2_159436c9d4777b1b", "pair_id": "pair_08f59d28799155b1", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(dim(H_μ))=z\\;\\land\\;z=(xy)", "inference_order": 1118} {"schema_version": "1.0", "item_id": "rcd2_bb69aa8cc62c4a2e", "pair_id": "pair_bcba132b8080ca41", "statement_latex": "(T2) - (T1) \\ge 0", "inference_order": 1119} {"schema_version": "1.0", "item_id": "rcd2_19c1225e47d56d54", "pair_id": "pair_bb54a11340fac5a0", "statement_latex": "\\forall z\\in D,\\;(z=(Δy)\\Longleftrightarrow z=(f'(x)Δx + εΔx))", "inference_order": 1120} {"schema_version": "1.0", "item_id": "rcd2_41fea515b95d58d5", "pair_id": "pair_97790b8d57345d2e", "statement_latex": "(0) \\\\ne (x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2})", "inference_order": 1121} {"schema_version": "1.0", "item_id": "rcd2_123f14a7f2633b9e", "pair_id": "pair_0bd39448aa802271", "statement_latex": "\\pi_{(n)}((sum_{d|n} mu(d)*a_{n/d}))=\\pi_{(n)}((0))", "inference_order": 1122} {"schema_version": "1.0", "item_id": "rcd2_04d14a2d515e1840", "pair_id": "pair_14267edea434eed5", "statement_latex": "\\delta_D((sum (6-i)p_i),(12))=0", "inference_order": 1123} {"schema_version": "1.0", "item_id": "rcd2_84dfbbc36744f196", "pair_id": "pair_9ba5614280e01f59", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(SMS^T)=z\\;\\land\\;z=(D ⊕ D)", "inference_order": 1124} {"schema_version": "1.0", "item_id": "rcd2_35b73db375a4b495", "pair_id": "pair_4a1d4788acd39d80", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((|Irr_{p'}(G)|))=(F((|Irr_{p'}(N_{G}(P))|))) + \\eta", "inference_order": 1125} {"schema_version": "1.0", "item_id": "rcd2_25b8eb8127e34d26", "pair_id": "pair_7da09b757d164396", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn)))=(F((2^{n-1}∏_{i=1}^{n}ε(Xi)))) + \\eta", "inference_order": 1126} {"schema_version": "1.0", "item_id": "rcd2_cec90a0bb91ea8ae", "pair_id": "pair_d6ab9edbd47b7b88", "statement_latex": "\\neg\\left(\\forall F:D\\to\\{0,1\\},\\;F((f(s(x))))=F((s(f(x))))\\right)", "inference_order": 1127} {"schema_version": "1.0", "item_id": "rcd2_10978a235b412221", "pair_id": "pair_d426b90947d1e90f", "statement_latex": "(K(A) = sup_{|z|) - (1} (|z|-1) ||(z-A)^-1||) > 0", "inference_order": 1128} {"schema_version": "1.0", "item_id": "rcd2_6314531c7a19d8aa", "pair_id": "pair_d8d284d46ea567fd", "statement_latex": "\\exists s\\in\\mathbb{R}_{>(|A+B|)", "inference_order": 1129} {"schema_version": "1.0", "item_id": "rcd2_8abc3c4b938ff082", "pair_id": "pair_db0c6626139fca0f", "statement_latex": "\\neg\\left(\\forall F:D\\to\\{0,1\\},\\;F((|Y^X/G|))=F(((1/|G|) * Σ_{g∈G} m^{c(g)}))\\right)", "inference_order": 1130} {"schema_version": "1.0", "item_id": "rcd2_b60dec5e2d42ce2f", "pair_id": "pair_bf9ee452434b20bd", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(n)\\\\nez\\;\\land\\;z=(p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k})", "inference_order": 1131} {"schema_version": "1.0", "item_id": "rcd2_525542abde309078", "pair_id": "pair_a7e5d240eac94cd0", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (1)\\Rightarrow t<(g)", "inference_order": 1132} {"schema_version": "1.0", "item_id": "rcd2_cbd0ea899c7ec55d", "pair_id": "pair_aa131a08fa4c213d", "statement_latex": "\\max\\{0,(P(‖f‖_T > 1 - α))-(α)\\}=0", "inference_order": 1133} {"schema_version": "1.0", "item_id": "rcd2_f0127f6912998dda", "pair_id": "pair_b3355fcff5887f3e", "statement_latex": "(w) \\in {(xyz)}", "inference_order": 1134} {"schema_version": "1.0", "item_id": "rcd2_a64017c2b02a4ea3", "pair_id": "pair_287ef432c524a888", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(F))+\\delta_D(z,(-V'(x))))=0", "inference_order": 1135} {"schema_version": "1.0", "item_id": "rcd2_ac3af4352833ad00", "pair_id": "pair_3e560aad4812e7e4", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((1/sqrt(p) * sum_{n\\\\ne0}^{p-1} exp(2*pi*i*n^2*q/p)))=F((e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))))", "inference_order": 1136} {"schema_version": "1.0", "item_id": "rcd2_98543c2a9525bff1", "pair_id": "pair_2fa4534bff250931", "statement_latex": "\\nexists S \\subseteq D: (((sum_{n \\in S} (1/n)) \\in S \\land (1) \\notin S) \\lor ((1) \\in S \\land (sum_{n \\in S} (1/n)) \\notin S))", "inference_order": 1137} {"schema_version": "1.0", "item_id": "rcd2_d45f1486dfb2f3f9", "pair_id": "pair_e13dfef30b697bce", "statement_latex": "((1/x) * sum_{n\\lex} |f(n) - A(x)|^2 \\le (2 + \\varepsilon(x)) * B(x)^2) \\land (0 = 0)", "inference_order": 1138} {"schema_version": "1.0", "item_id": "rcd2_4e80fd877c33c9be", "pair_id": "pair_c5ddbf8ea4914292", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;(liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)})\\le t\\Rightarrow (φ(Φ^{-1}(γ^n(A))))\\le t\\right)", "inference_order": 1139} {"schema_version": "1.0", "item_id": "rcd2_1b197f365cd76033", "pair_id": "pair_2d61c2731a48d894", "statement_latex": "((1/(1-alpha*z)),(product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}))\\in\\{(z,z):z\\in D\\}", "inference_order": 1140} {"schema_version": "1.0", "item_id": "rcd2_a2101a693d97c462", "pair_id": "pair_822f27f270a32f07", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(prod_{i} d_{i})+s=((Vol(M))) + \\eta", "inference_order": 1141} {"schema_version": "1.0", "item_id": "rcd2_e30669bda4940bea", "pair_id": "pair_bb570f1536c0034e", "statement_latex": "\\arctan(Pr[Intersection(X_i ≤ u_i)])\\le\\arctan(Pr[Intersection(Y_i ≤ u_i)])", "inference_order": 1142} {"schema_version": "1.0", "item_id": "rcd2_d5df18460cbc8020", "pair_id": "pair_720d2256c3d8ab52", "statement_latex": "\\nexists S \\subseteq D: ((((f^{-1})'(b)) \\in S \\land (1 / f'(a)) \\notin S) \\lor ((1 / f'(a)) \\in S \\land ((f^{-1})'(b)) \\notin S))", "inference_order": 1143} {"schema_version": "1.0", "item_id": "rcd2_a5bdc561039802de", "pair_id": "pair_b43386925572d9f2", "statement_latex": "\\{(\\zeta(s))\\}=\\{(\\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}})\\}", "inference_order": 1144} {"schema_version": "1.0", "item_id": "rcd2_f2f6b2975524b2f8", "pair_id": "pair_6a93474b1ecc6b41", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2))=F((z_1^2 + z_2^2 + ... + z_{16}^2))", "inference_order": 1145} {"schema_version": "1.0", "item_id": "rcd2_223e2d28caa08d61", "pair_id": "pair_8a16de5bc8b7376f", "statement_latex": "\\neg\\left((D ⊕ D) = (SMS^T)\\right)", "inference_order": 1146} {"schema_version": "1.0", "item_id": "rcd2_0e8b5934018610a2", "pair_id": "pair_50badbdc0b8507e4", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;((-log E[e^{-f(B)}]),z)=((inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]),z)\\right)", "inference_order": 1147} {"schema_version": "1.0", "item_id": "rcd2_9c3b1ae695c4e7f3", "pair_id": "pair_41f3f45ce3b64178", "statement_latex": "\\forall z\\in D,\\;((rank(A)),z)\\\\ne((rank([A|b])),z)", "inference_order": 1148} {"schema_version": "1.0", "item_id": "rcd2_a3218b641672cd96", "pair_id": "pair_cf827eac048fd266", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3))))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (P(|(1/n) * sum_{i=(1}^n X_i| > epsilon))\\}) + \\eta", "inference_order": 1149} {"schema_version": "1.0", "item_id": "rcd2_cb1e9e8b1f3c2814", "pair_id": "pair_de1e4b5f450a9cc1", "statement_latex": "(\\infty} a_n = sup {a_n}) - (lim_{n -) > 0", "inference_order": 1150} {"schema_version": "1.0", "item_id": "rcd2_356b21caa34d6606", "pair_id": "pair_347097297ab9c443", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (M^a = {x \\in M | f(x)) \\le x \\land x \\le (a})", "inference_order": 1151} {"schema_version": "1.0", "item_id": "rcd2_a1cd9b0fb4df8e09", "pair_id": "pair_0f26f316e9b7b3e0", "statement_latex": "\\neg\\left(((d/dx integral_{a(x)}^{b(x)} f(x,t) dt),(f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 1152} {"schema_version": "1.0", "item_id": "rcd2_d94dca54eb3f0710", "pair_id": "pair_7ce4664145e484c0", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(f(x)))+\\delta_D(z,(0 (mod p^n))))=0", "inference_order": 1153} {"schema_version": "1.0", "item_id": "rcd2_388ac31b6d540511", "pair_id": "pair_ad7da540fe04b892", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (prod_{i=1}^n ||v_i||)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|det(N)|)\\}", "inference_order": 1154} {"schema_version": "1.0", "item_id": "rcd2_6350289756401f51", "pair_id": "pair_fdfcaf1f654e1d01", "statement_latex": "\\{t\\in\\mathbb{R}:t\\le (|d(A,X) - d(B,X)|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (d(A,B))\\}", "inference_order": 1155} {"schema_version": "1.0", "item_id": "rcd2_d1eaa9c524bf4dcf", "pair_id": "pair_8ca052cac63c7087", "statement_latex": "\\nexists S \\subseteq D: (((x * (y * z) + y * (z * x) + z * (x * y)) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (x * (y * z) + y * (z * x) + z * (x * y)) \\notin S)) + \\eta", "inference_order": 1156} {"schema_version": "1.0", "item_id": "rcd2_a80cf1081bd6ca47", "pair_id": "pair_36428eb7373b7521", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (Δ)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (ζ(n) / 2^(n-1))\\}", "inference_order": 1157} {"schema_version": "1.0", "item_id": "rcd2_139791a8e845a3aa", "pair_id": "pair_48692e23ecfcc698", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(rank(A))\\\\nez\\;\\land\\;z=(rank([A|b]))", "inference_order": 1158} {"schema_version": "1.0", "item_id": "rcd2_672c640d21c42c2e", "pair_id": "pair_4415402255935d0e", "statement_latex": "(ker(d(f|_Y)_y) \\subseteq tau) \\land (0 = 0)", "inference_order": 1159} {"schema_version": "1.0", "item_id": "rcd2_81562b21b537047b", "pair_id": "pair_0d12ebb73f8d6a4b", "statement_latex": "\\delta_D((F_{n-1}*F_{n+1} - F_n^2),((-1)^n))=0", "inference_order": 1160} {"schema_version": "1.0", "item_id": "rcd2_83b3698f24359686", "pair_id": "pair_f8ca8facbda83359", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;((gamma / alpha) * ||u - v||)\\le t\\Rightarrow (||u - u_h||)\\le t\\right)", "inference_order": 1161} {"schema_version": "1.0", "item_id": "rcd2_2045edbbecbd8a44", "pair_id": "pair_6593f453672f9856", "statement_latex": "\\forall z\\in D,\\;((E),z)=((F(α)),z)", "inference_order": 1162} {"schema_version": "1.0", "item_id": "rcd2_05f8a13a9a09953b", "pair_id": "pair_46cb763ff9c42c8b", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((|A+B|))\\\\neF((|A+H|+|B+H|-|H|))", "inference_order": 1163} {"schema_version": "1.0", "item_id": "rcd2_d0f4d369fa758b42", "pair_id": "pair_39155343ca7c52d1", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((y_{n+1}))=F(((x-c_{n})y_{n}-d_{n}y_{n-1}))", "inference_order": 1164} {"schema_version": "1.0", "item_id": "rcd2_c4e498a54d518de7", "pair_id": "pair_bf53f0437a957cfa", "statement_latex": "\\{(-zeta'(s)/zeta(s))\\}=\\{(F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s)))\\}", "inference_order": 1165} {"schema_version": "1.0", "item_id": "rcd2_55c3c4a8494aedf8", "pair_id": "pair_e50f798376e786c5", "statement_latex": "\\neg\\left(\\{(((x_{1}+x_{2}+\\cdots +x_{m})^{n}),z):z\\in D\\}=\\{((\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}),z):z\\in D\\}\\right)", "inference_order": 1166} {"schema_version": "1.0", "item_id": "rcd2_d643e1a82c86e5e5", "pair_id": "pair_e59908088ec687ce", "statement_latex": "\\delta_D((|B_1|),(|B_2|))=0", "inference_order": 1167} {"schema_version": "1.0", "item_id": "rcd2_649860875608600b", "pair_id": "pair_f3a3d104be324795", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((f^{-1})'(b))=F((\\sup\\{f(x, \\theta) : x \\in C(\\theta)\\}))", "inference_order": 1168} {"schema_version": "1.0", "item_id": "rcd2_f75ec3554dd84866", "pair_id": "pair_6255afb17f28bab9", "statement_latex": "\\nexists S \\subseteq D: (((E) \\in S \\land (\\bigcup_{k}(a_{k},b_{k})) \\notin S) \\lor ((\\bigcup_{k}(a_{k},b_{k})) \\in S \\land (E) \\notin S))", "inference_order": 1169} {"schema_version": "1.0", "item_id": "rcd2_93df160222180fd3", "pair_id": "pair_295130b22ec63482", "statement_latex": "\\forall z\\in D,\\;(((x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}),z)=(((x-m){\\dbinom {x}{m}}),z)", "inference_order": 1170} {"schema_version": "1.0", "item_id": "rcd2_c982ac31e833a20c", "pair_id": "pair_addc4361b40a860a", "statement_latex": "\\neg\\left((d/|S|) - (Pr[P(r1, ..., rn) = 0]) \\ge 0\\right)", "inference_order": 1171} {"schema_version": "1.0", "item_id": "rcd2_ba901bad0f8fd953", "pair_id": "pair_4b1b11e25d848256", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((det(A)))\\\\neF((det(A_v^u) * det(A_tilde_v^u)))", "inference_order": 1172} {"schema_version": "1.0", "item_id": "rcd2_ee583f3e101490b0", "pair_id": "pair_902790a3150e177c", "statement_latex": "\\operatorname{dist}((||u - u_h||)-((gamma / alpha) * ||u - v||),(-\\infty,0])=(0) + \\eta", "inference_order": 1173} {"schema_version": "1.0", "item_id": "rcd2_b4a1afbad71560d0", "pair_id": "pair_9b583072ba908615", "statement_latex": "{(f(s(x)))} = {(s(f(x)))}", "inference_order": 1174} {"schema_version": "1.0", "item_id": "rcd2_ad2b52a88e1233cd", "pair_id": "pair_5947a149e20d5896", "statement_latex": "(n) - (sum_{j=1}^r d_j) < 0", "inference_order": 1175} {"schema_version": "1.0", "item_id": "rcd2_5be8fa486a3df452", "pair_id": "pair_5622612cabd9c563", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (-Σ p_i log q_i)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (-Σ p_i log p_i)\\}", "inference_order": 1176} {"schema_version": "1.0", "item_id": "rcd2_2ee622389ec0b8fb", "pair_id": "pair_eb688846f87767c6", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(sum_{k=(m}^{n} f_k Delta g_k))+\\delta_D(z,((f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k)))=0) + \\eta", "inference_order": 1177} {"schema_version": "1.0", "item_id": "rcd2_7e0320fae30d262d", "pair_id": "pair_885f5db7d4d7895a", "statement_latex": "\\{(product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}))\\}=\\{(sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n})\\}", "inference_order": 1178} {"schema_version": "1.0", "item_id": "rcd2_be902f33472a071c", "pair_id": "pair_ac63c54265fe44b3", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\sup\\{f(x, \\theta) : x \\in C(\\theta)\\}))", "inference_order": 1179} {"schema_version": "1.0", "item_id": "rcd2_ec6f4c2dd6493552", "pair_id": "pair_e5619142485f78af", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(W(r,k))+s=((2^{2^{r^{2^{2^{k+9}}}}})) + \\eta", "inference_order": 1180} {"schema_version": "1.0", "item_id": "rcd2_6373106f31e15a01", "pair_id": "pair_efe340cccb9dd62f", "statement_latex": "\\exists s\\in\\mathbb{R}_{>(0}\\;\\text{such that}\\;(∑_{n≥1}P(|S_{n}|>εn))+s=(∞)) + \\eta", "inference_order": 1181} {"schema_version": "1.0", "item_id": "rcd2_e3cb509ebf9ff516", "pair_id": "pair_ffe4a635941a01a2", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(V(K)^((n-1)/n) V(L)^(1/n))+s=(V_1(K, L))", "inference_order": 1182} {"schema_version": "1.0", "item_id": "rcd2_e22dc8dfed2bb9dd", "pair_id": "pair_6d5b5ca8467ab348", "statement_latex": "\\neg\\left(\\inf_{z\\in D}(\\delta_D(z,(Tor_1^A(C, M)))+\\delta_D(z,(0)))=0\\right)", "inference_order": 1183} {"schema_version": "1.0", "item_id": "rcd2_faf9a4052cf78f49", "pair_id": "pair_75efdfe7d24226c5", "statement_latex": "\\{(nu_p(x^n - y^n))\\}=(\\{(nu_p(x - y) + nu_p(n))\\}) + \\eta", "inference_order": 1184} {"schema_version": "1.0", "item_id": "rcd2_9d286672ddfae1b6", "pair_id": "pair_0669e1797f79f82a", "statement_latex": "\\neg\\left(\\forall F:D\\to\\{0,1\\},\\;F(((x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}))=F(((x-m){\\dbinom {x}{m}}))\\right)", "inference_order": 1185} {"schema_version": "1.0", "item_id": "rcd2_3ea278a9b6aad7a0", "pair_id": "pair_5e9a6b46177aa61f", "statement_latex": "\\delta_D((x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2),(z_1^2 + z_2^2 + ... + z_{16}^2))=0", "inference_order": 1186} {"schema_version": "1.0", "item_id": "rcd2_ead79d6c07914ca6", "pair_id": "pair_55a77df73dd25ae0", "statement_latex": "(1) - (sum_{k=0}^n a_k/binom(n,k)) \\ge 0", "inference_order": 1187} {"schema_version": "1.0", "item_id": "rcd2_66b3a30f40c0cb4c", "pair_id": "pair_cf62671ae060c8f6", "statement_latex": "\\delta_D(((\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2),(\\sum_{i0} M(t)e^{-ta}),(-\\infty,0])=0", "inference_order": 1231} {"schema_version": "1.0", "item_id": "rcd2_9da01090e14fe7af", "pair_id": "pair_de119f21b9fbc967", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(inf_{t>0} M(t)e^{-ta})\\le t\\Rightarrow (P(X ≤ a))\\le t", "inference_order": 1232} {"schema_version": "1.0", "item_id": "rcd2_41375a332c163790", "pair_id": "pair_4f47dc32a7f986ec", "statement_latex": "\\max\\{0,(\\limsup_{x → +\\infty} x^2 q(x))-(\\frac{1}{4})\\}=0\\;\\land\\;(\\limsup_{x → +\\infty} x^2 q(x))\\ne (\\frac{1}{4})", "inference_order": 1233} {"schema_version": "1.0", "item_id": "rcd2_0cc5999a19e27690", "pair_id": "pair_436f7fa6bb3eb389", "statement_latex": "\\nexists S \\subseteq D: (((|A+B|) \\in S \\land (|A+H|+|B+H|-|H|) \\notin S) \\lor ((|A+H|+|B+H|-|H|) \\in S \\land (|A+B|) \\notin S))", "inference_order": 1234} {"schema_version": "1.0", "item_id": "rcd2_ee78adfb683ee0e3", "pair_id": "pair_872c41e33ff1fbee", "statement_latex": "\\max\\{0,(Pr(max_{1>k≤n} |S_k| ≥ 3alpha))-(3 max_{1≤k≤n} Pr(|S_k| ≥ alpha))\\}=0", "inference_order": 1235} {"schema_version": "1.0", "item_id": "rcd2_c1b5a0b832e0471a", "pair_id": "pair_7fd0c915542b5b1c", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((E(T)))\\\\neF((n H_n))", "inference_order": 1236} {"schema_version": "1.0", "item_id": "rcd2_c1e74dbd6ddef6a7", "pair_id": "pair_f837b9d0966556ac", "statement_latex": "\\delta_D((p(z)),(\\omega p^*(z)))=0", "inference_order": 1237} {"schema_version": "1.0", "item_id": "rcd2_9256a7c53b6fdd6a", "pair_id": "pair_aa04f1669db18956", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (max_{|z|=1}|P'(z)|) \\le x \\land x \\le (n max_{|z|=1}|P(z)|)) + \\eta", "inference_order": 1238} {"schema_version": "1.0", "item_id": "rcd2_5f919da2067878c0", "pair_id": "pair_ade17c3dd001731b", "statement_latex": "\\forall z\\in D,\\;(z=(sum_{p=2}^{\\infty} 1/(p-1))\\Longleftrightarrow z=(1))", "inference_order": 1239} {"schema_version": "1.0", "item_id": "rcd2_9a8b9dd4ba9e3134", "pair_id": "pair_e5a51c0d7364c6e2", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;((f(x)),z)=((s),z)\\right)", "inference_order": 1240} {"schema_version": "1.0", "item_id": "rcd2_6ef558911f68423b", "pair_id": "pair_e2567581c7e01f0d", "statement_latex": "\\max\\{0,(1 + |a_{n-2}| + ... + |a_{0}|)-(|a_{n-1}|)\\}=0\\;\\land\\;(1 + |a_{n-2}| + ... + |a_{0}|)\\ne (|a_{n-1}|)", "inference_order": 1241} {"schema_version": "1.0", "item_id": "rcd2_1cb46f2e358e8ada", "pair_id": "pair_93d8a96189e9f4e0", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (dim_C H^i(X, F))\\Rightarrow t<(infinity)", "inference_order": 1242} {"schema_version": "1.0", "item_id": "rcd2_00f7292d85e84eaa", "pair_id": "pair_74edcb9be40f62eb", "statement_latex": "\\{(lim_{x→c} f(x)/g(x))\\}=\\{(\\sup\\{f(x, \\theta) : x \\in C(\\theta)\\})\\}", "inference_order": 1243} {"schema_version": "1.0", "item_id": "rcd2_5827d92d7cc6772d", "pair_id": "pair_42c402e05f82fc97", "statement_latex": "\\neg(\\neg (ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) =( 2^{n-1}∏_{i=1}^{n}ε(Xi)))) + \\eta", "inference_order": 1244} {"schema_version": "1.0", "item_id": "rcd2_448b410649f3402c", "pair_id": "pair_1d98097899389158", "statement_latex": "\\nexists S \\subseteq D: (((\\dot{q}) \\in S \\land (f(q,u)) \\notin S) \\lor ((f(q,u)) \\in S \\land (\\dot{q}) \\notin S))", "inference_order": 1245} {"schema_version": "1.0", "item_id": "rcd2_87f6a8843be67c9b", "pair_id": "pair_ae2eaa5aadd99986", "statement_latex": "\\nexists S \\subseteq D: (((w(S)) \\in S \\land (sum_{x \\in S} w(x)) \\notin S) \\lor ((sum_{x \\in S} w(x)) \\in S \\land (w(S)) \\notin S))", "inference_order": 1246} {"schema_version": "1.0", "item_id": "rcd2_9678baf8091d35df", "pair_id": "pair_aca411f09c7874cf", "statement_latex": "(\\infty) = (sum_{n \\in S} (1/n))", "inference_order": 1247} {"schema_version": "1.0", "item_id": "rcd2_9b4ec8d2c442e68c", "pair_id": "pair_a1806a2edc1ef3e7", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(dX_t))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(mu_t dt + sigma_t dB_t))", "inference_order": 1248} {"schema_version": "1.0", "item_id": "rcd2_bf24cb9649814781", "pair_id": "pair_725d3e8e971c8169", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (D^{-1}*sqrt(det(AA^T)))^(1/(N-M))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (max|X_j|)\\} + \\eta", "inference_order": 1249} {"schema_version": "1.0", "item_id": "rcd2_abfc580d0ff9b97c", "pair_id": "pair_9d6649795cef3025", "statement_latex": "(\\binom{k}{r} \\frac{r}{k}) \\in {(\\binom{k-1}{r-1})}", "inference_order": 1250} {"schema_version": "1.0", "item_id": "rcd2_b4084de9493facb4", "pair_id": "pair_235937b551ce04f6", "statement_latex": "\\frac{\\exp(1/2)}{\\exp(1/2)+\\exp(floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)))}<\\frac{1}{2}", "inference_order": 1251} {"schema_version": "1.0", "item_id": "rcd2_3ab01ef82febac18", "pair_id": "pair_fce99265c91fa270", "statement_latex": "\\neg\\left(\\exists k\\in\\mathbb{Z}:(ay)=(x)+k(m)\\right)", "inference_order": 1252} {"schema_version": "1.0", "item_id": "rcd2_292bbc9ad9a22281", "pair_id": "pair_52376eb897b5d805", "statement_latex": "\\forall z\\in D,\\;(z=(dS)\\Longleftrightarrow z=((dE + PdV) / T))", "inference_order": 1253} {"schema_version": "1.0", "item_id": "rcd2_678db6a4a076b3e3", "pair_id": "pair_831fe7de262e7108", "statement_latex": "((1 + ‖P‖) \\inf_{u ∈ U} ‖v - u‖) - (‖v - Pv‖) \\ge 0", "inference_order": 1254} {"schema_version": "1.0", "item_id": "rcd2_6af3100ebe91b319", "pair_id": "pair_41cb4f893c241a1b", "statement_latex": "\\frac{\\exp(|d(A,X) - d(B,X)|)}{\\exp(|d(A,X) - d(B,X)|)+\\exp(d(A,B))}\\le\\frac{1}{2}", "inference_order": 1255} {"schema_version": "1.0", "item_id": "rcd2_b8a1adbd0c47a076", "pair_id": "pair_e3a5ba8bb54919ae", "statement_latex": "\\{((gcd{(n-1 k-1), (n k+1), (n+1 k)}),z):z\\in D\\}=(\\{((gcd{(n-1 k), (n k-1), (n+1 k+1)}),z):z\\in D\\}) + \\eta", "inference_order": 1256} {"schema_version": "1.0", "item_id": "rcd2_229d3de8d637192c", "pair_id": "pair_f5d015ac9ee7670a", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(ρ_W(g) ∘ f))+\\delta_D(z,(f ∘ ρ_V(g))))=(0) + \\eta", "inference_order": 1257} {"schema_version": "1.0", "item_id": "rcd2_a57cc9249e9eaafb", "pair_id": "pair_ec45af887e152cd4", "statement_latex": "\\neg\\left(\\max\\{0,(c|A|^{1 + ε})-(max(|A + A|, |A * A|))\\}=0\\right)", "inference_order": 1258} {"schema_version": "1.0", "item_id": "rcd2_28783c94838eedbb", "pair_id": "pair_a25a80bfcb2c8c96", "statement_latex": "\\forall z\\in D,\\;(z=((a^2 - b^2)\\Longleftrightarrow z=(a + b)(a - b))) + \\eta_{1077}", "inference_order": 1259} {"schema_version": "1.0", "item_id": "rcd2_8b58ccee5ab5f1e8", "pair_id": "pair_9c9766233ff88a4f", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((gcd{(n-1 k-1), (n k+1), (n+1 k)}))=F((gcd{(n-1 k), (n k-1), (n+1 k+1)}))", "inference_order": 1260} {"schema_version": "1.0", "item_id": "rcd2_66e2c109960877e1", "pair_id": "pair_38f1b130ca294a46", "statement_latex": "\\max\\{0,(Pr(S_n > t))-(exp(-(sigma^2 / a^2) * h(at / sigma^2)))\\}=0", "inference_order": 1261} {"schema_version": "1.0", "item_id": "rcd2_2933ba103e909112", "pair_id": "pair_819870f48c4b1669", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;((H(t)),z)=((∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE),z)\\right)", "inference_order": 1262} {"schema_version": "1.0", "item_id": "rcd2_5aa698f378cd6601", "pair_id": "pair_77e33a77fea4e18a", "statement_latex": "\\nexists S \\subseteq D: (((F_{n-1}*F_{n+1} - F_n^2) \\in S \\land ((-1)^n) \\notin S) \\lor (((-1)^n) \\in S \\land (F_{n-1}*F_{n+1} - F_n^2) \\notin S))", "inference_order": 1263} {"schema_version": "1.0", "item_id": "rcd2_db85cf4c7f7aa9a1", "pair_id": "pair_1f7a89e3805792fd", "statement_latex": "(1) - (e*p*(d+1)) \\ge 0 + \\eta", "inference_order": 1264} {"schema_version": "1.0", "item_id": "rcd2_689822332f4d6b23", "pair_id": "pair_886f7b2a93943ef6", "statement_latex": "\\forall z\\in D,\\;(z\\\\ne(∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2)\\Longleftrightarrow z=(∑_{i=1}^{8} c_i^2))", "inference_order": 1265} {"schema_version": "1.0", "item_id": "rcd2_14ac499dae375732", "pair_id": "pair_0d0d34a0329826dd", "statement_latex": "\\{((G(k1, ..., km)),z):z\\in D\\}=(\\{(([x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i),z):z\\in D\\}) + \\eta", "inference_order": 1266} {"schema_version": "1.0", "item_id": "rcd2_e7a033d71b6d42b3", "pair_id": "pair_ff37a8a291b33407", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((max_matching_size))\\\\neF((1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)))", "inference_order": 1267} {"schema_version": "1.0", "item_id": "rcd2_8ad6cfd54fbf28a6", "pair_id": "pair_8f673fdbc5d59c2e", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (f_1(X)f_2(Y)) \\le x \\land x \\le (f_3(X ∨ Y)f_4(X ∧ Y))) + \\eta_{805}", "inference_order": 1268} {"schema_version": "1.0", "item_id": "rcd2_d5966020f8867886", "pair_id": "pair_8070c40f83ead364", "statement_latex": "\\frac{\\exp(|det(N)|)}{\\exp(|det(N)|)+\\exp(prod_{i=(1}^n ||v_i||)}\\le\\frac{1}{2}) + \\eta", "inference_order": 1269} {"schema_version": "1.0", "item_id": "rcd2_39f158d07c8a8bf1", "pair_id": "pair_f6921ffc186bb8e9", "statement_latex": "\\frac{\\exp(delta |X| |Y|)}{\\exp(delta |X| |Y|)+\\exp(e(X,Y))}\\\\ge\\frac{1}{2}", "inference_order": 1270} {"schema_version": "1.0", "item_id": "rcd2_7318c179491e0328", "pair_id": "pair_7e6c93f583fde729", "statement_latex": "\\neg\\left(\\inf_{z\\in D}(\\delta_D(z,(L))+\\delta_D(z,(K(α))))=0\\right)", "inference_order": 1271} {"schema_version": "1.0", "item_id": "rcd2_cc8f4ec765740b24", "pair_id": "pair_0c7315b82d4df5d0", "statement_latex": "(f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x)) \\land (0 = 0)", "inference_order": 1272} {"schema_version": "1.0", "item_id": "rcd2_bc815f2e7e777c70", "pair_id": "pair_7eb64502c96c1fde", "statement_latex": "\\forall k\\in\\mathbb{N}_{>0}\\exists N\\forall n\\ge N:d((lim_{n),(∞} |S_n(β)|^(1/n)))<1/k", "inference_order": 1273} {"schema_version": "1.0", "item_id": "rcd2_a33d2b8ea9a34098", "pair_id": "pair_eb7b50635dd78cc0", "statement_latex": "(\\{(det(I_m + AB))\\}\\times\\{(det(I_n + BA))\\})\\cap\\{(z,z):z\\in D\\}=\\{((det(I_m + AB)),(det(I_n + BA)))\\}", "inference_order": 1274} {"schema_version": "1.0", "item_id": "rcd2_95a59689f07826cd", "pair_id": "pair_29e773efb1e0c381", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(L))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(K(α)))", "inference_order": 1275} {"schema_version": "1.0", "item_id": "rcd2_ab54464c52621203", "pair_id": "pair_63f5bc0ae0226de5", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{k=0}^{n-1} floor(x+k/n)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(floor(n*x)))", "inference_order": 1276} {"schema_version": "1.0", "item_id": "rcd2_79129d6433467146", "pair_id": "pair_7a50c31bd734a8a4", "statement_latex": "\\neg\\left((\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger}))) \\in {(\\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha)}\\right)", "inference_order": 1277} {"schema_version": "1.0", "item_id": "rcd2_cc97e5dedfe4ffd0", "pair_id": "pair_792ccbd07ca1182e", "statement_latex": "\\exists s \\in \\mathbb{R}_{>0}: (limsup_{n-) \\\\ne (\\infty} E_n = intersection_{n=1 to \\infty} union_{k=n to \\infty} E_k) + s", "inference_order": 1278} {"schema_version": "1.0", "item_id": "rcd2_d69e63e6f0f560e2", "pair_id": "pair_9694af5e041e4865", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(ΔδT))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(0))", "inference_order": 1279} {"schema_version": "1.0", "item_id": "rcd2_1eca4e8561fe55cb", "pair_id": "pair_ff31293c235667d3", "statement_latex": "\\forall z\\in D,\\;(z=(mu(t))\\Longleftrightarrow z=(integral_{-infinity}^{t} A(s)p(s,t) ds))", "inference_order": 1280} {"schema_version": "1.0", "item_id": "rcd2_bfe3d3c8eda989e4", "pair_id": "pair_49ef87d06ac97d3a", "statement_latex": "\\nexists S \\subseteq D: (((mu(t)) \\in S \\land (integral_{-\\infty}^{t} A(s)p(s,t) ds) \\notin S) \\lor ((integral_{-\\infty}^{t} A(s)p(s,t) ds) \\in S \\land (mu(t)) \\notin S)) + \\eta", "inference_order": 1281} {"schema_version": "1.0", "item_id": "rcd2_c62238b8d614cb54", "pair_id": "pair_57f910e14cb9f634", "statement_latex": "\\nexists S \\subseteq D: (((-z\\eta'(s)/z\\eta(s)) \\in S \\land (F(s) - z\\eta(s)F(s)G(s) - z\\eta'(s)G(s) + (-z\\eta'(s)/z\\eta(s) - F(s))(1 - z\\eta(s)G(s))) \\notin S) \\lor ((F(s) - z\\eta(s)F(s)G(s) - z\\eta'(s)G(s) + (-z\\eta'(s)/z\\eta(s) - F(s))(1 - z\\eta(s)G(s))) \\in S \\land (-z\\eta'(s)/z\\eta(s)) \\notin S))", "inference_order": 1282} {"schema_version": "1.0", "item_id": "rcd2_a70539358d6b8895", "pair_id": "pair_ff9db8570754754f", "statement_latex": "\\nexists S \\subseteq D: ((((x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2)) \\in S \\land (z_1^2 + z_2^2 + ... + z_{16}^2) \\notin S) \\lor ((z_1^2 + z_2^2 + ... + z_{16}^2) \\in S \\land ((x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2)) \\notin S))", "inference_order": 1283} {"schema_version": "1.0", "item_id": "rcd2_6ab45faaab7baed8", "pair_id": "pair_87333e443d5700f6", "statement_latex": "((x+y)^n =( \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k) \\land (0 = 0)) + \\eta", "inference_order": 1284} {"schema_version": "1.0", "item_id": "rcd2_fe52ce3c0d962c16", "pair_id": "pair_a41af95998f42bee", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((a/p))=F((epsilon(pi_a)))", "inference_order": 1285} {"schema_version": "1.0", "item_id": "rcd2_de24c01ecd7aeb77", "pair_id": "pair_1e000eb3ff83cc9f", "statement_latex": "\\operatorname{softplus}((n)-(sum_{j=1}^r d_j))>\\log 2", "inference_order": 1286} {"schema_version": "1.0", "item_id": "rcd2_fbd185b52fde8bbe", "pair_id": "pair_68ee169b9b4ae4a9", "statement_latex": "(\\dim(E^*)) \\in {(\\text{card}(\\mathbb{K}^I))} + \\eta", "inference_order": 1287} {"schema_version": "1.0", "item_id": "rcd2_9a270db8379099c1", "pair_id": "pair_a57e4e3ce1ae86b9", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(cp * n)\\le t\\Rightarrow (r(G))\\le t", "inference_order": 1288} {"schema_version": "1.0", "item_id": "rcd2_4412091170965e01", "pair_id": "pair_29b324593a3b8806", "statement_latex": "((∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2) \\land (0 = 0)", "inference_order": 1289} {"schema_version": "1.0", "item_id": "rcd2_c6974771d988f042", "pair_id": "pair_59753ac1886d9034", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (sum_{k=0}^n a_k/binom(n,k)) \\le x \\land x \\le (1)) + \\eta_{381}", "inference_order": 1290} {"schema_version": "1.0", "item_id": "rcd2_005d0644e5e46849", "pair_id": "pair_d18cc22f96b3e3da", "statement_latex": "((delta |X| |Y|),(e(X,Y)))\\in\\{(u,v)\\in\\mathbb{R}^2:u>v\\}", "inference_order": 1291} {"schema_version": "1.0", "item_id": "rcd2_2dc0e04ae38e143b", "pair_id": "pair_3a7c7277f91ad886", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(α))+\\delta_D(z,(Pr_θ0(X ∈ R))))=0", "inference_order": 1292} {"schema_version": "1.0", "item_id": "rcd2_16710332b864dbbc", "pair_id": "pair_210668c7ba2dd451", "statement_latex": "\\neg\\left((\\{(\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k})\\}\\times\\{({\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n})\\})\\cap\\{(z,z):z\\in D\\}=\\{((\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}),({\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}))\\}\\right)", "inference_order": 1293} {"schema_version": "1.0", "item_id": "rcd2_dc4bfb7fac90b623", "pair_id": "pair_120692b011c53e49", "statement_latex": "\\delta_D((\\zeta(s)),(\\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}}))=0", "inference_order": 1294} {"schema_version": "1.0", "item_id": "rcd2_28ec816d4ab8603c", "pair_id": "pair_e3f050aa051278e9", "statement_latex": "\\frac{\\exp(r(G))}{\\exp(r(G))+\\exp(cp * n)}\\le\\frac{1}{2}", "inference_order": 1295} {"schema_version": "1.0", "item_id": "rcd2_80483b838cd774f6", "pair_id": "pair_fe30979f3da82c4f", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;((|A_d|),z)=((\\frac{w(d)}{d}X + R_d),z)\\right)", "inference_order": 1296} {"schema_version": "1.0", "item_id": "rcd2_a654532661c8c35e", "pair_id": "pair_5d6baa6f9ca79f17", "statement_latex": "\\sup_{x\\in D}\\mathbf{1}_{\\{prime p\\in D:n < p < 2*n\\}}(x)=1", "inference_order": 1297} {"schema_version": "1.0", "item_id": "rcd2_f538a7a3b898fd69", "pair_id": "pair_a94bf23ca7aa472c", "statement_latex": "\\{t\\in\\mathbb{R}:t\\le (AC * BD)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (AB * CD + BC * DA)\\} + \\eta", "inference_order": 1298} {"schema_version": "1.0", "item_id": "rcd2_0f3a2138109240b9", "pair_id": "pair_e9a28d2962be7d45", "statement_latex": "(\\{(ρ_W(g) ∘ f)\\}\\times\\{(f ∘ ρ_V(g))\\})\\cap\\{(z,z):z\\in D\\}=(\\{((ρ_W(g) ∘ f),(f ∘ ρ_V(g)))\\}) + \\eta", "inference_order": 1299} {"schema_version": "1.0", "item_id": "rcd2_807542cdbf7751a8", "pair_id": "pair_3e33883424fe3c1b", "statement_latex": "\\max\\{0,(|ω(n)−loglogn|)-(loglogn)^(1/2+ε)\\}=(0\\;\\land\\;(|ω(n)−loglogn|)\\ne (loglogn)^(1/2+ε)) + \\eta", "inference_order": 1300} {"schema_version": "1.0", "item_id": "rcd2_f9789c4f4a4f53c0", "pair_id": "pair_29e773efb1e0c381", "statement_latex": "\\neg\\left(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(L))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(K(α)))\\right)", "inference_order": 1301} {"schema_version": "1.0", "item_id": "rcd2_1d7cf339c224f272", "pair_id": "pair_886f7b2a93943ef6", "statement_latex": "\\forall z\\in D,\\;(z=(∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2)\\Longleftrightarrow z=(∑_{i=1}^{8} c_i^2))", "inference_order": 1302} {"schema_version": "1.0", "item_id": "rcd2_22816f19cd634ff2", "pair_id": "pair_a7e5d240eac94cd0", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (1)\\Rightarrow t>(g)", "inference_order": 1303} {"schema_version": "1.0", "item_id": "rcd2_9521c9bd4bd9f179", "pair_id": "pair_200ee8ecdac87cf8", "statement_latex": "\\nexists S \\subseteq D: (((Tor_1^A(C, M)) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (Tor_1^A(C, M)) \\notin S))", "inference_order": 1304} {"schema_version": "1.0", "item_id": "rcd2_850c547721551c95", "pair_id": "pair_846a62d66f9aee97", "statement_latex": "((dE + PdV) / T) = (dS)", "inference_order": 1305} {"schema_version": "1.0", "item_id": "rcd2_275de04eea35c9fd", "pair_id": "pair_9fdfd0daf6a4ccc0", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((∫ |∇u*|^p dx)-(∫ |∇u|^p dx))=0", "inference_order": 1306} {"schema_version": "1.0", "item_id": "rcd2_7c66fe04e18bf4c4", "pair_id": "pair_420a41a5c6a16a10", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((\\lim_{\\theta → \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx))=F((-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)))", "inference_order": 1307} {"schema_version": "1.0", "item_id": "rcd2_691cc755f1a93d5a", "pair_id": "pair_38fa35f82ced937e", "statement_latex": "\\{((U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C),z):z\\in D\\}=\\{((|\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C),z):z\\in D\\}", "inference_order": 1308} {"schema_version": "1.0", "item_id": "rcd2_0eb4892b9fc19f7e", "pair_id": "pair_7da09b757d164396", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn)))=F((2^{n-1}∏_{i=1}^{n}ε(Xi)))", "inference_order": 1309} {"schema_version": "1.0", "item_id": "rcd2_9d8de83eeff7c3e5", "pair_id": "pair_040362726de7e564", "statement_latex": "((constant_term(prod_{1≥i≠j≤n} (1-t_i/t_j)^{a_i})),(sum a_i)! / prod(a_i!))\\in\\{(z,z):z\\in D\\}", "inference_order": 1310} {"schema_version": "1.0", "item_id": "rcd2_af0b099bca616ed3", "pair_id": "pair_c9c142ef02b3aed0", "statement_latex": "(ker(d(f|_Y)_y))\\setminus (tau)=\\emptyset", "inference_order": 1311} {"schema_version": "1.0", "item_id": "rcd2_a66f422a07b2c53c", "pair_id": "pair_a4a5b9845b5f937d", "statement_latex": "\\neg\\left(\\frac{\\exp(||f(A)||)}{\\exp(||f(A)||)+\\exp(2 sup_{z in W(A)} |f(z)|)}\\le\\frac{1}{2}\\right)", "inference_order": 1312} {"schema_version": "1.0", "item_id": "rcd2_4fcfe661fdc3aead", "pair_id": "pair_0669e1797f79f82a", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F(((x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}))=F(((x-m){\\dbinom {x}{m}}))", "inference_order": 1313} {"schema_version": "1.0", "item_id": "rcd2_7dd5d847575ed667", "pair_id": "pair_5631533fe4bc5065", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (|f(z)|) \\le x \\land x \\le (|z|)) + \\eta_{609}", "inference_order": 1314} {"schema_version": "1.0", "item_id": "rcd2_b008c0d460b0c578", "pair_id": "pair_779853b36e5441ba", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(\\operatorname{Re} \\langle x, y \\rangle)=z\\;\\land\\;z=(\\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2))", "inference_order": 1315} {"schema_version": "1.0", "item_id": "rcd2_e2aa4bf92abc3fdc", "pair_id": "pair_1c29dcce825db149", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((rank(A)) \\in S \\land (rank([A|b])) \\notin S) \\lor ((rank([A|b])) \\in S \\land (rank(A)) \\notin S))\\right)", "inference_order": 1316} {"schema_version": "1.0", "item_id": "rcd2_e5c2ac94a3ec22da", "pair_id": "pair_f5908f2fb2d1554d", "statement_latex": "\\{((F(z,t)),z):z\\in D\\}=\\{((e^{g(z)t}),z):z\\in D\\}", "inference_order": 1317} {"schema_version": "1.0", "item_id": "rcd2_7cc0939406122ebc", "pair_id": "pair_281a2b9e19ea8ac8", "statement_latex": "\\max\\{0,(b - a - ε)-(μ(E))\\}=0\\;\\land\\;(b - a - ε)\\ne (μ(E))", "inference_order": 1318} {"schema_version": "1.0", "item_id": "rcd2_dda6064ebb91af30", "pair_id": "pair_daea5f5eeccf5793", "statement_latex": "((dim(H_μ)),(n!))\\in\\{(z,z):z\\in D\\}", "inference_order": 1319} {"schema_version": "1.0", "item_id": "rcd2_494838f73995c0c1", "pair_id": "pair_f3e521c12f23718a", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(rank(T) + nullity(T)))+\\delta_D(z,(dim(V))))=(0) + \\eta", "inference_order": 1320} {"schema_version": "1.0", "item_id": "rcd2_fffa2c2f5d8cf148", "pair_id": "pair_9fd002d37e657903", "statement_latex": "((product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2})),(sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}))\\in\\{(z,z):z\\in D\\}", "inference_order": 1321} {"schema_version": "1.0", "item_id": "rcd2_9325a3756913f5a4", "pair_id": "pair_8c3ac8f21239d7ac", "statement_latex": "(g) - (1) >( 0) + \\eta", "inference_order": 1322} {"schema_version": "1.0", "item_id": "rcd2_cade97c75e131eb2", "pair_id": "pair_26cecaf933bb5c40", "statement_latex": "\\neg\\left(\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(min(|A|+|B|-1, p))+s=(|A+B|)\\right)", "inference_order": 1323} {"schema_version": "1.0", "item_id": "rcd2_7b0732caa42abd70", "pair_id": "pair_d9e101fd12d6f890", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(nullity A))+\\delta_D(z,(nullity H)))=0", "inference_order": 1324} {"schema_version": "1.0", "item_id": "rcd2_6808157837ffdef0", "pair_id": "pair_5696551244d2e317", "statement_latex": "(a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}) = (y^{2})", "inference_order": 1325} {"schema_version": "1.0", "item_id": "rcd2_a070b3a2af167276", "pair_id": "pair_35733e1e74e4ecee", "statement_latex": "(a_0 + a_1X + ... + a_nX^n) = (F(X))", "inference_order": 1326} {"schema_version": "1.0", "item_id": "rcd2_9e2e3da6d990a813", "pair_id": "pair_432f76a3be5ea262", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(lim_{n→∞} σ_n(f,x))=z\\;\\land\\;z=(f(x))", "inference_order": 1327} {"schema_version": "1.0", "item_id": "rcd2_de5b3e7d4334c211", "pair_id": "pair_97db7e5b3265ad46", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(∫ |∇u|^p dx)\\le t\\Rightarrow (∫ |∇u*|^p dx)\\le t", "inference_order": 1328} {"schema_version": "1.0", "item_id": "rcd2_13e6cf76e18a8f90", "pair_id": "pair_d3437dba54f79f51", "statement_latex": "\\neg\\left(\\exists z\\in D\\;\\text{such that}\\;(H)=z\\;\\land\\;z=(H_0 + gV)\\right)", "inference_order": 1329} {"schema_version": "1.0", "item_id": "rcd2_3750474a51d76f54", "pair_id": "pair_7a56872f6b2b1e3b", "statement_latex": "(\\{(x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2})\\}\\times\\{(0)\\})\\cap\\{(z,z):z\\in D\\}=\\{((x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2}),(0))\\}", "inference_order": 1330} {"schema_version": "1.0", "item_id": "rcd2_404492ef3627b02a", "pair_id": "pair_257cf3083b7e1fab", "statement_latex": "(lim_{n -) - (\\infty} a_n =( sup {a_n}) > 0) + \\eta", "inference_order": 1331} {"schema_version": "1.0", "item_id": "rcd2_2ccc162e9253b2c5", "pair_id": "pair_f241db6d00659c0c", "statement_latex": "(g ∘ T) \\\\ne (f)", "inference_order": 1332} {"schema_version": "1.0", "item_id": "rcd2_e1fec499524060e9", "pair_id": "pair_792655a4bb0a9430", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (p(a,d)) < x \\land x < (c * d^L)", "inference_order": 1333} {"schema_version": "1.0", "item_id": "rcd2_0c929cee094cea58", "pair_id": "pair_981ac7ea7887d35b", "statement_latex": "\\{(lim_{n→∞} σ_n(f,x))\\}=\\{(f(x))\\}", "inference_order": 1334} {"schema_version": "1.0", "item_id": "rcd2_6415491885836691", "pair_id": "pair_22f49b5618a83d9b", "statement_latex": "\\delta_D((f(x)),(s))=0", "inference_order": 1335} {"schema_version": "1.0", "item_id": "rcd2_bb07a0d78b828d25", "pair_id": "pair_dfdab42120eecc82", "statement_latex": "((p),(a^2 + b^2 + c^2 + d^2))\\in\\{(z,z):z\\in D\\} + \\eta", "inference_order": 1336} {"schema_version": "1.0", "item_id": "rcd2_af3a3c04798ee679", "pair_id": "pair_d5d7a3cab87b8092", "statement_latex": "\\delta_D((nu_p(x^n - y^n)),(nu_p(x - y) + nu_p(n)))=0", "inference_order": 1337} {"schema_version": "1.0", "item_id": "rcd2_743aef1a912b170c", "pair_id": "pair_59c6d4998ca96891", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (exp(-(sigma^2 / a^2) * h(at / sigma^2)))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (Pr(S_n >( t))\\}) + \\eta", "inference_order": 1338} {"schema_version": "1.0", "item_id": "rcd2_4d200d2b3fdbde4f", "pair_id": "pair_aeca574e107f0226", "statement_latex": "\\forall z\\in D,\\;(z=((L(E, s))\\Longleftrightarrow z=(sum_{n=1}^∞ a_n / n^s))) + \\eta", "inference_order": 1339} {"schema_version": "1.0", "item_id": "rcd2_c975bcb15daabdbe", "pair_id": "pair_5ea14df46a8e7227", "statement_latex": "(Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2)) \\lor (0 = 1)", "inference_order": 1340} {"schema_version": "1.0", "item_id": "rcd2_bcd823d3016ef198", "pair_id": "pair_d2c51e84eca5d8bb", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;((x + y)/2)\\le t\\Rightarrow (sqrt(x*y))\\le t\\right)", "inference_order": 1341} {"schema_version": "1.0", "item_id": "rcd2_adc8358cfd0a8e84", "pair_id": "pair_ea569954e25af666", "statement_latex": "(prod_{i=1}^n (r_i!)^(1/r_i)) - (per(A)) \\ge 0", "inference_order": 1342} {"schema_version": "1.0", "item_id": "rcd2_d4673509932cb4e0", "pair_id": "pair_9eeea7927127851c", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (|alpha - beta|)\\Rightarrow t<(|alpha - alpha_i|)", "inference_order": 1343} {"schema_version": "1.0", "item_id": "rcd2_3269b9002160d6f7", "pair_id": "pair_c1006da8bb6ceb4f", "statement_latex": "\\neg\\left(\\neg(\\neg (1/sqrt(p) * sum_{n=0}^{p-1} \\exp(2*\\pi*i*n^2*q/p) = e^(1/4 * \\pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} \\exp(-\\pi*i*n^2*p/(2q))))\\right)", "inference_order": 1344} {"schema_version": "1.0", "item_id": "rcd2_e3f8312cec75081f", "pair_id": "pair_9589f37f1cd42b77", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (|Q(x)|)\\Rightarrow t<(\\epsilon)", "inference_order": 1345} {"schema_version": "1.0", "item_id": "rcd2_e273e50a5e56f4d5", "pair_id": "pair_10f901bf08dc807d", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(Ax))+\\delta_D(z,(λx)))=0", "inference_order": 1346} {"schema_version": "1.0", "item_id": "rcd2_3a61226e04116575", "pair_id": "pair_d9f6b6548a9b56b7", "statement_latex": "\\forall z\\in D,\\;((P[σ(x(t))] / P[σ(x_tilde(t))]),z)\\\\ne((e^σ[x(t)]),z)", "inference_order": 1347} {"schema_version": "1.0", "item_id": "rcd2_84998867e4a3d538", "pair_id": "pair_f47ee9dba8be9e6f", "statement_latex": "\\frac{\\exp(|F|)}{\\exp(|F|)+\\exp(max_{r: t+2r ≤( n} |F_{n,k,t,r}|)}\\le\\frac{1}{2}) + \\eta", "inference_order": 1348} {"schema_version": "1.0", "item_id": "rcd2_768aac7be8f4c363", "pair_id": "pair_a651dfff71df6b9e", "statement_latex": "((E),(nhv))\\in\\{(z,z):z\\in D\\}", "inference_order": 1349} {"schema_version": "1.0", "item_id": "rcd2_a796f3c6ecab9ac6", "pair_id": "pair_19f6c5675cf4159d", "statement_latex": "\\{((lim_{n→∞} σ_n(f,x)),z):z\\in D\\}\\\\ne\\{((f(x)),z):z\\in D\\}", "inference_order": 1350} {"schema_version": "1.0", "item_id": "rcd2_bda5136915527a43", "pair_id": "pair_c0f70df5fea8f0c8", "statement_latex": "\\forall z\\in D,\\;(z=(∫_{γ} ∇φ(r) · dr)\\Longleftrightarrow z=(φ(q) - φ(p)))", "inference_order": 1351} {"schema_version": "1.0", "item_id": "rcd2_27e07fcd1cdc9d62", "pair_id": "pair_faeba38f0690befe", "statement_latex": "\\frac{\\exp(ceil(n/m))}{\\exp(ceil(n/m))+\\exp(max_{1≤(i≤m} |B_i|)}\\le\\frac{1}{2}) + \\eta", "inference_order": 1352} {"schema_version": "1.0", "item_id": "rcd2_bdae8ff794ea0d25", "pair_id": "pair_677785bbb55c35ad", "statement_latex": "\\operatorname{softplus}((P(X_N - X_0 ≤ epsilon))-(exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2))))\\le\\log 2", "inference_order": 1353} {"schema_version": "1.0", "item_id": "rcd2_1dd11f73b8117925", "pair_id": "pair_0c7387c59209c4e8", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((f) \\in S \\land (g ∘ T) \\notin S) \\lor ((g ∘ T) \\in S \\land (f) \\notin S))\\right)", "inference_order": 1354} {"schema_version": "1.0", "item_id": "rcd2_16dc5d43d3c5ec9e", "pair_id": "pair_2a51358e4de7a7c8", "statement_latex": "\\sup_{\\varepsilon>0}\\inf_N\\mathbf{1}_{\\{\\forall n\\ge N:d((lim_{n),(∞} |S_n(β)|^(1/n)))<\\varepsilon\\}}=1", "inference_order": 1355} {"schema_version": "1.0", "item_id": "rcd2_99560e698f57ab3a", "pair_id": "pair_e9b08501032d2e29", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(∑ x_i) / (∑ (1-x_i))\\le t\\Rightarrow ((∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n})\\le t", "inference_order": 1356} {"schema_version": "1.0", "item_id": "rcd2_aa665a24c986e931", "pair_id": "pair_9a5043d1eba75496", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(x_i)=z\\;\\land\\;z=(\\text{conv}\\{w(X) : w \\in W\\})", "inference_order": 1357} {"schema_version": "1.0", "item_id": "rcd2_d53d490f7fd6e063", "pair_id": "pair_cc2b4c6aa590c852", "statement_latex": "\\neg\\left(\\delta_D((P(A ∩ B)),(P(B|A)*P(A)))=0\\right)", "inference_order": 1358} {"schema_version": "1.0", "item_id": "rcd2_b3880f3d52b86c85", "pair_id": "pair_8d68efd1c7b766f0", "statement_latex": "(x(A) * Product_{B \\in Gamma(A)} (1 - x(B))) - (Pr[A]) \\ge 0", "inference_order": 1359} {"schema_version": "1.0", "item_id": "rcd2_f533324a6de5e222", "pair_id": "pair_9e1a1f1abcf1ff9f", "statement_latex": "\\frac{\\exp(1)}{\\exp(1)+\\exp(|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2|)}\\le\\frac{1}{2}", "inference_order": 1360} {"schema_version": "1.0", "item_id": "rcd2_83252a735a450e80", "pair_id": "pair_8580def2e7d38229", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger})))=(z\\;\\land\\;z=(\\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha)) + \\eta", "inference_order": 1361} {"schema_version": "1.0", "item_id": "rcd2_c3fc3f6bd5b96e11", "pair_id": "pair_d04a99fc0847edc1", "statement_latex": "\\nexists S \\subseteq D: (((H) \\in S \\land (H_0 + gV) \\notin S) \\lor ((H_0 + gV) \\in S \\land (H) \\notin S))", "inference_order": 1362} {"schema_version": "1.0", "item_id": "rcd2_b30a476e59a213e1", "pair_id": "pair_779cbbe2829c9dda", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(E))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\bigcup_{k}(a_{k},b_{k})))", "inference_order": 1363} {"schema_version": "1.0", "item_id": "rcd2_98ecc5ef9c776bd2", "pair_id": "pair_5ce7af2c9aa579d9", "statement_latex": "\\nexists S \\subseteq D: (((gcd{(n-1 k-1), (n k+1), (n+1 k)}) \\in S \\land (gcd{(n-1 k), (n k-1), (n+1 k+1)}) \\notin S) \\lor ((gcd{(n-1 k), (n k-1), (n+1 k+1)}) \\in S \\land (gcd{(n-1 k-1), (n k+1), (n+1 k)}) \\notin S))", "inference_order": 1364} {"schema_version": "1.0", "item_id": "rcd2_282451c1b462ee09", "pair_id": "pair_c2fe54b2ef5733ba", "statement_latex": "\\nexists S \\subseteq D: (((\\nu(\\chi(1))) \\in S \\land (\\nu(|G:D|)) \\notin S) \\lor ((\\nu(|G:D|)) \\in S \\land (\\nu(\\chi(1))) \\notin S)) + \\eta", "inference_order": 1365} {"schema_version": "1.0", "item_id": "rcd2_e173c5769f0bed8d", "pair_id": "pair_f5a7a04438b94169", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{k=0}^m c_k 1_{N=k}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k))", "inference_order": 1366} {"schema_version": "1.0", "item_id": "rcd2_1fe19d54e14a0898", "pair_id": "pair_3636f01615b6c2c9", "statement_latex": "\\delta_D((a^n + b^n),(c^n))=(0) + \\eta", "inference_order": 1367} {"schema_version": "1.0", "item_id": "rcd2_ceeabe31a7bce07a", "pair_id": "pair_4121f4b00f46cbf5", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((w))=F((xyz))", "inference_order": 1368} {"schema_version": "1.0", "item_id": "rcd2_59b830ed1140151a", "pair_id": "pair_809ef9d8745abe70", "statement_latex": "\\frac{\\exp(|ω(n)−loglogn|)}{\\exp(|ω(n)−loglogn|)+\\exp(loglogn)^(1/2+ε)}>\\frac{1}{2}", "inference_order": 1369} {"schema_version": "1.0", "item_id": "rcd2_7603b65c7e27dce9", "pair_id": "pair_f2da3779253bea8d", "statement_latex": "\\neg\\left(((lim_{n → infinity} a_n),(sup {a_n}))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 1370} {"schema_version": "1.0", "item_id": "rcd2_5a220b7cfa7fa281", "pair_id": "pair_3e4a8382500f0cbc", "statement_latex": "(λE(f(X))) \\\\ne (E(Xf(X-1)))", "inference_order": 1371} {"schema_version": "1.0", "item_id": "rcd2_38e0732ea820b800", "pair_id": "pair_44dd3bfba602af9e", "statement_latex": "(\\infty} E_n = intersection_{n=1 to \\infty} union_{k=n to \\infty} E_k) - (limsup_{n-) < 0", "inference_order": 1372} {"schema_version": "1.0", "item_id": "rcd2_a88011a8227e86d3", "pair_id": "pair_87954a319988bae2", "statement_latex": "\\nexists S \\subseteq D: (((binom(m+n, k)_q) \\in S \\land (sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}) \\notin S) \\lor ((sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}) \\in S \\land (binom(m+n, k)_q) \\notin S)) + \\eta", "inference_order": 1373} {"schema_version": "1.0", "item_id": "rcd2_55cc7a0478b4614a", "pair_id": "pair_7d799a509162a9a7", "statement_latex": "\\sup_{\\varepsilon>0}\\inf_N\\mathbf{1}_{\\{\\forall n\\ge N:d((S_βα),(S_β1α1 * S_β2α2))<\\varepsilon\\}}\\\\ne1", "inference_order": 1374} {"schema_version": "1.0", "item_id": "rcd2_01a5ec8a733c6baf", "pair_id": "pair_043336a5b4a5a121", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,({\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}))", "inference_order": 1375} {"schema_version": "1.0", "item_id": "rcd2_49bfa0ea95f38b85", "pair_id": "pair_5fa8ce53c14ba09f", "statement_latex": "\\neg\\left(((P_{n-1}(x)P_{n+1}(x)),(P_n(x)^2))\\in\\{(u,v)\\in\\mathbb{R}^2:u\\le v\\}\\setminus\\{(u,u):u\\in\\mathbb{R}\\}\\right)", "inference_order": 1376} {"schema_version": "1.0", "item_id": "rcd2_85f2b7fb846a1e3f", "pair_id": "pair_f71eba9daccd9c6a", "statement_latex": "\\max\\{0,(P(|sum_{i=1}^n a_i X_i| ≥ k))-(2 inf_{0 ≤ c ≤ k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz)\\}=0", "inference_order": 1377} {"schema_version": "1.0", "item_id": "rcd2_a52b6da84ee4e45d", "pair_id": "pair_2e9bf6d5d9ff0314", "statement_latex": "\\{((Ax),z):z\\in D\\}=(\\{((λx),z):z\\in D\\}) + \\eta", "inference_order": 1378} {"schema_version": "1.0", "item_id": "rcd2_391d8faac86ec551", "pair_id": "pair_cf827eac048fd266", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3))))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (P(|(1/n) * sum_{i=1}^n X_i| > epsilon))\\}", "inference_order": 1379} {"schema_version": "1.0", "item_id": "rcd2_f4958c1685c71da9", "pair_id": "pair_20da720f0e4d38e4", "statement_latex": "\\arctan(det(M))\\le\\arctan(det(A)*det(C))", "inference_order": 1380} {"schema_version": "1.0", "item_id": "rcd2_b0dd3b915de94d37", "pair_id": "pair_464dac91dc103256", "statement_latex": "-(j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)) < -((sum_i a_i*c_i)*(sum_j b_j*d_j) \\\\ne (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i)", "inference_order": 1381} {"schema_version": "1.0", "item_id": "rcd2_0038d302c8389325", "pair_id": "pair_bbd3abee35ebf21e", "statement_latex": "\\{((P(A|B)),z):z\\in D\\}\\\\ne\\{((P(B|A)*P(A)/P(B)),z):z\\in D\\}", "inference_order": 1382} {"schema_version": "1.0", "item_id": "rcd2_a224e8b53c19b83a", "pair_id": "pair_bf2f027430e9633e", "statement_latex": "\\{(([[u, 0], [0, u^-1]]),z):z\\in D\\}=\\{((e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)),z):z\\in D\\}", "inference_order": 1383} {"schema_version": "1.0", "item_id": "rcd2_c2f396de0c53428d", "pair_id": "pair_03a574c6fb4079df", "statement_latex": "((∑_{n≥(1}P(|S_{n}|>εn)),(∞))\\in\\{(u,v)\\in\\mathbb{R}^2:u\\le v\\}\\setminus\\{(u,u):u\\in\\mathbb{R}\\}) + \\eta", "inference_order": 1384} {"schema_version": "1.0", "item_id": "rcd2_bc4e74f9d8a7e259", "pair_id": "pair_1947ad972f7c1ba0", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(z=(N^{(i-1)})\\Longleftrightarrow z=(\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}))\\right)", "inference_order": 1385} {"schema_version": "1.0", "item_id": "rcd2_91e8e8ff50e8990b", "pair_id": "pair_12f08c4d59e1c177", "statement_latex": "\\neg\\left(((\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1}),(1))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 1386} {"schema_version": "1.0", "item_id": "rcd2_23d9eadad982bcc8", "pair_id": "pair_275cb9fd797f7710", "statement_latex": "\\nexists S \\subseteq D: (((F(X)) \\in S \\land (a_0 + a_1X + ... + a_nX^n) \\notin S) \\lor ((a_0 + a_1X + ... + a_nX^n) \\in S \\land (F(X)) \\notin S))", "inference_order": 1387} {"schema_version": "1.0", "item_id": "rcd2_dcf7226473eb7275", "pair_id": "pair_a21f487b6b2da905", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;({m+n \\choose r})\\\\nez\\;\\land\\;z=(\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k})", "inference_order": 1388} {"schema_version": "1.0", "item_id": "rcd2_3d052674afea3d0c", "pair_id": "pair_d97a1e0e47ce3a0c", "statement_latex": "\\neg\\left(\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(∫ |∇u*|^p dx)+s=(∫ |∇u|^p dx)\\right)", "inference_order": 1389} {"schema_version": "1.0", "item_id": "rcd2_7f75e07c198829ec", "pair_id": "pair_db5c0ee214b85abe", "statement_latex": "\\exists c \\in D: (binom(m+n, k)_q) = c \\land (sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}) = c", "inference_order": 1390} {"schema_version": "1.0", "item_id": "rcd2_bf54e7e1851c563b", "pair_id": "pair_8373302242bff73f", "statement_latex": "\\neg\\left((P(A|B)) \\in {(P(B|A)*P(A)/P(B))}\\right)", "inference_order": 1391} {"schema_version": "1.0", "item_id": "rcd2_90d48bcb975e08ad", "pair_id": "pair_d8361ccc1d5ada75", "statement_latex": "-(limsup_{n-) < -(\\infty} E_n \\\\ne intersection_{n=1 to \\infty} union_{k=n to \\infty} E_k)", "inference_order": 1392} {"schema_version": "1.0", "item_id": "rcd2_c99f5649989aada9", "pair_id": "pair_f2f41320160ca368", "statement_latex": "\\{(H)\\}\\\\ne\\{(∑_{x,y} ψ^†(x) F(x, y) ψ(y))\\}", "inference_order": 1393} {"schema_version": "1.0", "item_id": "rcd2_4249c824540c2276", "pair_id": "pair_0edb7de1ef392869", "statement_latex": "\\nexists S \\subseteq D: (((sum (6-i)p_i) \\in S \\land (12) \\notin S) \\lor ((12) \\in S \\land (sum (6-i)p_i) \\notin S))", "inference_order": 1394} {"schema_version": "1.0", "item_id": "rcd2_113bcf81c3648a06", "pair_id": "pair_5b29caf652e00f94", "statement_latex": "\\max\\{0,(1/2)-(floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)))\\}=0\\;\\land\\;(1/2)\\ne (floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)))", "inference_order": 1395} {"schema_version": "1.0", "item_id": "rcd2_e7235ceb938ed6dc", "pair_id": "pair_f8b788585d43e2ba", "statement_latex": "(α = Pr_θ0(X ∈ R)) \\land (0 = 0)", "inference_order": 1396} {"schema_version": "1.0", "item_id": "rcd2_5938d37836ca37f5", "pair_id": "pair_57eba0d377a8c710", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (E[\\exp(lambda*(X-E[X]))]) \\le x \\land x \\le (\\exp(lambda^2*(b-a)^2/8))", "inference_order": 1397} {"schema_version": "1.0", "item_id": "rcd2_af47569c9f42aeb9", "pair_id": "pair_41bcdf363b54858e", "statement_latex": "\\operatorname{dist}((sum_{k\\\\ne0}^n a_k/binom(n,k))-(1),(-\\infty,0])=0", "inference_order": 1398} {"schema_version": "1.0", "item_id": "rcd2_6e8e2e3717c1d9d9", "pair_id": "pair_9a870181ab8b9f1c", "statement_latex": "(E(Xf(X-1))) \\in {(λE(f(X)))} + \\eta", "inference_order": 1399} {"schema_version": "1.0", "item_id": "rcd2_de6198d4ab81cc34", "pair_id": "pair_e2424768e723319d", "statement_latex": "(integral f(x) dLambda(x)) = (E[sum_{x \\in N} f(x)])", "inference_order": 1400} {"schema_version": "1.0", "item_id": "rcd2_a294ca1d5b1a95e9", "pair_id": "pair_6cc7e5e1c6446264", "statement_latex": "\\forall z\\in D,\\;((\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q}),z)=((\\infty),z)", "inference_order": 1401} {"schema_version": "1.0", "item_id": "rcd2_aa68eaee28e257c4", "pair_id": "pair_b37e9d154ec9ff23", "statement_latex": "\\{(F(X))\\}=\\{(\\text{conv}\\{w(X) : w \\in W\\})\\}", "inference_order": 1402} {"schema_version": "1.0", "item_id": "rcd2_f9a6d103ff491438", "pair_id": "pair_26e3b4724fe998e0", "statement_latex": "\\frac{\\exp(v)}{\\exp(v)+\\exp(b)}\\le\\frac{1}{2}", "inference_order": 1403} {"schema_version": "1.0", "item_id": "rcd2_480f1394ca8c8975", "pair_id": "pair_31b0375c651dec34", "statement_latex": "\\frac{\\exp(||u - u_h||)}{\\exp(||u - u_h||)+\\exp((gamma / alpha) * ||u - v||)}\\le\\frac{1}{2}", "inference_order": 1404} {"schema_version": "1.0", "item_id": "rcd2_039661508b85f17a", "pair_id": "pair_aa09e3aed77c9d3d", "statement_latex": "\\{((sum_{i=0}^n binom(n,i)),z):z\\in D\\}=\\{((2^n),z):z\\in D\\}", "inference_order": 1405} {"schema_version": "1.0", "item_id": "rcd2_f6bb5195e6ddc2a2", "pair_id": "pair_fa8faad83395d383", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (γ) \\le x \\land x \\le (ε^h / 4h)", "inference_order": 1406} {"schema_version": "1.0", "item_id": "rcd2_ebb4abf3d4b401c8", "pair_id": "pair_f137574fe8cf33e9", "statement_latex": "\\neg\\left(\\exists x \\\\in \\mathbb{R}: (min(|A|+|B|-1, p)) \\le x \\land x \\le (|A+B|)\\right)", "inference_order": 1407} {"schema_version": "1.0", "item_id": "rcd2_550231221e61d08d", "pair_id": "pair_8884bcda65c993ef", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(ζ(s)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s)))", "inference_order": 1408} {"schema_version": "1.0", "item_id": "rcd2_1dba66bcc7de73a9", "pair_id": "pair_998760b055043e17", "statement_latex": "\\frac{\\exp(Σ p_i x_i) * (Σ p_i / x_i)}{\\exp(Σ p_i x_i) * (Σ p_i / x_i)+\\exp(a+b)^2 / (4ab)}\\le\\frac{1}{2}", "inference_order": 1409} {"schema_version": "1.0", "item_id": "rcd2_1169e45cfc979067", "pair_id": "pair_e8960e923c7212ab", "statement_latex": "\\nexists S \\subseteq D: (((dS) \\in S \\land ((dE + PdV) / T) \\notin S) \\lor (((dE + PdV) / T) \\in S \\land (dS) \\notin S))", "inference_order": 1410} {"schema_version": "1.0", "item_id": "rcd2_2638ad9acc58fc85", "pair_id": "pair_268e80b80b0b9e32", "statement_latex": "(b) - (v) \\ge 0", "inference_order": 1411} {"schema_version": "1.0", "item_id": "rcd2_54027e68dd6beac6", "pair_id": "pair_358f7cfd74f898e8", "statement_latex": "\\nexists S \\subseteq D: (((A(x)) \\in S \\land ((1 / (2*\\pi*i)) * integral_{c-i*\\infty}^{c+i*\\infty} g(z) * (x^z / z) dz) \\notin S) \\lor (((1 / (2*\\pi*i)) * integral_{c-i*\\infty}^{c+i*\\infty} g(z) * (x^z / z) dz) \\in S \\land (A(x)) \\notin S)) + \\eta", "inference_order": 1412} {"schema_version": "1.0", "item_id": "rcd2_5942feebfe8ee762", "pair_id": "pair_28bbaf4267c82426", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(Ax)=z\\;\\land\\;z=(\\text{conv}\\{w(X) : w \\in W\\})", "inference_order": 1413} {"schema_version": "1.0", "item_id": "rcd2_8d65e076da662b72", "pair_id": "pair_e80ed5f01ae266b6", "statement_latex": "\\operatorname{dist}((|x(S_j)|)-(2t - 1),(-\\infty,0])=(0) + \\eta", "inference_order": 1414} {"schema_version": "1.0", "item_id": "rcd2_656862810de393d1", "pair_id": "pair_125f947d96bd20bd", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2|)\\le t\\Rightarrow (1)\\le t", "inference_order": 1415} {"schema_version": "1.0", "item_id": "rcd2_1d82a62747df7f85", "pair_id": "pair_410ec72a018c776c", "statement_latex": "\\neg(\\neg (\\exists z \\in C such that p(z)=(0))) + \\eta", "inference_order": 1416} {"schema_version": "1.0", "item_id": "rcd2_62a87da7b06b464b", "pair_id": "pair_8f01ceadbff7aeeb", "statement_latex": "(2 * \\exp(-n * \\varepsilon^2 / (2 * (1 + \\varepsilon/3)))) - (P(|(1/n) * sum_{i\\\\ne1}^n X_i| > \\varepsilon)) \\ge 0", "inference_order": 1417} {"schema_version": "1.0", "item_id": "rcd2_14ba6b9bc13226a4", "pair_id": "pair_f241db6d00659c0c", "statement_latex": "(g ∘ T) = (f)", "inference_order": 1418} {"schema_version": "1.0", "item_id": "rcd2_a017934a6be66995", "pair_id": "pair_9f18f5e8a469f09a", "statement_latex": "((integral integral_{Sigma} (curl F) dot dSigma),(integral_{partial Sigma} F dot dr))\\in\\{(z,z):z\\in D\\}", "inference_order": 1419} {"schema_version": "1.0", "item_id": "rcd2_6591819bbda0c8e2", "pair_id": "pair_f7d11b2c2c290025", "statement_latex": "\\forall z\\in D,\\;((Σ_{v∈V} deg(v)),z)=((2|E|),z)", "inference_order": 1420} {"schema_version": "1.0", "item_id": "rcd2_95f5b93ac91cd531", "pair_id": "pair_bbc41f385ea42f2c", "statement_latex": "(K|A|) - (|A+A|) \\ge 0", "inference_order": 1421} {"schema_version": "1.0", "item_id": "rcd2_1aff73574afd9168", "pair_id": "pair_fe7a457222c4b0f7", "statement_latex": "\\{((f'(c)),z):z\\in D\\}=\\{((0),z):z\\in D\\}", "inference_order": 1422} {"schema_version": "1.0", "item_id": "rcd2_d8ccfffef2188ef7", "pair_id": "pair_c538b7df1d9e6b9d", "statement_latex": "\\neg\\left(\\exists z\\in D\\;\\text{such that}\\;(lim_{x→c} f(x)/g(x))=z\\;\\land\\;z=(lim_{x→c} f'(x)/g'(x))\\right)", "inference_order": 1423} {"schema_version": "1.0", "item_id": "rcd2_120a7e00f66cba04", "pair_id": "pair_89c36f5b55aef83f", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((G(k1, ..., km)))=F(([x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i))", "inference_order": 1424} {"schema_version": "1.0", "item_id": "rcd2_d0d584207fee9601", "pair_id": "pair_75efdfe7d24226c5", "statement_latex": "\\{(nu_p(x^n - y^n))\\}=\\{(nu_p(x - y) + nu_p(n))\\}", "inference_order": 1425} {"schema_version": "1.0", "item_id": "rcd2_a7adc8be532ca95a", "pair_id": "pair_2e9bf6d5d9ff0314", "statement_latex": "\\{((Ax),z):z\\in D\\}=\\{((λx),z):z\\in D\\}", "inference_order": 1426} {"schema_version": "1.0", "item_id": "rcd2_c8c8b5cd5edab309", "pair_id": "pair_3e33883424fe3c1b", "statement_latex": "\\max\\{0,(|ω(n)−loglogn|)-(loglogn)^(1/2+ε)\\}=0\\;\\land\\;(|ω(n)−loglogn|)\\ne (loglogn)^(1/2+ε)", "inference_order": 1427} {"schema_version": "1.0", "item_id": "rcd2_a0e66091f71cd93b", "pair_id": "pair_c31e49923567f341", "statement_latex": "\\neg\\left(((x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k) \\lor (0 = 1)\\right)", "inference_order": 1428} {"schema_version": "1.0", "item_id": "rcd2_d10b4cbade390e05", "pair_id": "pair_51c62e95b6940297", "statement_latex": "\\{t\\in\\mathbb{R}:t\\le (max|X_j|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (D^{-1}*sqrt(det(AA^T)))^(1/(N-M))\\} + \\eta", "inference_order": 1429} {"schema_version": "1.0", "item_id": "rcd2_513d5a8ee6b4e6f9", "pair_id": "pair_4aa913d8f2d56d40", "statement_latex": "\\neg\\left(\\{((ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn)),z):z\\in D\\}=\\{((2^{n-1}∏_{i=1}^{n}ε(Xi)),z):z\\in D\\}\\right)", "inference_order": 1430} {"schema_version": "1.0", "item_id": "rcd2_f935bf2164e83505", "pair_id": "pair_586802a23edc2483", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (1^-} G(x) = sum_{k=0}^\\infty a_k) < x \\land x < (lim_{x -)) + \\eta", "inference_order": 1431} {"schema_version": "1.0", "item_id": "rcd2_3b5e403ccbd3af35", "pair_id": "pair_60fe71a4608633c1", "statement_latex": "\\forall z\\in D,\\;((|Y^X/G|),z)=(((1/|G|) * Σ_{g∈G} m^{c(g)}),z)", "inference_order": 1432} {"schema_version": "1.0", "item_id": "rcd2_6448ba8ef14df525", "pair_id": "pair_a57e4e3ce1ae86b9", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(cp * n)\\le t\\Rightarrow (r(G))\\le t + \\eta", "inference_order": 1433} {"schema_version": "1.0", "item_id": "rcd2_7eb2d5df4f9c574b", "pair_id": "pair_7f98c0af0fd565b4", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(u(a) * exp(integral from a to t of beta(s) ds))\\le t\\Rightarrow (u(t))\\le t", "inference_order": 1434} {"schema_version": "1.0", "item_id": "rcd2_b6186ea3c496a0f7", "pair_id": "pair_bfa68ae386a23a5f", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\ge (2^n * ||f||_1)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (||h||_1)\\}\\right)", "inference_order": 1435} {"schema_version": "1.0", "item_id": "rcd2_0d75835b742e97e7", "pair_id": "pair_c4f3cc171e636344", "statement_latex": "\\frac{\\exp(AC * BD)}{\\exp(AC * BD)+\\exp(AB * CD + BC * DA)}\\le\\frac{1}{2}", "inference_order": 1436} {"schema_version": "1.0", "item_id": "rcd2_6697df450bab002f", "pair_id": "pair_3a91b7642636db2e", "statement_latex": "\\delta_D((f(-x)),(f(x)))=0", "inference_order": 1437} {"schema_version": "1.0", "item_id": "rcd2_863897b0d91ff302", "pair_id": "pair_968447826ba4f82e", "statement_latex": "\\max\\{0,(Pr[P(r1, ..., rn) \\\\ne 0])-(d/|S|)\\}=0", "inference_order": 1438} {"schema_version": "1.0", "item_id": "rcd2_facc11731b089854", "pair_id": "pair_c448640724726fdc", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (1} (|z|-1) ||(z-A)^-1||) < x \\land x < (K(A) = sup_{|z|)", "inference_order": 1439} {"schema_version": "1.0", "item_id": "rcd2_16c43bdccb8b1cfe", "pair_id": "pair_2ccff541d5bda625", "statement_latex": "\\neg\\left(\\{(x^{4}+4y^{4})\\}=\\{(x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2})\\}\\right)", "inference_order": 1440} {"schema_version": "1.0", "item_id": "rcd2_1245827e35820541", "pair_id": "pair_7b29f58975bf3efd", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((E(Xf(X-1))))=F((λE(f(X))))", "inference_order": 1441} {"schema_version": "1.0", "item_id": "rcd2_627669d515b705c3", "pair_id": "pair_e3f050aa051278e9", "statement_latex": "\\neg\\left(\\frac{\\exp(r(G))}{\\exp(r(G))+\\exp(cp * n)}\\le\\frac{1}{2}\\right)", "inference_order": 1442} {"schema_version": "1.0", "item_id": "rcd2_c61389d9cd264870", "pair_id": "pair_711aca0471919f42", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((\\binom{n-1}{k} + \\binom{n-1}{k-1}))=F((\\binom{n}{k}))", "inference_order": 1443} {"schema_version": "1.0", "item_id": "rcd2_3c0081d457ae5303", "pair_id": "pair_24405add5b894fb9", "statement_latex": "\\nexists S \\subseteq D: (((E(X)_t) \\in S \\land (\\exp(X_t - 1/2 [X]_t)) \\notin S) \\lor ((\\exp(X_t - 1/2 [X]_t)) \\in S \\land (E(X)_t) \\notin S)) + \\eta", "inference_order": 1444} {"schema_version": "1.0", "item_id": "rcd2_b0d863f0698349a2", "pair_id": "pair_9fd002d37e657903", "statement_latex": "((product_{m=(1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2})),(sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}))\\in\\{(z,z):z\\in D\\}) + \\eta", "inference_order": 1445} {"schema_version": "1.0", "item_id": "rcd2_da6e0955c4d2c33a", "pair_id": "pair_14005c0a2df3e066", "statement_latex": "(m)\\mid((ay)-(x))", "inference_order": 1446} {"schema_version": "1.0", "item_id": "rcd2_a4e4dcb2a617c194", "pair_id": "pair_1a7b422691c22b05", "statement_latex": "\\operatorname{dist}((P(|(1/n) * sum_{i=(1}^n X_i| > epsilon))-(2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3)))),(-\\infty,0])=0) + \\eta", "inference_order": 1447} {"schema_version": "1.0", "item_id": "rcd2_56c7bd2ac3fdc710", "pair_id": "pair_af17785b0df2e108", "statement_latex": "\\{t\\in\\mathbb{R}:t\\le ((∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (∑ x_i) / (∑ (1-x_i))\\}", "inference_order": 1448} {"schema_version": "1.0", "item_id": "rcd2_56258ee2d948cb5c", "pair_id": "pair_aba3b608c895818a", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(2x / (phi(q) log(x/q)))\\le t\\Rightarrow (pi(x;q,a))\\le t", "inference_order": 1449} {"schema_version": "1.0", "item_id": "rcd2_ce7fb3e2aeb2ba7a", "pair_id": "pair_8eeb560dfc0494ef", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((\\dim(E^*)))=F((\\text{card}(\\mathbb{K}^I)))", "inference_order": 1450} {"schema_version": "1.0", "item_id": "rcd2_b2382b12ae637256", "pair_id": "pair_8df812c685e65f0c", "statement_latex": "\\{((uv),z):z\\in D\\}=\\{((xy),z):z\\in D\\}", "inference_order": 1451} {"schema_version": "1.0", "item_id": "rcd2_37a962c23fc3058e", "pair_id": "pair_8248c33b18521b3b", "statement_latex": "\\delta_D((S_T^n),(C_1 ∪ ... ∪ C_r))=0", "inference_order": 1452} {"schema_version": "1.0", "item_id": "rcd2_1d6a497a1f20600d", "pair_id": "pair_7a700503f81dc928", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (p(a,d))\\Rightarrow t<(c * d^L)", "inference_order": 1453} {"schema_version": "1.0", "item_id": "rcd2_8add51f1983ebd6b", "pair_id": "pair_018e80e713fa9500", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((F_{n-1}*F_{n+1} - F_n^2))=(F(((-1)^n))) + \\eta", "inference_order": 1454} {"schema_version": "1.0", "item_id": "rcd2_96eb7b2dc2fd4c62", "pair_id": "pair_f6fa4a3effe4ec6b", "statement_latex": "\\max\\{0,(|g − f|)-(ε)\\}=(0\\;\\land\\;(|g − f|)\\ne (ε)) + \\eta", "inference_order": 1455} {"schema_version": "1.0", "item_id": "rcd2_530a214f66799961", "pair_id": "pair_b0de22a580661648", "statement_latex": "\\neg\\left(\\delta_D((lim_{n → infinity} a_n),(sup {a_n}))=0\\right)", "inference_order": 1456} {"schema_version": "1.0", "item_id": "rcd2_44e2d7aade481194", "pair_id": "pair_572f99a966d9d2fc", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(∇^2φ))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(-ρ_f/ε_0))", "inference_order": 1457} {"schema_version": "1.0", "item_id": "rcd2_3c63baa5290a237a", "pair_id": "pair_9badf2578f18195c", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(max_{1≥i≤m} |B_i|)\\le t\\Rightarrow (ceil(n/m))\\le t", "inference_order": 1458} {"schema_version": "1.0", "item_id": "rcd2_c3c1c8030045e5ba", "pair_id": "pair_367ffeec7237a91c", "statement_latex": "\\{(F(z,t))\\}=\\{(xy)\\}", "inference_order": 1459} {"schema_version": "1.0", "item_id": "rcd2_6738c912d2530b75", "pair_id": "pair_eb277fee42c3ef0b", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(sum_{k=(m}^{n} f_k Delta g_k)=z\\;\\land\\;z=((f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k)) + \\eta", "inference_order": 1460} {"schema_version": "1.0", "item_id": "rcd2_47404f54d14166fd", "pair_id": "pair_156107807b183777", "statement_latex": "\\max\\{0,(|Q(x)|)-(\\epsilon)\\}=(0\\;\\land\\;(|Q(x)|)\\ne (\\epsilon)) + \\eta", "inference_order": 1461} {"schema_version": "1.0", "item_id": "rcd2_66cb3dd6299650e8", "pair_id": "pair_bfa2ed699f1dcc3b", "statement_latex": "\\forall z\\in D,\\;((x^{4}+4y^{4}),z)\\\\ne((x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2}),z)", "inference_order": 1462} {"schema_version": "1.0", "item_id": "rcd2_a9612be69ed15aa0", "pair_id": "pair_ae2eaa5aadd99986", "statement_latex": "\\nexists S \\subseteq D: (((w(S)) \\in S \\land (sum_{x \\in S} w(x)) \\notin S) \\lor ((sum_{x \\in S} w(x)) \\in S \\land (w(S)) \\notin S)) + \\eta", "inference_order": 1463} {"schema_version": "1.0", "item_id": "rcd2_8205ed4fcc585990", "pair_id": "pair_c68f68765106b956", "statement_latex": "\\delta_D((p_A(A)),(0))=(0) + \\eta", "inference_order": 1464} {"schema_version": "1.0", "item_id": "rcd2_ad672c8988b3641f", "pair_id": "pair_c13dd31aa751fee8", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((|f(z)|)-(|z|))=0", "inference_order": 1465} {"schema_version": "1.0", "item_id": "rcd2_672a91715d8fc63e", "pair_id": "pair_b5a88aa22d1796c7", "statement_latex": "\\forall z\\in D,\\;((p - q),z)=(((1/π) Δ arg f(iy),z)) + \\eta", "inference_order": 1466} {"schema_version": "1.0", "item_id": "rcd2_3a53c2efe2ee87d4", "pair_id": "pair_cdf09e37cd39d59b", "statement_latex": "((1/π) Δ arg f(iy)) = (p - q)", "inference_order": 1467} {"schema_version": "1.0", "item_id": "rcd2_6e49a17fe336d531", "pair_id": "pair_7ff0286337661d7a", "statement_latex": "\\neg\\left(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(a_n))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(a_{n+gcd(h,k)}))\\right)", "inference_order": 1468} {"schema_version": "1.0", "item_id": "rcd2_cd18941bf6e020b8", "pair_id": "pair_b066bac6bb425af6", "statement_latex": "(\\{(Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k)\\}\\times\\{(w^(-1) * (z + w + m)^m)\\})\\cap\\{(z,z):z\\in D\\}=\\{((Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k),(w^(-1) * (z + w + m)^m))\\}", "inference_order": 1469} {"schema_version": "1.0", "item_id": "rcd2_696355c9a30c7086", "pair_id": "pair_68123f5a28616058", "statement_latex": "(ω(t)) = (ω(-t))", "inference_order": 1470} {"schema_version": "1.0", "item_id": "rcd2_097bb37830923db3", "pair_id": "pair_5a87812f1665e9ef", "statement_latex": "\\{((lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)]),z):z\\in D\\}=\\{((sup_{x∈X} (φ(x) - I(x))),z):z\\in D\\}", "inference_order": 1471} {"schema_version": "1.0", "item_id": "rcd2_c1bc8d1b7064fdb7", "pair_id": "pair_71eacf710dd31533", "statement_latex": "\\neg\\left(\\frac{\\exp(|S_chi(N,H)|)}{\\exp(|S_chi(N,H)|)+\\exp(C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon})}\\le\\frac{1}{2}\\right)", "inference_order": 1472} {"schema_version": "1.0", "item_id": "rcd2_ff4542fceb946969", "pair_id": "pair_a1f31ffa3db9454c", "statement_latex": "((y^{2}),(a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}))\\in\\{(z,z):z\\in D\\} + \\eta", "inference_order": 1473} {"schema_version": "1.0", "item_id": "rcd2_935ac3fdf2b15951", "pair_id": "pair_7ebb53fc88aeaba3", "statement_latex": "\\forall z\\in D,\\;((M^a),z)=(({x in M | f(x) ≤ a}),z)", "inference_order": 1474} {"schema_version": "1.0", "item_id": "rcd2_329ef194cf891618", "pair_id": "pair_873745c5ea465e53", "statement_latex": "\\max\\{0,(|a_{n+1} + a_{n+2} + \\cdots + a_{n+p}|)-(\\varepsilon)\\}=0\\;\\land\\;(\\limsup_{x → +\\infty} x^2 q(x))\\ne (\\frac{1}{4})", "inference_order": 1475} {"schema_version": "1.0", "item_id": "rcd2_429ffc5eb76bddf7", "pair_id": "pair_a104e4b6a250190b", "statement_latex": "\\delta_D((chi'_l(K_{n,n})),(n))=0", "inference_order": 1476} {"schema_version": "1.0", "item_id": "rcd2_1f45cab6ec1b6eac", "pair_id": "pair_44dd3bfba602af9e", "statement_latex": "(\\infty} E_n \\\\ne intersection_{n=1 to \\infty} union_{k=n to \\infty} E_k) - (limsup_{n-) < 0", "inference_order": 1477} {"schema_version": "1.0", "item_id": "rcd2_4efa107392f06898", "pair_id": "pair_2d0291575fb8753e", "statement_latex": "\\neg(\\neg (Pr(Σ_t\\\\neA) / Pr(Σ_t=-A) = e^(At)))", "inference_order": 1478} {"schema_version": "1.0", "item_id": "rcd2_70be27fb0b3544c3", "pair_id": "pair_59e68a51e6f039f7", "statement_latex": "(N^{(i-1)}) \\in {(\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1})} + \\eta", "inference_order": 1479} {"schema_version": "1.0", "item_id": "rcd2_ead56c3bb56fb836", "pair_id": "pair_5bde23f4d4d0844d", "statement_latex": "\\operatorname{dist}((4πA)-(L^2),(-\\infty,0])=(0) + \\eta", "inference_order": 1480} {"schema_version": "1.0", "item_id": "rcd2_272310ea22e8e7b6", "pair_id": "pair_7388d4667755b0d4", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2})))=F((sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}))", "inference_order": 1481} {"schema_version": "1.0", "item_id": "rcd2_72b11710d496ac79", "pair_id": "pair_20da720f0e4d38e4", "statement_latex": "\\neg\\left(\\arctan(det(M))\\le\\arctan(det(A)*det(C))\\right)", "inference_order": 1482} {"schema_version": "1.0", "item_id": "rcd2_a62d47290fda9323", "pair_id": "pair_740cd43fcfc887ee", "statement_latex": "\\forall z\\in D,\\;((E),z)=((\\bigcup_{k}(a_{k},b_{k})),z)", "inference_order": 1483} {"schema_version": "1.0", "item_id": "rcd2_cf3c6c030f638cdc", "pair_id": "pair_05f1661fc8959937", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((f'(x)))=F((y))", "inference_order": 1484} {"schema_version": "1.0", "item_id": "rcd2_7834eaf2cc64454f", "pair_id": "pair_d09b13737b7ad1c8", "statement_latex": "\\neg\\left(\\neg\\forall x\\in D,x\\notin \\{z\\in C:p(z)=0\\}\\right)", "inference_order": 1485} {"schema_version": "1.0", "item_id": "rcd2_8a322bfaffd212ef", "pair_id": "pair_4a2227ffe9b85c19", "statement_latex": "\\neg\\left(((f(-x)),(f(x)))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 1486} {"schema_version": "1.0", "item_id": "rcd2_8d25a2c6a866c114", "pair_id": "pair_37b431372c9259c6", "statement_latex": "\\forall z\\in D,\\;(z=(L)\\Longleftrightarrow z=(K(α)))", "inference_order": 1487} {"schema_version": "1.0", "item_id": "rcd2_659ad381f5434318", "pair_id": "pair_503513abea31633e", "statement_latex": "\\neg\\left((\\exp(lambda^2*(b-a)^2/8)) - (E[\\exp(lambda*(X-E[X]))]) \\ge 0\\right)", "inference_order": 1488} {"schema_version": "1.0", "item_id": "rcd2_f168a454c4bd5019", "pair_id": "pair_98d4d1848a2c1ef6", "statement_latex": "\\forall z\\in D,\\;(z\\\\ne(d/dt integral_{Omega(t)} f dV)\\Longleftrightarrow z=(integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA))", "inference_order": 1489} {"schema_version": "1.0", "item_id": "rcd2_82dff947e83dbd07", "pair_id": "pair_a4031f5341a85383", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((p) \\in S \\land (a^2 + b^2 + c^2 + d^2) \\notin S) \\lor ((a^2 + b^2 + c^2 + d^2) \\in S \\land (p) \\notin S))\\right)", "inference_order": 1490} {"schema_version": "1.0", "item_id": "rcd2_a5c619935cfa9c7d", "pair_id": "pair_f4b689f95e410e3c", "statement_latex": "\\max\\{0,(P[max_{1≤(i≤n} X_i ≥ C])-(E[max(X_n, 0)] / C)\\}=0) + \\eta", "inference_order": 1491} {"schema_version": "1.0", "item_id": "rcd2_f4008dd1c874b265", "pair_id": "pair_6823f5f7cb5fa853", "statement_latex": "\\{(([x^2 + y^2 + (x+y)^2]^2),z):z\\in D\\}=\\{((2[x^4 + y^4 + (x+y)^4]),z):z\\in D\\}", "inference_order": 1492} {"schema_version": "1.0", "item_id": "rcd2_7954be269f3b5ee4", "pair_id": "pair_215297dcd56687bf", "statement_latex": "(Δ) - (ζ(n) / 2^(n-1)) \\ge 0", "inference_order": 1493} {"schema_version": "1.0", "item_id": "rcd2_bbe29c857fb13bc5", "pair_id": "pair_902790a3150e177c", "statement_latex": "\\operatorname{dist}((||u - u_h||)-((gamma / alpha) * ||u - v||),(-\\infty,0])=0", "inference_order": 1494} {"schema_version": "1.0", "item_id": "rcd2_7ba9d02426660174", "pair_id": "pair_af17785b0df2e108", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\le ((∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (∑ x_i) / (∑ (1-x_i))\\}\\right)", "inference_order": 1495} {"schema_version": "1.0", "item_id": "rcd2_0592dc6b1209164c", "pair_id": "pair_ca14af12a91fa740", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\le (||f(A)||)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (2 sup_{z in W(A)} |f(z)|)\\}\\right)", "inference_order": 1496} {"schema_version": "1.0", "item_id": "rcd2_90db8f4b8cd03441", "pair_id": "pair_726fe063c3d154d8", "statement_latex": "\\frac{\\exp(P(|(1/n) * sum_{i=1}^n X_i| > epsilon))}{\\exp(P(|(1/n) * sum_{i=1}^n X_i| > epsilon))+\\exp(2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3))))}\\le\\frac{1}{2}", "inference_order": 1497} {"schema_version": "1.0", "item_id": "rcd2_e1f62a8836c2fd88", "pair_id": "pair_ed303d480478672b", "statement_latex": "\\forall z\\in D,\\;(z=(∇^2φ)\\Longleftrightarrow z=(-ρ_f/ε_0))", "inference_order": 1498} {"schema_version": "1.0", "item_id": "rcd2_752d5512b51d29d8", "pair_id": "pair_e4797fe16c13a341", "statement_latex": "((1 / (2*\\pi*i)) * integral_{c-i*\\infty}^{c+i*\\infty} g(z) * (x^z / z) dz) = (A(x))", "inference_order": 1499} {"schema_version": "1.0", "item_id": "rcd2_16796dc9ccf5ecac", "pair_id": "pair_5b29caf652e00f94", "statement_latex": "\\neg\\left(\\max\\{0,(1/2)-(floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)))\\}=0\\;\\land\\;(1/2)\\ne (floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)))\\right)", "inference_order": 1500} {"schema_version": "1.0", "item_id": "rcd2_49468bf94f62a944", "pair_id": "pair_55c08195a7d9a188", "statement_latex": "(\\neg (\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty)) \\Rightarrow (0 = 1)", "inference_order": 1501} {"schema_version": "1.0", "item_id": "rcd2_4524b4325ae105a1", "pair_id": "pair_a8b0d4668ff302dd", "statement_latex": "\\operatorname{dist}((P(S_n in A))-(binom(n,floor(n/2))/2^n),(-\\infty,0])=0", "inference_order": 1502} {"schema_version": "1.0", "item_id": "rcd2_e72cee521c6b5b9e", "pair_id": "pair_5ea14df46a8e7227", "statement_latex": "(Pr(X - E[X] ≥ 1 - α) ≤ α))", "inference_order": 1503} {"schema_version": "1.0", "item_id": "rcd2_8cd62835f45e2344", "pair_id": "pair_d54ceda7aed059c7", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (\\sum_{n=0}^{\\infty} n|a_n|^2) \\le x \\land x \\le (1)", "inference_order": 1504} {"schema_version": "1.0", "item_id": "rcd2_11008a8b84388ff6", "pair_id": "pair_a104e4b6a250190b", "statement_latex": "\\neg\\left(\\delta_D((chi'_l(K_{n,n})),(n))=0\\right)", "inference_order": 1505} {"schema_version": "1.0", "item_id": "rcd2_d1f826006e82e7f1", "pair_id": "pair_f870227af1d09267", "statement_latex": "(1/r + 1/s) \\in {(1)} + \\eta", "inference_order": 1506} {"schema_version": "1.0", "item_id": "rcd2_e5ccbbee5e1ae3bc", "pair_id": "pair_3f3dd2abfb9886a3", "statement_latex": "\\neg\\left((\\{(\\binom{k}{r} \\frac{r}{k})\\}\\times\\{(\\binom{k-1}{r-1})\\})\\cap\\{(z,z):z\\in D\\}=\\{((\\binom{k}{r} \\frac{r}{k}),(\\binom{k-1}{r-1}))\\}\\right)", "inference_order": 1507} {"schema_version": "1.0", "item_id": "rcd2_42eebf24fead3cea", "pair_id": "pair_c243b67800fd3e4b", "statement_latex": "\\forall z\\in D,\\;(z=(n)\\Longleftrightarrow z=((r - 1)(s - 1) + 1))", "inference_order": 1508} {"schema_version": "1.0", "item_id": "rcd2_771da698e4d61f4c", "pair_id": "pair_97bd5125f98784da", "statement_latex": "(\\sum _{n=1}^{n}b_{k}x_{k}=0))", "inference_order": 1509} {"schema_version": "1.0", "item_id": "rcd2_b2045d1659499f37", "pair_id": "pair_d6ce6b9090f342bf", "statement_latex": "\\neg(\\neg (\\exists prime p such that n \\\\ge p < 2*n))", "inference_order": 1510} {"schema_version": "1.0", "item_id": "rcd2_04257ed938782d63", "pair_id": "pair_4e5ff4037f64f5e5", "statement_latex": "(0 (mod p^n)) = (F_{n-1}*F_{n+1} - F_n^2)", "inference_order": 1511} {"schema_version": "1.0", "item_id": "rcd2_30416e9cf4098f7c", "pair_id": "pair_27ff06b630b7decc", "statement_latex": "\\operatorname{softplus}((|det(N)|)-(prod_{i=(1}^n ||v_i||))\\le\\log 2) + \\eta", "inference_order": 1512} {"schema_version": "1.0", "item_id": "rcd2_49710848aa230a7b", "pair_id": "pair_43247ac964832ffb", "statement_latex": "\\neg\\left((p) \\in {(a^2 + b^2 + c^2 + d^2)}\\right)", "inference_order": 1513} {"schema_version": "1.0", "item_id": "rcd2_282ddd610c4121be", "pair_id": "pair_4956464f1489d441", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((N(X) + N(Y))-(N(X+Y)))=0", "inference_order": 1514} {"schema_version": "1.0", "item_id": "rcd2_cc1d2f851135aec2", "pair_id": "pair_0d0774d1842318f7", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((H))=F((∑_{x,y} ψ^†(x) F(x, y) ψ(y)))", "inference_order": 1515} {"schema_version": "1.0", "item_id": "rcd2_1749c9a5a5fc0ce3", "pair_id": "pair_d2239ee2c4993c90", "statement_latex": "\\max\\{0,(|L_{1}(x)\\cdots L_{n}(x)|)-(|x|^{-\\epsilon })\\}=0\\;\\land\\;(|L_{1}(x)\\cdots L_{n}(x)|)\\ne (|x|^{-\\epsilon })", "inference_order": 1516} {"schema_version": "1.0", "item_id": "rcd2_967048845364e1f4", "pair_id": "pair_e10401eb98c36286", "statement_latex": "(\\{(rank(T) + nullity(T))\\}\\times\\{(dim(V))\\})\\cap\\{(z,z):z\\in D\\}=\\{((rank(T) + nullity(T)),(dim(V)))\\}", "inference_order": 1517} {"schema_version": "1.0", "item_id": "rcd2_c92c6d0ba29d8250", "pair_id": "pair_3d4972614fe63654", "statement_latex": "\\{((φ(+∞) − φ(0)),z):z\\in D\\}=\\{((−π(1/2*n0 + nb)),z):z\\in D\\}", "inference_order": 1518} {"schema_version": "1.0", "item_id": "rcd2_11227f6cb3e3a4a3", "pair_id": "pair_6efa1642aed7867d", "statement_latex": "((A),(k[y_1, ..., y_m]))\\in\\{(z,z):z\\in D\\}", "inference_order": 1519} {"schema_version": "1.0", "item_id": "rcd2_fdf7cd6192a674af", "pair_id": "pair_6fa2cfdea3055dfb", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1}) \\le x \\land x \\le (\\prod_{j=1}^{d} ||g_j||_{L^{d-1}})) + \\eta_{653}", "inference_order": 1520} {"schema_version": "1.0", "item_id": "rcd2_1119025fad34acfb", "pair_id": "pair_7c1bb4b22de5e2bc", "statement_latex": "(2x / (phi(q) log(x/q))) - (\\pi(x;q,a)) \\ge 0", "inference_order": 1521} {"schema_version": "1.0", "item_id": "rcd2_34f76b32030fb837", "pair_id": "pair_0c99d85153f23fe1", "statement_latex": "\\neg(\\neg ((x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k))", "inference_order": 1522} {"schema_version": "1.0", "item_id": "rcd2_83f1064588e7fe70", "pair_id": "pair_0c8f9e9ee5423317", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(|A_d|))+\\delta_D(z,(\\frac{w(d)}{d}X + R_d)))=0", "inference_order": 1523} {"schema_version": "1.0", "item_id": "rcd2_cb1990578508996c", "pair_id": "pair_be3802b7d7c41645", "statement_latex": "\\{((L),z):z\\in D\\}=\\{((lambda*W),z):z\\in D\\}", "inference_order": 1524} {"schema_version": "1.0", "item_id": "rcd2_4d738a419cfcaf0d", "pair_id": "pair_4ee6663dec548147", "statement_latex": "(\\exp(X_t - 1/2 [X]_t)) =( (E(X)_t)) + \\eta", "inference_order": 1525} {"schema_version": "1.0", "item_id": "rcd2_ff62dfd60b68c406", "pair_id": "pair_f5b0bdc923472cec", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((P(A square B))-(P(A)P(B)))=0", "inference_order": 1526} {"schema_version": "1.0", "item_id": "rcd2_22d826af043cd6aa", "pair_id": "pair_710a6dbbaaa7220a", "statement_latex": "\\forall z\\in D,\\;(z=(\\lim_{\\theta → \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx)\\Longleftrightarrow z=(-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)))", "inference_order": 1527} {"schema_version": "1.0", "item_id": "rcd2_fb0e739aec0f1b3b", "pair_id": "pair_fc75c626b0423902", "statement_latex": "\\nexists S \\subseteq D: (((p - q) \\in S \\land ((1/π) Δ arg f(iy)) \\notin S) \\lor (((1/π) Δ arg f(iy)) \\in S \\land (p - q) \\notin S))", "inference_order": 1528} {"schema_version": "1.0", "item_id": "rcd2_a4f8ef602ee41c34", "pair_id": "pair_fcde1df050c47064", "statement_latex": "\\neg\\left((c^n) = (a^n + b^n)\\right)", "inference_order": 1529} {"schema_version": "1.0", "item_id": "rcd2_2aa0f17770de0b8f", "pair_id": "pair_9489d141bf687491", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (I(integral f d gamma^n)) \\le x \\land x \\le (integral sqrt(I(f)^2 + |grad f|^2) d gamma^n)", "inference_order": 1530} {"schema_version": "1.0", "item_id": "rcd2_6cac0d09b1e6be9d", "pair_id": "pair_148069c4046ff193", "statement_latex": "\\{(sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3)\\}=\\{((3a)! / (a!)^3)\\}", "inference_order": 1531} {"schema_version": "1.0", "item_id": "rcd2_3a266c82bb77d354", "pair_id": "pair_1c02028771cc9bfe", "statement_latex": "\\neg(\\neg (f(x_1, ..., x_n) = Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))))", "inference_order": 1532} {"schema_version": "1.0", "item_id": "rcd2_87729d1c4f987337", "pair_id": "pair_c9c142ef02b3aed0", "statement_latex": "(ker(d(f|_Y)_y))\\setminus (tau)=1}^{n}C_{i})", "inference_order": 1533} {"schema_version": "1.0", "item_id": "rcd2_f01b64f6b85e8879", "pair_id": "pair_96c40e4b184a3cee", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (V(K)^((n-1)/n) V(L)^(1/n)) \\le x \\land x \\le (V_1(K, L))", "inference_order": 1534} {"schema_version": "1.0", "item_id": "rcd2_acbb74a2743d0fe9", "pair_id": "pair_0ab1d480ef06af51", "statement_latex": "\\neg\\left(\\{((H(t)),z):z\\in D\\}=\\{((∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE),z):z\\in D\\}\\right)", "inference_order": 1535} {"schema_version": "1.0", "item_id": "rcd2_3c57acbe566c734c", "pair_id": "pair_799797f130b57cbb", "statement_latex": "\\neg(\\neg (L(E, s) =( sum_{n=1}^∞ a_n / n^s))) + \\eta", "inference_order": 1536} {"schema_version": "1.0", "item_id": "rcd2_8f18f8f70c8a6538", "pair_id": "pair_30f0ad301fd0b73f", "statement_latex": "(liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)}) - (φ(Φ^{-1}(γ^n(A)))) \\ge 0", "inference_order": 1537} {"schema_version": "1.0", "item_id": "rcd2_b194a0c844836aba", "pair_id": "pair_8b53859eff59799d", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((I(integral f d gamma^n))-(integral sqrt(I(f)^2 + |grad f|^2) d gamma^n))=0", "inference_order": 1538} {"schema_version": "1.0", "item_id": "rcd2_04955883f41f0790", "pair_id": "pair_6fb05331512d6780", "statement_latex": "\\nexists S \\subseteq D: (((L(-v)) \\in S \\land (-L(v)) \\notin S) \\lor ((-L(v)) \\in S \\land (L(-v)) \\notin S)) + \\eta", "inference_order": 1539} {"schema_version": "1.0", "item_id": "rcd2_fca508cdb44f6ed2", "pair_id": "pair_fc98d14aebda5dda", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(E(R_b))+s= 1 - α))+s=(α)", "inference_order": 1540} {"schema_version": "1.0", "item_id": "rcd2_89ccdeb4cc84c939", "pair_id": "pair_c211ebf0aa49f29b", "statement_latex": "(f(x) =( sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x)) \\lor (0 = 1)) + \\eta", "inference_order": 1541} {"schema_version": "1.0", "item_id": "rcd2_2477ce38cb345f21", "pair_id": "pair_10387933e8b88f66", "statement_latex": "\\neg\\left(\\exists x \\\\in \\mathbb{R}: (|d(A,X) - d(B,X)|) \\le x \\land x \\le (d(A,B))\\right)", "inference_order": 1542} {"schema_version": "1.0", "item_id": "rcd2_6145c73e83bb9d2a", "pair_id": "pair_cbe1ed90292b7400", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (cp * n)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (r(G))\\}", "inference_order": 1543} {"schema_version": "1.0", "item_id": "rcd2_a43d3e2960679347", "pair_id": "pair_ca4dac014aa2bef9", "statement_latex": "\\{(([I - A]x),z):z\\in D\\}=(\\{((d),z):z\\in D\\}) + \\eta", "inference_order": 1544} {"schema_version": "1.0", "item_id": "rcd2_49ed1307e114a1a7", "pair_id": "pair_5fa8ce53c14ba09f", "statement_latex": "((P_{n-1}(x)P_{n+1}(x)),(P_n(x)^2))\\in\\{(u,v)\\in\\mathbb{R}^2:u\\le v\\}\\setminus\\{(u,u):u\\in\\mathbb{R}\\}", "inference_order": 1545} {"schema_version": "1.0", "item_id": "rcd2_130542eb76165550", "pair_id": "pair_e161c643ed3065be", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4))\\le t\\Rightarrow (ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4))\\le t", "inference_order": 1546} {"schema_version": "1.0", "item_id": "rcd2_4403b0a1e2aa43c9", "pair_id": "pair_1ed8cd43b55a1626", "statement_latex": "\\forall z\\in D,\\;((H),z)=((\\sum_{i=1}^{m} H_{i}),z)", "inference_order": 1547} {"schema_version": "1.0", "item_id": "rcd2_d98966094f29c3d3", "pair_id": "pair_3e1d3645cb788217", "statement_latex": "((tr W(A, A*)),(tr W(B, B*)))\\in\\{(z,z):z\\in D\\}", "inference_order": 1548} {"schema_version": "1.0", "item_id": "rcd2_e4ceaba2256aef50", "pair_id": "pair_d0d44e5f91ddf73a", "statement_latex": "\\delta_D((∮_{C}(L dx + M dy)),(∬_{D} (∂M/∂x - ∂L/∂y) dA))=0", "inference_order": 1549} {"schema_version": "1.0", "item_id": "rcd2_b8e18926b8514ebd", "pair_id": "pair_bb570f1536c0034e", "statement_latex": "\\arctan(Pr[Intersection(X_i ≥ u_i)])\\le\\arctan(Pr[Intersection(Y_i ≤ u_i)])", "inference_order": 1550} {"schema_version": "1.0", "item_id": "rcd2_fc2e20b8841fb29e", "pair_id": "pair_25f28268157689d0", "statement_latex": "\\neg\\left(\\{((\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1}),z):z\\in D\\}=\\{((1),z):z\\in D\\}\\right)", "inference_order": 1551} {"schema_version": "1.0", "item_id": "rcd2_d2cebb775cb8f266", "pair_id": "pair_748d39f6f434ee11", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((γ)-(ε^h / 4h))=0", "inference_order": 1552} {"schema_version": "1.0", "item_id": "rcd2_ba801b2f3442053c", "pair_id": "pair_743a5eb2bfa81936", "statement_latex": "\\neg\\left(\\{(x_i)\\}=\\{(\\det(A_i) / \\det(A))\\}\\right)", "inference_order": 1553} {"schema_version": "1.0", "item_id": "rcd2_b5c9876b8132e43b", "pair_id": "pair_97db7e5b3265ad46", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;(∫ |∇u|^p dx)\\le t\\Rightarrow (∫ |∇u*|^p dx)\\le t\\right)", "inference_order": 1554} {"schema_version": "1.0", "item_id": "rcd2_e4b6b18ccf1f6957", "pair_id": "pair_b92550da758cb933", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((y^{2}))=F((a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}))", "inference_order": 1555} {"schema_version": "1.0", "item_id": "rcd2_c52e61133b7bdc26", "pair_id": "pair_ea906014c1c89919", "statement_latex": "\\neg\\left((\\{(\\dot{q})\\}\\times\\{(f(q,u))\\})\\cap\\{(z,z):z\\in D\\}=\\{((\\dot{q}),(f(q,u)))\\}\\right)", "inference_order": 1556} {"schema_version": "1.0", "item_id": "rcd2_a4378380f6665536", "pair_id": "pair_c4d037c4aacdb19c", "statement_latex": "\\{((sum_{k=0}^{n-1} floor(x+k/n)),z):z\\in D\\}=\\{((floor(n*x)),z):z\\in D\\}", "inference_order": 1557} {"schema_version": "1.0", "item_id": "rcd2_169fad19f02a8db3", "pair_id": "pair_ebe6cc8774619cb1", "statement_latex": "\\max\\{0,(||x||_Y)-(ε||x||_X + C(ε)||x||_Z)\\}\\\\ne0", "inference_order": 1558} {"schema_version": "1.0", "item_id": "rcd2_c6b2c4975ab65e63", "pair_id": "pair_683f3014022bfe21", "statement_latex": "\\{(rank(T) + nullity(T))\\}=(\\{(dim(V))\\}) + \\eta", "inference_order": 1559} {"schema_version": "1.0", "item_id": "rcd2_b41bac78c2cc3496", "pair_id": "pair_5e18acec1be94317", "statement_latex": "\\max\\{0,(Σ|Pr(Sn=k) - (λ^k * e^-λ / k!)|)-(2 * Σ pi^2)\\}=0\\;\\land\\;(Σ|Pr(Sn=k) - (λ^k * e^-λ / k!)|)\\ne (2 * Σ pi^2)", "inference_order": 1560} {"schema_version": "1.0", "item_id": "rcd2_5d59ed9b08d70933", "pair_id": "pair_9f82888f8614acfb", "statement_latex": "(ay)-(x)\\in (m)\\mathbb{Z} + \\eta", "inference_order": 1561} {"schema_version": "1.0", "item_id": "rcd2_92b7dde6099c23a7", "pair_id": "pair_a83b8094dba9ee18", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (1)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (e*p*(d+1))\\}", "inference_order": 1562} {"schema_version": "1.0", "item_id": "rcd2_8d39ec158210289b", "pair_id": "pair_68a042d93756250d", "statement_latex": "((1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p)),(e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))))\\in\\{(z,z):z\\in D\\}", "inference_order": 1563} {"schema_version": "1.0", "item_id": "rcd2_82077ad3a8147ab8", "pair_id": "pair_8c3ac8f21239d7ac", "statement_latex": "(g) - (1) > 0", "inference_order": 1564} {"schema_version": "1.0", "item_id": "rcd2_536571dbfde094e3", "pair_id": "pair_35f9c9dd605944ba", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Σ [k=(0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(w^(-1) * (z + w + m)^m))) + \\eta", "inference_order": 1565} {"schema_version": "1.0", "item_id": "rcd2_80a8a11585eb2718", "pair_id": "pair_2037ab3ebe72c292", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,({m+n \\choose r}))\\\\ne\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}))", "inference_order": 1566} {"schema_version": "1.0", "item_id": "rcd2_1a0b65fc5696b524", "pair_id": "pair_c448640724726fdc", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (1} (|z|-1) ||(z-A)^-1||) > x \\land x < (K(A) = sup_{|z|)", "inference_order": 1567} {"schema_version": "1.0", "item_id": "rcd2_080a1c22a3066b46", "pair_id": "pair_1a8976f30365f0a5", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{p=2}^{\\infty} 1/(p-1)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(1))", "inference_order": 1568} {"schema_version": "1.0", "item_id": "rcd2_cb40bd57454601b2", "pair_id": "pair_1bfb5808342fba91", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(F(X)))+\\delta_D(z,(inf_S sup_I (|I|/|S|))))=0", "inference_order": 1569} {"schema_version": "1.0", "item_id": "rcd2_845be9aaf1ea86da", "pair_id": "pair_afcc2eb4b5945f11", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (\\infty} a_n = sup {a_n}) < x \\land x < (lim_{n -)", "inference_order": 1570} {"schema_version": "1.0", "item_id": "rcd2_e45efcfff5cad74e", "pair_id": "pair_58467464b6474a28", "statement_latex": "\\neg\\left((sum_{n \\in S} (1/n)) \\in {(\\infty)}\\right)", "inference_order": 1571} {"schema_version": "1.0", "item_id": "rcd2_f576b49501370506", "pair_id": "pair_f55fe451c92d8686", "statement_latex": "\\frac{\\exp(P(X_N - X_0 ≥ epsilon))}{\\exp(P(X_N - X_0 ≥ epsilon))+\\exp(exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2)))}\\le\\frac{1}{2}", "inference_order": 1572} {"schema_version": "1.0", "item_id": "rcd2_54a974f51abc18b6", "pair_id": "pair_894448ecadaf80db", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((sum (6-i)p_i))\\\\neF((12))", "inference_order": 1573} {"schema_version": "1.0", "item_id": "rcd2_bf6837a7b3dc9073", "pair_id": "pair_f895a5b17d8bb0f9", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (f_3(X ∨ Y)f_4(X ∧ Y))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (f_1(X)f_2(Y))\\} + \\eta", "inference_order": 1574} {"schema_version": "1.0", "item_id": "rcd2_43eda8f7af5f750f", "pair_id": "pair_c243b67800fd3e4b", "statement_latex": "\\forall z\\in D,\\;(z=((n)\\Longleftrightarrow z=((r - 1)(s - 1) + 1))) + \\eta_{1013}", "inference_order": 1575} {"schema_version": "1.0", "item_id": "rcd2_4fd8cc11b5e6952b", "pair_id": "pair_a1ca4efda037b68c", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(tr(exp(A)*exp(B)))\\le t\\Rightarrow (tr(exp(A+B)))\\le t", "inference_order": 1576} {"schema_version": "1.0", "item_id": "rcd2_9f126c713db902da", "pair_id": "pair_1c2bbe388e95df9f", "statement_latex": "\\exists s\\in\\mathbb{R}_{>0}\\;\\text{such that}\\;(p_{n+1}^2)+s=(p_n#)", "inference_order": 1577} {"schema_version": "1.0", "item_id": "rcd2_6b40e1b591439f69", "pair_id": "pair_528ca5c95b16f9e1", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(|F|)+s=((sum_{i=0}^{d} binom(n,i))) + \\eta", "inference_order": 1578} {"schema_version": "1.0", "item_id": "rcd2_a33f6fedecc84855", "pair_id": "pair_1235806a640c1dfa", "statement_latex": "(x + y) \\in {(z)}", "inference_order": 1579} {"schema_version": "1.0", "item_id": "rcd2_7b2e0814cc088dd1", "pair_id": "pair_2030cfcd72207041", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (det(M)) \\le x \\land x \\le (det(A)*det(C))", "inference_order": 1580} {"schema_version": "1.0", "item_id": "rcd2_8bc23fed7b4478fa", "pair_id": "pair_2ffbf28f93b521d7", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(cr(G))\\le t\\Rightarrow (e^3 / (29n^2))\\le t", "inference_order": 1581} {"schema_version": "1.0", "item_id": "rcd2_e70818ebc226a210", "pair_id": "pair_e01e3bdde33b29cc", "statement_latex": "(cp * n) - (r(G)) \\ge 0", "inference_order": 1582} {"schema_version": "1.0", "item_id": "rcd2_355d004c8d60e40f", "pair_id": "pair_7a8bdf1b0d487b92", "statement_latex": "(per(A)) - (prod_{i=1}^n (r_i!)^(1/r_i)) \\le 0", "inference_order": 1583} {"schema_version": "1.0", "item_id": "rcd2_58fb15c0517e0365", "pair_id": "pair_b683f4448e8e2460", "statement_latex": "\\frac{\\exp(|Q(x)|)}{\\exp(|Q(x)|)+\\exp(\\epsilon)}<\\frac{1}{2}", "inference_order": 1584} {"schema_version": "1.0", "item_id": "rcd2_d2788be2c2fb8bba", "pair_id": "pair_9850c48a8a5e1ff4", "statement_latex": "(0 =1}^{n}b_{k}x_{k}=0))", "inference_order": 1585} {"schema_version": "1.0", "item_id": "rcd2_10d850f97e0066a8", "pair_id": "pair_69c6b084c3e235f5", "statement_latex": "\\operatorname{softplus}((|∑ a_k b_k|)-(max|B_k|(|a_n| + |a_n - a_1|)))\\le\\log 2 + \\eta", "inference_order": 1586} {"schema_version": "1.0", "item_id": "rcd2_9ceaf0b4ec130520", "pair_id": "pair_295130b22ec63482", "statement_latex": "\\forall z\\in D,\\;(((x+m+1)\\sum _{i=((xy),z)", "inference_order": 1587} {"schema_version": "1.0", "item_id": "rcd2_33e1c332ed9b118d", "pair_id": "pair_2317938c849f9486", "statement_latex": "\\forall z\\in D,\\;(z=(\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S})\\Longleftrightarrow z=(\\int_{V} (\\nabla \\cdot \\mathbf{F}) dV))", "inference_order": 1588} {"schema_version": "1.0", "item_id": "rcd2_95a01f841579162d", "pair_id": "pair_628b2c47674928b0", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(f'(x)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(y))", "inference_order": 1589} {"schema_version": "1.0", "item_id": "rcd2_3c86dcbbe005e857", "pair_id": "pair_c306c2b696171f5c", "statement_latex": "(1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = (F_{n-1}*F_{n+1} - F_n^2)", "inference_order": 1590} {"schema_version": "1.0", "item_id": "rcd2_3bdd2ba9fb10a26a", "pair_id": "pair_f6d6fbfb6288ecb1", "statement_latex": "(f(x)) = (f(-x))", "inference_order": 1591} {"schema_version": "1.0", "item_id": "rcd2_6a3cbdc5fe37db32", "pair_id": "pair_01f4fb41742ea63d", "statement_latex": "\\delta_D((lim_{n→∞} σ_n(f,x)),(f(x)))\\\\ne0", "inference_order": 1592} {"schema_version": "1.0", "item_id": "rcd2_a002b4305929b955", "pair_id": "pair_5edcc4aa3b14e778", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((|A_d|) \\in S \\land (\\frac{w(d)}{d}X + R_d) \\notin S) \\lor ((\\frac{w(d)}{d}X + R_d) \\in S \\land (|A_d|) \\notin S))\\right)", "inference_order": 1593} {"schema_version": "1.0", "item_id": "rcd2_1f6e18036eca8262", "pair_id": "pair_cb7eb92aa7efd83b", "statement_latex": "\\delta_D((B),(SAS^T))=(0) + \\eta", "inference_order": 1594} {"schema_version": "1.0", "item_id": "rcd2_2009d7d6444c3f9d", "pair_id": "pair_6c570176bc82bfdd", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(z=(dX_t)\\Longleftrightarrow z=(mu_t dt + sigma_t dB_t))\\right)", "inference_order": 1595} {"schema_version": "1.0", "item_id": "rcd2_2fb8d0510250ad14", "pair_id": "pair_43247ac964832ffb", "statement_latex": "(p) \\in {(a^2 + b^2 + c^2 + d^2)}", "inference_order": 1596} {"schema_version": "1.0", "item_id": "rcd2_099ebc6bfb394ff4", "pair_id": "pair_ad27237074b33b17", "statement_latex": "(-L(v)) = (L(-v))", "inference_order": 1597} {"schema_version": "1.0", "item_id": "rcd2_ba7258ee7f351a8a", "pair_id": "pair_eddac251b309ec5a", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(([I - A]x),z)=((d),z)\\right)", "inference_order": 1598} {"schema_version": "1.0", "item_id": "rcd2_05db1eec20fd3815", "pair_id": "pair_dd53470da43b5872", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (1) \\le x \\land x \\le (|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2|)", "inference_order": 1599} {"schema_version": "1.0", "item_id": "rcd2_18704e558d6f72a0", "pair_id": "pair_25f28268157689d0", "statement_latex": "\\{((\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1}),z):z\\in D\\}=\\{((1),z):z\\in D\\}", "inference_order": 1600} {"schema_version": "1.0", "item_id": "rcd2_e76eb95395bfaa9a", "pair_id": "pair_a4a5b9845b5f937d", "statement_latex": "\\frac{\\exp(||f(A)||)}{\\exp(||f(A)||)+\\exp(2 sup_{z in W(A)} |f(z)|)}\\le\\frac{1}{2}", "inference_order": 1601} {"schema_version": "1.0", "item_id": "rcd2_8e225d951e57d279", "pair_id": "pair_3b5e5e6f6aa930c7", "statement_latex": "\\operatorname{dist}((ζ(n) / 2^(n-1))-(Δ),(-\\infty,0])=(0) + \\eta", "inference_order": 1602} {"schema_version": "1.0", "item_id": "rcd2_965ebde59d450e3f", "pair_id": "pair_729f16a737b947c1", "statement_latex": "\\delta_D(((x+m+1)\\sum _{i\\\\ne0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}),((x-m){\\dbinom {x}{m}}))=0", "inference_order": 1603} {"schema_version": "1.0", "item_id": "rcd2_59658ff6baea6c1d", "pair_id": "pair_41cb4f893c241a1b", "statement_latex": "\\frac{\\exp(|d(A,X) - d(B,X)|)}{\\exp(|d(A,X) - d(B,X)|)+\\exp(d(A,B))}\\le\\frac{1}{2} + \\eta", "inference_order": 1604} {"schema_version": "1.0", "item_id": "rcd2_fef0264c6c40dc7c", "pair_id": "pair_fe0f99160ba6c2bb", "statement_latex": "(x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2}) \\in {(0)}", "inference_order": 1605} {"schema_version": "1.0", "item_id": "rcd2_e7408d3202c34d43", "pair_id": "pair_2b5aa7091a4b7fff", "statement_latex": "\\{((|A+B|),z):z\\in D\\}=\\{((xy),z):z\\in D\\}", "inference_order": 1606} {"schema_version": "1.0", "item_id": "rcd2_fa480fa8d5ee7540", "pair_id": "pair_e0bdcc9e40a780ce", "statement_latex": "\\forall z\\in D,\\;(z=(α)\\Longleftrightarrow z=(Pr_θ0(X ∈ R)))", "inference_order": 1607} {"schema_version": "1.0", "item_id": "rcd2_cea93580048cb44f", "pair_id": "pair_642cde1a49451c59", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (∑_{n≥ 1 - α))\\le t", "inference_order": 1608} {"schema_version": "1.0", "item_id": "rcd2_e7eb3b3a77498584", "pair_id": "pair_805903dbc220473e", "statement_latex": "\\operatorname{dist}((e*p*(d+1))-(1),(-\\infty,0])= 1 - α))-(α),(-\\infty,0])=0", "inference_order": 1609} {"schema_version": "1.0", "item_id": "rcd2_24b1ed83b1c0e544", "pair_id": "pair_4807a28ce3c3b23d", "statement_latex": "\\max\\{0,(Pr[Intersection(X_i ≤ u_i)])-(Pr[Intersection(Y_i ≤ u_i)])\\}=0", "inference_order": 1610} {"schema_version": "1.0", "item_id": "rcd2_c4b17c696c3ed646", "pair_id": "pair_0280c5b83f0aa6bb", "statement_latex": "\\{((\\zeta(s)),z):z\\in D\\}=\\{(((-1)^n),z):z\\in D\\}", "inference_order": 1611} {"schema_version": "1.0", "item_id": "rcd2_f31f1a01e3f7a4d6", "pair_id": "pair_73bd98d264980c33", "statement_latex": "\\neg(\\neg (P(X ≥ a) ≤ \\inf_{t>0} M(t)e^{-ta}))", "inference_order": 1612} {"schema_version": "1.0", "item_id": "rcd2_3ec4def4724de4f8", "pair_id": "pair_9d7260a39f19ad12", "statement_latex": "\\neg(\\neg (f(x) =( sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x)))) + \\eta", "inference_order": 1613} {"schema_version": "1.0", "item_id": "rcd2_5f3ef3a979003a96", "pair_id": "pair_e2567581c7e01f0d", "statement_latex": "\\max\\{0,(1 + |a_{n-2}| + ... + |a_{0}|)-(|a_{n-1}|)\\}=(0\\;\\land\\;(1 + |a_{n-2}| + ... + |a_{0}|)\\ne (|a_{n-1}|)) + \\eta", "inference_order": 1614} {"schema_version": "1.0", "item_id": "rcd2_e3a3fb3da1381a04", "pair_id": "pair_5175bf60f1e27ef7", "statement_latex": "((T^2),(-1))\\in\\{(z,z):z\\in D\\} + \\eta", "inference_order": 1615} {"schema_version": "1.0", "item_id": "rcd2_b57d2c5ddd051acf", "pair_id": "pair_c171d4610f21437c", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c)))+\\delta_D(z,(Integral transform+Spectral terms)))=(0) + \\eta", "inference_order": 1616} {"schema_version": "1.0", "item_id": "rcd2_bdb6e042d5211797", "pair_id": "pair_7cf348df58b7fdbc", "statement_latex": "\\{((sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3),z):z\\in D\\}=\\{(((3a)! / (a!)^3),z):z\\in D\\}", "inference_order": 1617} {"schema_version": "1.0", "item_id": "rcd2_5f12f245474d1776", "pair_id": "pair_f0a840eb2d31a7dc", "statement_latex": "\\neg(\\neg (Pr(max_{1\\lek\\len} |S_k| \\ge 3alpha) \\le 3 max_{1\\lek\\len} Pr(|S_k| \\ge alpha)))", "inference_order": 1618} {"schema_version": "1.0", "item_id": "rcd2_31d083bf12326a85", "pair_id": "pair_2037ab3ebe72c292", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,({m+n \\choose r}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}))", "inference_order": 1619} {"schema_version": "1.0", "item_id": "rcd2_11a570a33330040f", "pair_id": "pair_ddebb02c8603f247", "statement_latex": "\\{(E)\\}=\\{(F(α))\\}", "inference_order": 1620} {"schema_version": "1.0", "item_id": "rcd2_18420ffd27816a6a", "pair_id": "pair_aaa6f5cb5879f8db", "statement_latex": "\\max\\{0,(R)-(C)\\}=0\\;\\land\\;(R)\\ne (C)", "inference_order": 1621} {"schema_version": "1.0", "item_id": "rcd2_40056d2cdd27b1af", "pair_id": "pair_1f605285e008e05b", "statement_latex": "\\frac{\\exp(Pr[Intersection(X_i ≥ u_i)])}{\\exp(Pr[Intersection(X_i ≤ u_i)])+\\exp(Pr[Intersection(Y_i ≤ u_i)])}\\le\\frac{1}{2}", "inference_order": 1622} {"schema_version": "1.0", "item_id": "rcd2_a97e2b0f603d1a97", "pair_id": "pair_78be29b93d3819f4", "statement_latex": "\\neg\\left(\\inf_{z\\in D}(\\delta_D(z,([x^2 + y^2 + (x+y)^2]^2))+\\delta_D(z,(2[x^4 + y^4 + (x+y)^4])))=0\\right)", "inference_order": 1623} {"schema_version": "1.0", "item_id": "rcd2_de892c5278040ba5", "pair_id": "pair_47cd0880e2c6b666", "statement_latex": "(∇^2φ) \\in {(-ρ_f/ε_0)}", "inference_order": 1624} {"schema_version": "1.0", "item_id": "rcd2_1fd441b632a16712", "pair_id": "pair_a5e676aa0d30ba65", "statement_latex": "\\frac{\\exp(P(|sum_{i=1}^n a_i X_i| ≥ k))}{\\exp(P(|sum_{i=1}^n a_i X_i| ≥ k))+\\exp(2 inf_{0 ≤ c ≤ k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz)}\\le\\frac{1}{2}", "inference_order": 1625} {"schema_version": "1.0", "item_id": "rcd2_5b8871f88e409e7c", "pair_id": "pair_0391e512a99b5f30", "statement_latex": "(M^a = {x \\in M | f(x)) - (a}) \\le 0", "inference_order": 1626} {"schema_version": "1.0", "item_id": "rcd2_ef3cec19cbb31dfa", "pair_id": "pair_07921742684851e2", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(2 sup_{z in W(A)} |f(z)|)\\le t\\Rightarrow (||f(A)||)\\le t + \\eta", "inference_order": 1627} {"schema_version": "1.0", "item_id": "rcd2_e8048ba6ea38d6d8", "pair_id": "pair_020a74b5d4894a69", "statement_latex": "\\neg\\left(((ax + by),(gcd(a, b)))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 1628} {"schema_version": "1.0", "item_id": "rcd2_954a0a325b6c0fab", "pair_id": "pair_bb5799da15ad86d4", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Ax))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(0))", "inference_order": 1629} {"schema_version": "1.0", "item_id": "rcd2_535473af68767ad4", "pair_id": "pair_344c6fd9517324ec", "statement_latex": "(\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty) \\lor (0 = 1)", "inference_order": 1630} {"schema_version": "1.0", "item_id": "rcd2_bc43fec177079186", "pair_id": "pair_677785bbb55c35ad", "statement_latex": "\\operatorname{softplus}((P(X_N - X_0 ≥ epsilon))-(exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2))))\\le\\log 2", "inference_order": 1631} {"schema_version": "1.0", "item_id": "rcd2_ec90d5e82442cdb8", "pair_id": "pair_93604e3d52ef2226", "statement_latex": "(\\{(nullity A)\\}\\times\\{(nullity H)\\})\\cap\\{(z,z):z\\in D\\}=(\\{((nullity A),(nullity H))\\}) + \\eta", "inference_order": 1632} {"schema_version": "1.0", "item_id": "rcd2_c82469a938d49b43", "pair_id": "pair_7d416be7d7d48d75", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(z=(a^n + b^n)\\Longleftrightarrow z=(c^n))\\right)", "inference_order": 1633} {"schema_version": "1.0", "item_id": "rcd2_f2e44ddc4fc48fd2", "pair_id": "pair_a62523058af97a53", "statement_latex": "\\{(\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q})\\}=\\{(\\infty)\\}", "inference_order": 1634} {"schema_version": "1.0", "item_id": "rcd2_0645d74f886b867e", "pair_id": "pair_babc97b6c4670ef5", "statement_latex": "\\operatorname{dist}((-Σ p_i log p_i)-(-Σ p_i log q_i),(-\\infty,0])=0", "inference_order": 1635} {"schema_version": "1.0", "item_id": "rcd2_3b74cdf1e78d09ff", "pair_id": "pair_af63bd76204893fb", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\binom{k}{r} \\frac{r}{k}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\binom{k-1}{r-1}))", "inference_order": 1636} {"schema_version": "1.0", "item_id": "rcd2_4db16798d75e74be", "pair_id": "pair_bc5d05283db70598", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(|B_1|)\\\\nez\\;\\land\\;z=(|B_2|)", "inference_order": 1637} {"schema_version": "1.0", "item_id": "rcd2_9cc4d30bb8a537e5", "pair_id": "pair_ff510f9821b755ed", "statement_latex": "\\operatorname{dist}((‖v - Pv‖)-((1 + ‖P‖) inf_{u ∈ U} ‖v - u‖),(-\\infty,0])=0", "inference_order": 1638} {"schema_version": "1.0", "item_id": "rcd2_05f96c89bf4db343", "pair_id": "pair_9d6a22ad23393af9", "statement_latex": "\\neg\\left((|B_1|) \\in {(|B_2|)}\\right)", "inference_order": 1639} {"schema_version": "1.0", "item_id": "rcd2_4d0ab27a2efcd4f9", "pair_id": "pair_956f3ac218ead5ab", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((a^2 - b^2) \\in S \\land ((a + b)(a - b)) \\notin S) \\lor (((a + b)(a - b)) \\in S \\land (a^2 - b^2) \\notin S))\\right)", "inference_order": 1640} {"schema_version": "1.0", "item_id": "rcd2_25457e44bff2e48e", "pair_id": "pair_6a473075088c53e2", "statement_latex": "\\delta_D((ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn)),(2^{n-1}∏_{i\\\\ne1}^{n}ε(Xi)))=0", "inference_order": 1641} {"schema_version": "1.0", "item_id": "rcd2_3bae6f8ae58da046", "pair_id": "pair_148200ef83bb52e4", "statement_latex": "\\operatorname{Id}_D[(\\bigcap _{k=0}\\;\\text{such that}\\;(0)+s=(D(A) is dense in X, A is dissipative, and A − λ0I is surjective for some λ0)", "inference_order": 1642} {"schema_version": "1.0", "item_id": "rcd2_dc394818607c617b", "pair_id": "pair_c920a0de5224242c", "statement_latex": "\\frac{\\exp(P[max_{1>i≤n} X_i ≥ C])}{\\exp(P[max_{1≤i≤n} X_i ≥ C])+\\exp(E[max(X_n, 0)] / C)}\\le\\frac{1}{2}", "inference_order": 1643} {"schema_version": "1.0", "item_id": "rcd2_6315221e1d97f578", "pair_id": "pair_bdc3c65ce4bd2baa", "statement_latex": "\\neg(\\neg (Σ_{v∈V} deg(v) = 2|E|))", "inference_order": 1644} {"schema_version": "1.0", "item_id": "rcd2_4fe53f7165c7551e", "pair_id": "pair_71efd2d5e88b0066", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(p - q))\\\\ne\\operatorname{argmin}_{z\\in D}\\delta_D(z,(1/π) Δ arg f(iy))", "inference_order": 1645} {"schema_version": "1.0", "item_id": "rcd2_53cbe730fe1eeff7", "pair_id": "pair_7e7fa721dcd1adab", "statement_latex": "(\\operatorname{Re} \\langle x, y \\rangle) \\in {(\\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2))}", "inference_order": 1646} {"schema_version": "1.0", "item_id": "rcd2_76f1337a81cca68d", "pair_id": "pair_7b03ecbcf8eac7c2", "statement_latex": "\\forall z\\in D,\\;(z=(\\binom{n-1}{k} + \\binom{n-1}{k-1})\\Longleftrightarrow z=(\\binom{n}{k}))", "inference_order": 1647} {"schema_version": "1.0", "item_id": "rcd2_e3a9a72c7c69707d", "pair_id": "pair_347097297ab9c443", "statement_latex": "\\exists x \\\\notin \\mathbb{R}: (M^a = {x \\in M | f(x)) \\le x \\land x \\le (a})", "inference_order": 1648} {"schema_version": "1.0", "item_id": "rcd2_89e4f5212d5f52b7", "pair_id": "pair_329b05ca30d70c53", "statement_latex": "([a]) - ([b] for a majorizing b) \\ge 0 + \\eta", "inference_order": 1649} {"schema_version": "1.0", "item_id": "rcd2_628ca4d862b3a7d3", "pair_id": "pair_5a6ae949eb4abd45", "statement_latex": "(\\exp(-u^2 / (2σ_T^2))) - (P(‖f‖_T >( E(‖f‖_T) + u)) \\ge 0) + \\eta", "inference_order": 1650} {"schema_version": "1.0", "item_id": "rcd2_27f747fe39062872", "pair_id": "pair_66cef25a22d5e7ff", "statement_latex": "\\{((H),z):z\\in D\\}=\\{((\\sum_{i=1}^{m} H_{i}),z):z\\in D\\}", "inference_order": 1651} {"schema_version": "1.0", "item_id": "rcd2_c5cb6c4b8fcc1922", "pair_id": "pair_41c3ac1ea9c66336", "statement_latex": "\\operatorname{dist}((Pr(max_{1≤k≤n} |S_k| ≥ 3alpha))-(3 max_{1≤k≤n} Pr(|S_k| ≥ alpha)),(-\\infty,0])=0", "inference_order": 1652} {"schema_version": "1.0", "item_id": "rcd2_feb88b0b2eb63d8d", "pair_id": "pair_962bea36dbde5c4c", "statement_latex": "\\{(E)\\}=\\{(nhv)\\}", "inference_order": 1653} {"schema_version": "1.0", "item_id": "rcd2_81c496984f84ad89", "pair_id": "pair_4480985b4fafc15e", "statement_latex": "\\{((sum_{p=2}^{\\infty} 1/(p-1)),z):z\\in D\\}=\\{((1),z):z\\in D\\}", "inference_order": 1654} {"schema_version": "1.0", "item_id": "rcd2_dcbad1c47bd3859e", "pair_id": "pair_275cb9fd797f7710", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((F(X)) \\in S \\land (a_0 + a_1X + ... + a_nX^n) \\notin S) \\lor ((a_0 + a_1X + ... + a_nX^n) \\in S \\land (F(X)) \\notin S))\\right)", "inference_order": 1655} {"schema_version": "1.0", "item_id": "rcd2_362828da69b3d131", "pair_id": "pair_2ccff541d5bda625", "statement_latex": "\\{(x^{4}+4y^{4})\\}=\\{(x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2})\\}", "inference_order": 1656} {"schema_version": "1.0", "item_id": "rcd2_c4d8f368127dfc85", "pair_id": "pair_7d5e55987f94dfcb", "statement_latex": "\\forall k\\in\\mathbb{N}_{>(0}\\exists N\\forall n\\ge N:d((S_βα),(S_β1α1 * S_β2α2))<1/k) + \\eta", "inference_order": 1657} {"schema_version": "1.0", "item_id": "rcd2_10eba4e42c35f710", "pair_id": "pair_14ee2bca39442186", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((||u - u_h||)-((gamma / alpha) * ||u - v||))=0", "inference_order": 1658} {"schema_version": "1.0", "item_id": "rcd2_242a77368186110a", "pair_id": "pair_cde26e99353fc302", "statement_latex": "\\neg\\left((2^n * ||f||_1) - (||h||_1) \\ge 0\\right)", "inference_order": 1659} {"schema_version": "1.0", "item_id": "rcd2_060dd019ecb0ade0", "pair_id": "pair_e883f79bcd902127", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(tr W(A, A*)))\\\\ne\\operatorname{argmin}_{z\\in D}\\delta_D(z,(tr W(B, B*)))", "inference_order": 1660} {"schema_version": "1.0", "item_id": "rcd2_86ac961161f49bff", "pair_id": "pair_ccaf4fb48857e447", "statement_latex": "\\max\\{0,(Pr[A])-(x(A) * Product_{B in Gamma(A)} (1 - x(B)))\\}\\\\ne0", "inference_order": 1661} {"schema_version": "1.0", "item_id": "rcd2_3a34dbd814fe58a8", "pair_id": "pair_7b420fc2c9bb2375", "statement_latex": "\\neg\\left((chi'_l(K_{n,n})) \\in {(n)}\\right)", "inference_order": 1662} {"schema_version": "1.0", "item_id": "rcd2_3b2ca4f417a508bb", "pair_id": "pair_c708422eeac02032", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (1/2)\\Rightarrow t<(floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)))", "inference_order": 1663} {"schema_version": "1.0", "item_id": "rcd2_dff80eca668d6807", "pair_id": "pair_ea9ccbbf300e1c52", "statement_latex": "\\forall z\\in D,\\;((sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3),z)=(((3a)! / (a!)^3),z)", "inference_order": 1664} {"schema_version": "1.0", "item_id": "rcd2_945cc8d5d91358a2", "pair_id": "pair_1e79477188b29c98", "statement_latex": "\\{t\\in\\mathbb{R}:t\\le (\\sum_{n=0}^{\\infty} n|a_n|^2)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (1)\\}", "inference_order": 1665} {"schema_version": "1.0", "item_id": "rcd2_7c38fda65bfa0db4", "pair_id": "pair_d375eb3e055f3d5e", "statement_latex": "\\max\\{0,(delta |X| |Y|)-(e(X,Y))\\}=(0\\;\\land\\;(delta |X| |Y|)\\ne (e(X,Y))) + \\eta", "inference_order": 1666} {"schema_version": "1.0", "item_id": "rcd2_15b2bb7ab5f34365", "pair_id": "pair_0bfcd78a3e6c3b4d", "statement_latex": "\\neg\\left(((a^n + b^n),(c^n))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 1667} {"schema_version": "1.0", "item_id": "rcd2_955f40bf3e48d698", "pair_id": "pair_3636f01615b6c2c9", "statement_latex": "\\delta_D((a^n + b^n),(c^n))=0", "inference_order": 1668} {"schema_version": "1.0", "item_id": "rcd2_a844a9812f1b6e84", "pair_id": "pair_2112a0f0f8a38df0", "statement_latex": "\\max\\{0,(|\\xi - \\alpha |)-(CH(\\alpha )^{-3}\\max(1, \\xi ^{2}))\\}=0\\;\\land\\;(|\\xi - \\alpha |)\\ne (CH(\\alpha )^{-3}\\max(1, \\xi ^{2}))", "inference_order": 1669} {"schema_version": "1.0", "item_id": "rcd2_e34940714634d8c1", "pair_id": "pair_80be17254d69dd8f", "statement_latex": "(\\{(x_i)\\}\\times\\{(\\det(A_i) / \\det(A))\\})\\cap\\{(z,z):z\\in D\\}=(\\{((x_i),(\\det(A_i) / \\det(A)))\\}) + \\eta", "inference_order": 1670} {"schema_version": "1.0", "item_id": "rcd2_1c3690c4a8953654", "pair_id": "pair_1044962ebaf31746", "statement_latex": "\\{(f'(c))\\}=\\{(0)\\}", "inference_order": 1671} {"schema_version": "1.0", "item_id": "rcd2_5f519281fd6250dc", "pair_id": "pair_d4467766bca8edf1", "statement_latex": "\\exists x\\in D:x\\in \\{prime p\\in D:n <(D^{-1}*sqrt(det(AA^T)))^(1/(N-M))", "inference_order": 1672} {"schema_version": "1.0", "item_id": "rcd2_63986ecbe2221a77", "pair_id": "pair_9ba5614280e01f59", "statement_latex": "\\neg\\left(\\exists z\\in D\\;\\text{such that}\\;(SMS^T)=z\\;\\land\\;z=(D ⊕ D)\\right)", "inference_order": 1673} {"schema_version": "1.0", "item_id": "rcd2_8d08172b8d642901", "pair_id": "pair_7e71ab7048df4664", "statement_latex": "\\neg\\left((\\{(|A+B|)\\}\\times\\{(|A+H|+|B+H|-|H|)\\})\\cap\\{(z,z):z\\in D\\}=\\{((|A+B|),(|A+H|+|B+H|-|H|))\\}\\right)", "inference_order": 1674} {"schema_version": "1.0", "item_id": "rcd2_695c009e0cc9c41d", "pair_id": "pair_48692e23ecfcc698", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(rank(A))=z\\;\\land\\;z=(rank([A|b]))", "inference_order": 1675} {"schema_version": "1.0", "item_id": "rcd2_3cf5db7a0efa4712", "pair_id": "pair_b13f62c3d632b667", "statement_latex": "(c * d^L) - (p(a,d)) > 0", "inference_order": 1676} {"schema_version": "1.0", "item_id": "rcd2_33a36e40fb580f2e", "pair_id": "pair_473a8de9b12f1e51", "statement_latex": "\\max\\{0,(e^3 / (29n^2))-(cr(G))\\}=(0) + \\eta", "inference_order": 1677} {"schema_version": "1.0", "item_id": "rcd2_66f17ffc8d6beec8", "pair_id": "pair_0ad3cff2078e78cf", "statement_latex": "\\neg\\left(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,((3a)! / (a!)^3))\\right)", "inference_order": 1678} {"schema_version": "1.0", "item_id": "rcd2_07e0209488f83e97", "pair_id": "pair_981ac7ea7887d35b", "statement_latex": "\\{(lim_{n→∞} σ_n(f,x))\\}=\\{(\\sup\\{f(x, \\theta) : x \\in C(\\theta)\\})\\}", "inference_order": 1679} {"schema_version": "1.0", "item_id": "rcd2_55fddcf3da6b761b", "pair_id": "pair_8f01ceadbff7aeeb", "statement_latex": "(2 * \\exp(-n * \\varepsilon^2 / (2 * (1 + \\varepsilon/3)))) - (P(|(1/n) * sum_{i=1}^n X_i| > \\varepsilon)) \\ge 0", "inference_order": 1680} {"schema_version": "1.0", "item_id": "rcd2_fb412d7b06a0652c", "pair_id": "pair_8b4a6674e371cf5d", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((Δy))=F((f'(x)Δx + εΔx))", "inference_order": 1681} {"schema_version": "1.0", "item_id": "rcd2_dae4b3c7432ca41f", "pair_id": "pair_24469cc3742ff7ed", "statement_latex": "(f(q,u)) = (\\dot{q})", "inference_order": 1682} {"schema_version": "1.0", "item_id": "rcd2_6ebdd7266840b928", "pair_id": "pair_eb1a13750942f975", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(N(T + H) - N(T))\\le t\\Rightarrow (cH log T)\\le t", "inference_order": 1683} {"schema_version": "1.0", "item_id": "rcd2_471d4653b2ca566b", "pair_id": "pair_481cb20b2e9dadd1", "statement_latex": "\\delta_D((E),(nhv))=0", "inference_order": 1684} {"schema_version": "1.0", "item_id": "rcd2_ba4d6a89d699b6b0", "pair_id": "pair_e516c185aa9fe588", "statement_latex": "\\nexists S \\subseteq D: (((E[sum_{x \\in N} f(x)]) \\in S \\land (integral f(x) dLambda(x)) \\notin S) \\lor ((integral f(x) dLambda(x)) \\in S \\land (E[sum_{x \\in N} f(x)]) \\notin S))", "inference_order": 1685} {"schema_version": "1.0", "item_id": "rcd2_8f8f6366aba160d2", "pair_id": "pair_afd68796dbb78e84", "statement_latex": "(a}) \\in (M^a = {x \\in M | f(x)) + [0,\\infty)", "inference_order": 1686} {"schema_version": "1.0", "item_id": "rcd2_c7df54e27e7a2cfb", "pair_id": "pair_7d9520052832bbf0", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(n))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(p1 + p2 + p3))", "inference_order": 1687} {"schema_version": "1.0", "item_id": "rcd2_84e521ed634c6baa", "pair_id": "pair_fe7a457222c4b0f7", "statement_latex": "\\neg\\left(\\{((f'(c)),z):z\\in D\\}=\\{((0),z):z\\in D\\}\\right)", "inference_order": 1688} {"schema_version": "1.0", "item_id": "rcd2_58f3137d5d64e47d", "pair_id": "pair_70cdd0c285c9c5b5", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((f'(c)))\\\\neF((f(b)-f(a))/(b-a))", "inference_order": 1689} {"schema_version": "1.0", "item_id": "rcd2_be3bee22515c77e5", "pair_id": "pair_f6fa4a3effe4ec6b", "statement_latex": "\\max\\{0,(|g − f|)-(ε)\\}=0\\;\\land\\;(|g − f|)\\ne (ε)", "inference_order": 1690} {"schema_version": "1.0", "item_id": "rcd2_809eb1606b4abc2b", "pair_id": "pair_94b5f5455c4449c7", "statement_latex": "\\delta_D((w),(xyz))=0", "inference_order": 1691} {"schema_version": "1.0", "item_id": "rcd2_cc55aa98c64e83f2", "pair_id": "pair_67c8e180c3350005", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(Pr(max_{1≤k≤n} |S_k| ≥ 3alpha))+s=(3 max_{1≤k≤n} Pr(|S_k| ≥ alpha))", "inference_order": 1692} {"schema_version": "1.0", "item_id": "rcd2_835af8e3c13d0f44", "pair_id": "pair_d0848f4f91f7a840", "statement_latex": "(ω(-t)) \\in {(ω(t))}", "inference_order": 1693} {"schema_version": "1.0", "item_id": "rcd2_f2fc3a110c25bb79", "pair_id": "pair_8badc6c944195e45", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (P(‖f‖_T > E(‖f‖_T) + u)) \\le x \\land x \\le (\\exp(-u^2 / (2σ_T^2)))", "inference_order": 1694} {"schema_version": "1.0", "item_id": "rcd2_07e2a9f9a49de255", "pair_id": "pair_e3a5ba8bb54919ae", "statement_latex": "\\{((gcd{(n-1 k-1), (n k+1), (n+1 k)}),z):z\\in D\\}=\\{((gcd{(n-1 k), (n k-1), (n+1 k+1)}),z):z\\in D\\}", "inference_order": 1695} {"schema_version": "1.0", "item_id": "rcd2_04d1a549b3b5ee8c", "pair_id": "pair_367ab875787ea9c9", "statement_latex": "(0}^{\\infty }C_{k}\\neq \\\\emptyset) \\\\ne (\\bigcap _{k)", "inference_order": 1696} {"schema_version": "1.0", "item_id": "rcd2_9f8eca8e4a1312e4", "pair_id": "pair_215297dcd56687bf", "statement_latex": "(Δ) - (ζ(n) / 2^(n-1)) \\ge 0 + \\eta", "inference_order": 1697} {"schema_version": "1.0", "item_id": "rcd2_75d5a241c0387bcd", "pair_id": "pair_77811628e6f7fb51", "statement_latex": "\\nexists S \\subseteq D: (((P(A|B)) \\in S \\land (P(B|A)*P(A)/P(B)) \\notin S) \\lor ((P(B|A)*P(A)/P(B)) \\in S \\land (P(A|B)) \\notin S)) + \\eta", "inference_order": 1698} {"schema_version": "1.0", "item_id": "rcd2_0d6ffd8895ca1ecc", "pair_id": "pair_066e3d4b0012675f", "statement_latex": "\\{((Δy),z):z\\in D\\}\\\\ne\\{((f'(x)Δx + εΔx),z):z\\in D\\}", "inference_order": 1699} {"schema_version": "1.0", "item_id": "rcd2_70869e1b784557e1", "pair_id": "pair_062553c5cbe7755c", "statement_latex": "\\operatorname{dist}((W(r,k))-(2^{2^{r^{2^{2^{k+9}}}}}),(-\\infty,0])=0", "inference_order": 1700} {"schema_version": "1.0", "item_id": "rcd2_67ffb5b5c755da8b", "pair_id": "pair_00067de84bb18e39", "statement_latex": "\\forall z\\in D,\\;(z=(det(A))\\Longleftrightarrow z=(det(A_v^u) * det(A_tilde_v^u)))", "inference_order": 1701} {"schema_version": "1.0", "item_id": "rcd2_0ce6baf8a0914e53", "pair_id": "pair_6684444833625b35", "statement_latex": "\\forall z\\in D,\\;(z\\\\ne(n)\\Longleftrightarrow z=(p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}))", "inference_order": 1702} {"schema_version": "1.0", "item_id": "rcd2_21c97b5599a059bd", "pair_id": "pair_254cdc6b98c35f0b", "statement_latex": "(Pr(X - E[X] ≥ λ) ≥ σ^2 / (σ^2 + λ^2)) \\land (0 = 0)", "inference_order": 1703} {"schema_version": "1.0", "item_id": "rcd2_588ea1712adf0a19", "pair_id": "pair_cd3eead18148d8af", "statement_latex": "\\frac{\\exp(P(∪_{i=1}^∞ A_i))}{\\exp(P(∪_{i=1}^∞ A_i))+\\exp(∑_{i=1}^∞ P(A_i))}\\le\\frac{1}{2}", "inference_order": 1704} {"schema_version": "1.0", "item_id": "rcd2_20c6b8b64298e7d9", "pair_id": "pair_e218da2a1eb4d109", "statement_latex": "\\exists x \\\\in 1 - α))-(α),(-\\infty,0])=0", "inference_order": 1705} {"schema_version": "1.0", "item_id": "rcd2_0dee5bee8ffd71a4", "pair_id": "pair_d9f6b6548a9b56b7", "statement_latex": "\\forall z\\in D,\\;((P[σ(x(t))] / P[σ(x_tilde(t))]),z)=((e^σ[x(t)]),z)", "inference_order": 1706} {"schema_version": "1.0", "item_id": "rcd2_4fe38d3d526d8349", "pair_id": "pair_6bcbd22b959d46d2", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge ([a])\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge ([b] for a majorizing b)\\}", "inference_order": 1707} {"schema_version": "1.0", "item_id": "rcd2_2ae96603a2bafdc4", "pair_id": "pair_ddebb02c8603f247", "statement_latex": "\\{(E)\\}=\\{(\\text{conv}\\{w(X) : w \\in W\\})\\}", "inference_order": 1708} {"schema_version": "1.0", "item_id": "rcd2_10b22d6f2547a3e9", "pair_id": "pair_99b4eda2197da60e", "statement_latex": "\\{(a^2 - b^2)\\}=\\{(a + b)(a - b)\\}", "inference_order": 1709} {"schema_version": "1.0", "item_id": "rcd2_6896bec9df54ff65", "pair_id": "pair_44a1173943624fe6", "statement_latex": "\\max\\{0,(|∫ Dφ(y)(x) dμ(y)|)-(C(x) ‖φ‖_{∞})\\}=(0) + \\eta", "inference_order": 1710} {"schema_version": "1.0", "item_id": "rcd2_b7602ba0d388bfa4", "pair_id": "pair_f6921ffc186bb8e9", "statement_latex": "\\frac{\\exp(delta |X| |Y|)}{\\exp(delta |X| |Y|)+\\exp(e(X,Y))}<\\frac{1}{2}", "inference_order": 1711} {"schema_version": "1.0", "item_id": "rcd2_5da7ef165ff800e3", "pair_id": "pair_873745c5ea465e53", "statement_latex": "\\max\\{0,(|a_{n+1} + a_{n+2} + \\cdots + a_{n+p}|)-(\\varepsilon)\\}=0\\;\\land\\;(|a_{n+1} + a_{n+2} + \\cdots + a_{n+p}|)\\ne (\\varepsilon)", "inference_order": 1712} {"schema_version": "1.0", "item_id": "rcd2_e1ca0a3702104132", "pair_id": "pair_5d53a0fbbef35593", "statement_latex": "\\neg(\\neg (|F| \\le max_{r: t+2r \\le n} |F_{n,k,t,r}|)) + \\eta", "inference_order": 1713} {"schema_version": "1.0", "item_id": "rcd2_b0f2bc0ce3495ac6", "pair_id": "pair_0d2903de582c3d80", "statement_latex": "\\nexists S \\subseteq D: (((f(x)) \\in S \\land (0 (mod p^n)) \\notin S) \\lor ((0 (mod p^n)) \\in S \\land (f(x)) \\notin S)) + \\eta", "inference_order": 1714} {"schema_version": "1.0", "item_id": "rcd2_a78b46b58ac8a9bb", "pair_id": "pair_da0d76d660f5d44b", "statement_latex": "\\nexists S \\subseteq D: (((\\lim_{\\th\\eta \\to \\infty}\\frac{1}{\\th\\eta}\\log\\int_{A}e^{-\\th\\eta\\varphi(x)}\\,dx) \\in S \\land (-\\mathop{\\mathrm{ess\\,inf}}_{x\\\\in A}\\varphi(x)) \\notin S) \\lor ((-\\mathop{\\mathrm{ess\\,inf}}_{x\\\\in A}\\varphi(x)) \\in S \\land (\\lim_{\\th\\eta \\to \\infty}\\frac{1}{\\th\\eta}\\log\\int_{A}e^{-\\th\\eta\\varphi(x)}\\,dx) \\notin S))", "inference_order": 1715} {"schema_version": "1.0", "item_id": "rcd2_5832c6c1ab5c5f03", "pair_id": "pair_32c294284fc38a6f", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (exp(-u^2 / (2σ_T^2)))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (P(‖f‖_T > E(‖f‖_T) + u))\\}", "inference_order": 1716} {"schema_version": "1.0", "item_id": "rcd2_5a99deca5d584e2a", "pair_id": "pair_1ddbab4a4f3f9fb7", "statement_latex": "\\{((max_matching_size),z):z\\in D\\}=\\{((1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)),z):z\\in D\\}", "inference_order": 1717} {"schema_version": "1.0", "item_id": "rcd2_c57a5c5226a20234", "pair_id": "pair_f3c111bc249621b9", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(sum_{i=0}^n binom(n,i)))+\\delta_D(z,(2^n)))=0", "inference_order": 1718} {"schema_version": "1.0", "item_id": "rcd2_722af4bdc94e6510", "pair_id": "pair_8373302242bff73f", "statement_latex": "(P(A|B)) \\in {(P(B|A)*P(A)/P(B))}", "inference_order": 1719} {"schema_version": "1.0", "item_id": "rcd2_0ca051fa53581be8", "pair_id": "pair_7b9a84984f8b7874", "statement_latex": "\\neg\\left((\\neg ((x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k)) \\Rightarrow (0 = 1)\\right)", "inference_order": 1720} {"schema_version": "1.0", "item_id": "rcd2_67aee65f62f83258", "pair_id": "pair_98b2d2e88a6f921e", "statement_latex": "\\forall z\\in D,\\;(z=(ζ(s))\\Longleftrightarrow z=(Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s)))", "inference_order": 1721} {"schema_version": "1.0", "item_id": "rcd2_b2a4d6719c742303", "pair_id": "pair_c5427071f71c1c20", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((P(S_n in A))-(binom(n,floor(n/2))/2^n))=0", "inference_order": 1722} {"schema_version": "1.0", "item_id": "rcd2_5596d036f65cd345", "pair_id": "pair_276940381402d97e", "statement_latex": "\\{((f'(c)),z):z\\in D\\}=\\{((\\sup\\{f(x, \\theta) : x \\in C(\\theta)\\}),z):z\\in D\\}", "inference_order": 1723} {"schema_version": "1.0", "item_id": "rcd2_9e868fe41c2c3ae3", "pair_id": "pair_0cb3f78da8289508", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\ge (|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (1)\\}\\right)", "inference_order": 1724} {"schema_version": "1.0", "item_id": "rcd2_e15660d9614163eb", "pair_id": "pair_e187266170ddccd4", "statement_latex": "((Av),(rv))\\in\\{(z,z):z\\in D\\}", "inference_order": 1725} {"schema_version": "1.0", "item_id": "rcd2_53b92fc1e715f58c", "pair_id": "pair_29cce6f17d240808", "statement_latex": "(S_βα → S_β1α1 * S_β2α2) \\land (0 = product_{i=1}^n (E_{ii} + n - i)))", "inference_order": 1726} {"schema_version": "1.0", "item_id": "rcd2_aba436c467a5e995", "pair_id": "pair_eb302a17ad6ce10a", "statement_latex": "\\frac{\\exp(vol(s φ K + t ψ L)^(1/n))}{\\exp(vol(s φ K + t ψ L)^(1/n))+\\exp(C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n)))}\\le\\frac{1}{2}", "inference_order": 1727} {"schema_version": "1.0", "item_id": "rcd2_8e5e3a43d1299b16", "pair_id": "pair_35f9c9dd605944ba", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(w^(-1) * (z + w + m)^m))", "inference_order": 1728} {"schema_version": "1.0", "item_id": "rcd2_7941eae54953ab9a", "pair_id": "pair_e92e652fefccd66a", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (3 max_{1≤k≤n} Pr(|S_k| ≥ alpha))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (Pr(max_{1≤k≤n} |S_k| ≥ 3alpha))\\}", "inference_order": 1729} {"schema_version": "1.0", "item_id": "rcd2_a1879207a9efd310", "pair_id": "pair_fe30979f3da82c4f", "statement_latex": "\\forall z\\in D,\\;((|A_d|),z)=((\\frac{w(d)}{d}X + R_d),z)", "inference_order": 1730} {"schema_version": "1.0", "item_id": "rcd2_5fa89d377e1a57c4", "pair_id": "pair_eff6336bc71d360d", "statement_latex": "\\{(n)\\}=\\{(p1 + p2 + p3)\\}", "inference_order": 1731} {"schema_version": "1.0", "item_id": "rcd2_455f721a8f2489fd", "pair_id": "pair_8129b00b78a45cc6", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\ge (d/|S|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (Pr[P(r1, ..., rn) = 0])\\}\\right)", "inference_order": 1732} {"schema_version": "1.0", "item_id": "rcd2_5cf8b92080e13ffb", "pair_id": "pair_8869a246a5096af4", "statement_latex": "(N}c^{-r}K(m) \\mid (∑_{c) - (0 ,n,c)g(4π√mn/c)=Integral transform+Spectral terms)", "inference_order": 1733} {"schema_version": "1.0", "item_id": "rcd2_ef72d0c3410264be", "pair_id": "pair_e8f2a58c3f06a49c", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(x_i))+\\delta_D(z,(\\det(A_i) / \\det(A))))=(0) + \\eta", "inference_order": 1734} {"schema_version": "1.0", "item_id": "rcd2_bc989abfea1cfaee", "pair_id": "pair_1e000eb3ff83cc9f", "statement_latex": "\\operatorname{softplus}((n)-(sum_{j=1}^r d_j))<\\log 2", "inference_order": 1735} {"schema_version": "1.0", "item_id": "rcd2_c51244eb68e95e40", "pair_id": "pair_878b39957ab4020f", "statement_latex": "\\forall z\\in D,\\;((∇^2φ),z)=(((-ρ_f/ε_0),z)) + \\eta", "inference_order": 1736} {"schema_version": "1.0", "item_id": "rcd2_5b07e2335b3c5434", "pair_id": "pair_2953f21c0ffd15d1", "statement_latex": "\\max\\{0,(ceil(n/m))-(max_{1>i≤m} |B_i|)\\}=0", "inference_order": 1737} {"schema_version": "1.0", "item_id": "rcd2_694d953276e28553", "pair_id": "pair_23fb666d8efccf82", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (per(A)) \\le x \\land x \\le (prod_{i=1}^n (r_i!)^(1/r_i))) + \\eta", "inference_order": 1738} {"schema_version": "1.0", "item_id": "rcd2_9d23bdc611de5443", "pair_id": "pair_affd2ac491f91cee", "statement_latex": "\\frac{\\exp(Pr(|X-mu| ≥( k*sigma))}{\\exp(Pr(|X-mu| ≥ k*sigma))+\\exp(1/k^2)}\\le\\frac{1}{2}) + \\eta", "inference_order": 1739} {"schema_version": "1.0", "item_id": "rcd2_b5604e13f5bda8cc", "pair_id": "pair_75a4a2fbc700bc00", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\le (|e(S,T) - (d/n)|S||T||)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (λ√|S||T|)\\}\\right)", "inference_order": 1740} {"schema_version": "1.0", "item_id": "rcd2_4fbf9b9024363f83", "pair_id": "pair_7d9520052832bbf0", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(n))=(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(p1 + p2 + p3))) + \\eta", "inference_order": 1741} {"schema_version": "1.0", "item_id": "rcd2_7b33f4dcafd93c1b", "pair_id": "pair_f5f44625a814d1f5", "statement_latex": "\\nexists S \\subseteq D: (((f'(c)) \\in S \\land ((f(b)-f(a))/(b-a)) \\notin S) \\lor (((f(b)-f(a))/(b-a)) \\in S \\land (f'(c)) \\notin S)) + \\eta", "inference_order": 1742} {"schema_version": "1.0", "item_id": "rcd2_42028074736de97b", "pair_id": "pair_f21db2737cd917c7", "statement_latex": "\\delta_D((f'(x)),(y))=0", "inference_order": 1743} {"schema_version": "1.0", "item_id": "rcd2_c5a15b6c476b38a4", "pair_id": "pair_bc457141b046daeb", "statement_latex": "\\frac{\\exp(|g − f|)}{\\exp(|g − f|)+\\exp(ε)}<(\\frac{1}{2}) + \\eta", "inference_order": 1744} {"schema_version": "1.0", "item_id": "rcd2_a446dbc5393f827e", "pair_id": "pair_1f633c1c0a3d7930", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (|ω(n)−loglogn|)\\Rightarrow t\\\\ge(loglogn)^(1/2+ε)", "inference_order": 1745} {"schema_version": "1.0", "item_id": "rcd2_2f69a3acdb4519eb", "pair_id": "pair_9b0937b4e2d33589", "statement_latex": "\\neg\\left(\\exists x \\\\in \\mathbb{R}: (|A+A|) \\le x \\land x \\le (K|A|)\\right)", "inference_order": 1746} {"schema_version": "1.0", "item_id": "rcd2_7332a36585688e05", "pair_id": "pair_bfa2ed699f1dcc3b", "statement_latex": "\\forall z\\in D,\\;((x^{4}+4y^{4}),z)=((x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2}),z)", "inference_order": 1747} {"schema_version": "1.0", "item_id": "rcd2_5a1f91ae720f585a", "pair_id": "pair_1943c3d61b76f3fa", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((f(x_1, ..., x_n)))=F((Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))))", "inference_order": 1748} {"schema_version": "1.0", "item_id": "rcd2_bde311b499ebdb06", "pair_id": "pair_fbc21740d6f87c50", "statement_latex": "(\\pi(x;q,a)) - (2x / (phi(q) log(x/q))) \\le 0", "inference_order": 1749} {"schema_version": "1.0", "item_id": "rcd2_532dbdd0a63c0476", "pair_id": "pair_3260dfce9480992e", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (λ√|S||T|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|e(S,T) - (d/n)|S||T||)\\} + \\eta", "inference_order": 1750} {"schema_version": "1.0", "item_id": "rcd2_3aaf027e839a19ef", "pair_id": "pair_019ca791a0885a9d", "statement_latex": "\\frac{\\exp((1/x) * sum_{n≤(x} |f(n) - A(x)|^2)}{\\exp((1/x) * sum_{n≤x} |f(n) - A(x)|^2)+\\exp((2 + epsilon(x)) * B(x)^2)}\\le\\frac{1}{2}) + \\eta", "inference_order": 1751} {"schema_version": "1.0", "item_id": "rcd2_5a7bd1367845302a", "pair_id": "pair_e03b8c25feba0441", "statement_latex": "(\\{((\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2)\\}\\times\\{(\\sum_{i0}\\inf_N\\mathbf{1}_{\\{\\forall n\\ge N:d((S_βα),(S_β1α1 * S_β2α2))<\\varepsilon\\}}=1", "inference_order": 1769} {"schema_version": "1.0", "item_id": "rcd2_0e7544fa0308cb6f", "pair_id": "pair_eb2509002ddac5e3", "statement_latex": "(|Irr_{p'}(G)|) \\in {(|Irr_{p'}(N_{G}(P))|)}", "inference_order": 1770} {"schema_version": "1.0", "item_id": "rcd2_eab103abb55017b1", "pair_id": "pair_0969b5fda62bd73e", "statement_latex": "\\neg\\left(\\{(∇^2φ)\\}=\\{(-ρ_f/ε_0)\\}\\right)", "inference_order": 1771} {"schema_version": "1.0", "item_id": "rcd2_46f36bc88d1d6aed", "pair_id": "pair_29e0c41493a7ab1d", "statement_latex": "(\\binom{n-1}{k} + \\binom{n-1}{k-1}) \\in {(\\binom{n}{k})} + \\eta", "inference_order": 1772} {"schema_version": "1.0", "item_id": "rcd2_24859b870420ccf5", "pair_id": "pair_bf9ee452434b20bd", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(n)=z\\;\\land\\;z=(p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k})", "inference_order": 1773} {"schema_version": "1.0", "item_id": "rcd2_18ea570ac1e86a55", "pair_id": "pair_f869b06a961e0d31", "statement_latex": "\\{((k e_k),z):z\\in D\\}=\\{((\\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i),z):z\\in D\\}", "inference_order": 1774} {"schema_version": "1.0", "item_id": "rcd2_f7c3b41a213805f3", "pair_id": "pair_29b324593a3b8806", "statement_latex": "((∑_{i\\\\ne1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2) \\land (0 = 0)", "inference_order": 1775} {"schema_version": "1.0", "item_id": "rcd2_e58874c65a5f734e", "pair_id": "pair_87c8fefc366d1e8b", "statement_latex": "\\exists k\\in\\mathbb{Z}:(sum_{d|n} mu(d)*a_{n/d})-(0)=k(n)", "inference_order": 1776} {"schema_version": "1.0", "item_id": "rcd2_3347097ed4f0632e", "pair_id": "pair_d0935628b756ed51", "statement_latex": "(ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) =( 2^{n-1}∏_{i=1}^{n}ε(Xi)) \\land (0 = 0)) + \\eta", "inference_order": 1777} {"schema_version": "1.0", "item_id": "rcd2_2036e8f41d0bdae8", "pair_id": "pair_729f16a737b947c1", "statement_latex": "\\delta_D(((x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}),((x-m){\\dbinom {x}{m}}))=0", "inference_order": 1778} {"schema_version": "1.0", "item_id": "rcd2_f8b3b6fbbee2c75a", "pair_id": "pair_5514b90ff8c47f1e", "statement_latex": "\\neg\\left(\\operatorname{softplus}((det(M))-(det(A)*det(C)))\\le\\log 2\\right)", "inference_order": 1779} {"schema_version": "1.0", "item_id": "rcd2_6bbd506b88f42bcc", "pair_id": "pair_e13fbeff4d9e3328", "statement_latex": "\\operatorname{dist}((||x||_Y)-(ε||x||_X + C(ε)||x||_Z),(-\\infty,0])=0", "inference_order": 1780} {"schema_version": "1.0", "item_id": "rcd2_a110d3eda998e413", "pair_id": "pair_85e50878687fc032", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: ((\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2 = \\sum_{i) < x \\land x < (j} (a_i b_j - a_j b_i)^2)) + \\eta", "inference_order": 1781} {"schema_version": "1.0", "item_id": "rcd2_d490cefb0bb1130e", "pair_id": "pair_98f1de39177d238e", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(f(x)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(0 (mod p^n)))", "inference_order": 1782} {"schema_version": "1.0", "item_id": "rcd2_0eac8de4b3aec383", "pair_id": "pair_14719f341355549f", "statement_latex": "(1) = (F_{n-1}*F_{n+1} - F_n^2)", "inference_order": 1783} {"schema_version": "1.0", "item_id": "rcd2_7869af97cf76c9a6", "pair_id": "pair_85e50878687fc032", "statement_latex": "\\exists x \\\\in \\mathbb{R}: ((\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2 = \\sum_{i) < x \\land x < (j} (a_i b_j - a_j b_i)^2)", "inference_order": 1784} {"schema_version": "1.0", "item_id": "rcd2_618b2ea269c90a5a", "pair_id": "pair_327df2f670cd234e", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q})=z\\;\\land\\;z=(\\infty)", "inference_order": 1785} {"schema_version": "1.0", "item_id": "rcd2_cf9715e5d821143a", "pair_id": "pair_d84be77019a62a05", "statement_latex": "\\forall z\\in D,\\;(z=(cap(Z))\\Longleftrightarrow z=(H(p/(1+2^s))))", "inference_order": 1786} {"schema_version": "1.0", "item_id": "rcd2_7ed208b2b3419ecf", "pair_id": "pair_283e659e7592b775", "statement_latex": "(∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE) = (H(t))", "inference_order": 1787} {"schema_version": "1.0", "item_id": "rcd2_46ea84216709210f", "pair_id": "pair_d824e5e6ea6df6b5", "statement_latex": "\\forall z\\in D,\\;(z=(x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2})\\Longleftrightarrow z=(0))", "inference_order": 1788} {"schema_version": "1.0", "item_id": "rcd2_81d0947b2c828d85", "pair_id": "pair_0ad3cff2078e78cf", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,((3a)! / (a!)^3))", "inference_order": 1789} {"schema_version": "1.0", "item_id": "rcd2_53dae4d7ba6c0d2c", "pair_id": "pair_bb54a11340fac5a0", "statement_latex": "\\forall z\\in D,\\;(z=((Δy)\\Longleftrightarrow z=(f'(x)Δx + εΔx))) + \\eta_{897}", "inference_order": 1790} {"schema_version": "1.0", "item_id": "rcd2_e492ef52e7947e98", "pair_id": "pair_36d5ca99df216edc", "statement_latex": "(\\{(p(t))\\}\\times\\{(t^6 + c_4t^4 + c_2t^2 + c_1t + c_0)\\})\\cap\\{(z,z):z\\in D\\}=(\\{((p(t)),(t^6 + c_4t^4 + c_2t^2 + c_1t + c_0))\\}) + \\eta", "inference_order": 1791} {"schema_version": "1.0", "item_id": "rcd2_9bc532fdf747294c", "pair_id": "pair_77f68d6f59de7e01", "statement_latex": "\\delta_D((1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p)),(e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))))=0", "inference_order": 1792} {"schema_version": "1.0", "item_id": "rcd2_c01b7117f9afbc8c", "pair_id": "pair_4b8b3cefd925c91d", "statement_latex": "(ε||x||_X + C(ε)||x||_Z) - (||x||_Y) \\ge 0", "inference_order": 1793} {"schema_version": "1.0", "item_id": "rcd2_cbbf9f35eabe4fb3", "pair_id": "pair_1f633c1c0a3d7930", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (|ω(n)−loglogn|)\\Rightarrow t<(loglogn)^(1/2+ε)", "inference_order": 1794} {"schema_version": "1.0", "item_id": "rcd2_14175dd4112e6df2", "pair_id": "pair_a792d1771917cb18", "statement_latex": "(0 = product_{j=1}^{\\infty} (1/(1-z^j))^{M(alpha, j)}))", "inference_order": 1795} {"schema_version": "1.0", "item_id": "rcd2_82095829f3422637", "pair_id": "pair_f970287820f14ed2", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((min(|A|+|B|-1, p))-(|A+B|))\\\\ne0", "inference_order": 1796} {"schema_version": "1.0", "item_id": "rcd2_da268ddca6204a1c", "pair_id": "pair_e59daa177b5fc2ed", "statement_latex": "\\neg\\left(\\exists x \\\\in \\mathbb{R}: (|S_chi(N,H)|) \\le x \\land x \\le (C_{r,\\varepsilon} H^{1-1/r} q^{(r+1)/(4r^2)+\\varepsilon})\\right)", "inference_order": 1797} {"schema_version": "1.0", "item_id": "rcd2_fdad63a87b9df775", "pair_id": "pair_ce257c82b19333ab", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((P(X ≥ a))-(E[X] / a))=0", "inference_order": 1798} {"schema_version": "1.0", "item_id": "rcd2_757ea1cc449befad", "pair_id": "pair_69758197e88ffedf", "statement_latex": "\\operatorname{dist}((P(X ≥ 1 - α))-(α),(-\\infty,0])=0", "inference_order": 1799} {"schema_version": "1.0", "item_id": "rcd2_6c287e51690c6850", "pair_id": "pair_e7f697050a6e2088", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(N^{(i-1)})=z\\;\\land\\;z=(\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1})", "inference_order": 1800} {"schema_version": "1.0", "item_id": "rcd2_68c6251dac3ab253", "pair_id": "pair_7e71b108ff772b01", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(a^n + b^n))+\\delta_D(z,(c^n)))=0", "inference_order": 1801} {"schema_version": "1.0", "item_id": "rcd2_22b08799f867e5d9", "pair_id": "pair_f4d0ec8be08cfec8", "statement_latex": "(0) =( (ΔδT)) + \\eta", "inference_order": 1802} {"schema_version": "1.0", "item_id": "rcd2_d62fb74741d9d786", "pair_id": "pair_ea569954e25af666", "statement_latex": "(prod_{i=(1}^n (r_i!)^(1/r_i)) - (per(A)) \\ge 0) + \\eta", "inference_order": 1803} {"schema_version": "1.0", "item_id": "rcd2_e252d6877d9bca9c", "pair_id": "pair_2df0ca9e2ff72f6a", "statement_latex": "\\delta_D((dX_t),(mu_t dt + sigma_t dB_t))=(0) + \\eta", "inference_order": 1804} {"schema_version": "1.0", "item_id": "rcd2_aa55396e7a68b228", "pair_id": "pair_5d53a0fbbef35593", "statement_latex": "\\neg(\\neg (|F| \\le max_{r: t+2r \\le n} |F_{n,k,t,r}|))", "inference_order": 1805} {"schema_version": "1.0", "item_id": "rcd2_55149e537bb52846", "pair_id": "pair_667524cf50f41f7b", "statement_latex": "(n) \\mid (sum_{d|n} mu(d)*a_{n/d}) - (0) + \\eta", "inference_order": 1806} {"schema_version": "1.0", "item_id": "rcd2_ed29770f805dfaa0", "pair_id": "pair_7e2e9a745a168e87", "statement_latex": "\\neg\\left(\\inf_{z\\in D}(\\delta_D(z,(L))+\\delta_D(z,(lambda*W)))=0\\right)", "inference_order": 1807} {"schema_version": "1.0", "item_id": "rcd2_af16c482c95cdb78", "pair_id": "pair_7319e7abb0b643e0", "statement_latex": "(j} (a_i b_j - a_j b_i)^2) - ((\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2 \\\\ne \\sum_{i) > 0", "inference_order": 1808} {"schema_version": "1.0", "item_id": "rcd2_f2ac2affe38d22f4", "pair_id": "pair_46cdbfbda51a15a4", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2)))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (P(X_N - X_0 ≥ epsilon))\\}", "inference_order": 1809} {"schema_version": "1.0", "item_id": "rcd2_2b0e43b626063df8", "pair_id": "pair_aa10ecd99f42dac6", "statement_latex": "\\{(∫[0,∞] x^{s-1} f(x) dx)\\}=\\{(Γ(s) φ(-s))\\}", "inference_order": 1810} {"schema_version": "1.0", "item_id": "rcd2_49c591676dc9cefa", "pair_id": "pair_b5bd4e1dbf31333b", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (tr(exp(A)*exp(B)))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (tr(exp(A+B)))\\}", "inference_order": 1811} {"schema_version": "1.0", "item_id": "rcd2_2c25a6c89c0357d8", "pair_id": "pair_daea5f5eeccf5793", "statement_latex": "\\neg\\left(((dim(H_μ)),(n!))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 1812} {"schema_version": "1.0", "item_id": "rcd2_8c8636fee8205870", "pair_id": "pair_7b29f58975bf3efd", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((E(Xf(X-1))))\\\\neF((λE(f(X))))", "inference_order": 1813} {"schema_version": "1.0", "item_id": "rcd2_71fa4d0a88309b7b", "pair_id": "pair_683f3014022bfe21", "statement_latex": "\\{(rank(T) + nullity(T))\\}=\\{(dim(V))\\}", "inference_order": 1814} {"schema_version": "1.0", "item_id": "rcd2_2fa263315a3976f3", "pair_id": "pair_f7b21b018e6dccc9", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((|F|)-(max_{r: t+2r ≤ n} |F_{n,k,t,r}|))=0", "inference_order": 1815} {"schema_version": "1.0", "item_id": "rcd2_decc735c497f3351", "pair_id": "pair_68ee169b9b4ae4a9", "statement_latex": "(\\dim(E^*)) \\in {(\\text{card}(\\mathbb{K}^I))}", "inference_order": 1816} {"schema_version": "1.0", "item_id": "rcd2_faa315f3a7a9c8cf", "pair_id": "pair_0d9e5643c8b56395", "statement_latex": "\\max\\{0,(d_Y(f(x), f(y)))-(ε)\\}=(0\\;\\land\\;(d_Y(f(x), f(y)))\\ne (ε)) + \\eta", "inference_order": 1817} {"schema_version": "1.0", "item_id": "rcd2_1ec90c2cc2eeb89f", "pair_id": "pair_a81c683ac76fb875", "statement_latex": "(N / t^2) - (Pr(sqrt((X-μ)^T * V^-1 * (X-μ)) > t)) \\ge 0", "inference_order": 1818} {"schema_version": "1.0", "item_id": "rcd2_cf12b9e3240deead", "pair_id": "pair_7a700503f81dc928", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (p(a,d))\\Rightarrow t<((c * d^L)) + \\eta", "inference_order": 1819} {"schema_version": "1.0", "item_id": "rcd2_f52910c9ef25febc", "pair_id": "pair_94c0f099de0db246", "statement_latex": "\\tanh(Pr[Intersection(X_i ≤0\\;\\land\\;(P_{n-1}(x)P_{n+1}(x))\\ne (P_n(x)^2)", "inference_order": 1820} {"schema_version": "1.0", "item_id": "rcd2_e107d9746c6b878e", "pair_id": "pair_ac63c54265fe44b3", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\int_{V} (\\nabla \\cdot \\mathbf{F}) dV))", "inference_order": 1821} {"schema_version": "1.0", "item_id": "rcd2_0b4084ae550b26dc", "pair_id": "pair_4e5ff4037f64f5e5", "statement_latex": "(0 (mod p^n)) = (f(x))", "inference_order": 1822} {"schema_version": "1.0", "item_id": "rcd2_8c40a070bc60595b", "pair_id": "pair_f431c511d91c78c1", "statement_latex": "\\{(x * (y * z) + y * (z * x) + z * (x * y))\\}=\\{(0)\\}", "inference_order": 1823} {"schema_version": "1.0", "item_id": "rcd2_ba78fd1785a63251", "pair_id": "pair_eff4e9e313f52575", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((1/(1-alpha*z)))=(F((product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}))) + \\eta", "inference_order": 1824} {"schema_version": "1.0", "item_id": "rcd2_f19c28495a7a9c45", "pair_id": "pair_a84876c548af586a", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(max_{r: t+2r ≤ n} |F_{n,k,t,r}|)\\le t\\Rightarrow (|F|)\\le t", "inference_order": 1825} {"schema_version": "1.0", "item_id": "rcd2_1ba34d11172bbc57", "pair_id": "pair_4559f90b503b45e0", "statement_latex": "\\forall z\\in D,\\;((∫_{γ} ∇φ(r) · dr),z)=((φ(q) - φ(p)),z)", "inference_order": 1826} {"schema_version": "1.0", "item_id": "rcd2_469f7d82e4f620fa", "pair_id": "pair_505e8d63edcbd0a4", "statement_latex": "\\neg\\left((S_T^n) \\in {(C_1 ∪ ... ∪ C_r)}\\right)", "inference_order": 1827} {"schema_version": "1.0", "item_id": "rcd2_884099ae89e156b1", "pair_id": "pair_ea95bf36467431eb", "statement_latex": "\\max\\{0,(p_{n+1}^2)-(p_n#)\\}=1}^r d_j)-(n)\\}=0\\;\\land\\;(sum_{j=1}^r d_j)\\ne (n)", "inference_order": 1828} {"schema_version": "1.0", "item_id": "rcd2_a11ace3716918907", "pair_id": "pair_d71ebd0879b11a85", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (E[vol_n(X)^(1/n)]) \\le x \\land x \\le ((vol_n(E[X]))^(1/n))", "inference_order": 1829} {"schema_version": "1.0", "item_id": "rcd2_aff78b3ee4c134d6", "pair_id": "pair_5168bcf668cab3b7", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\ge (a+b)^2 / (4ab)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (Σ p_i x_i) * (Σ p_i / x_i)\\}\\right)", "inference_order": 1830} {"schema_version": "1.0", "item_id": "rcd2_79d9cce6d53dc0bc", "pair_id": "pair_5f57452fd80102a7", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((lim_{N→inf} (1/(2N+1)) sum_{n=F((\\sup\\{f(x, \\theta) : x \\in C(\\theta)\\}))", "inference_order": 1831} {"schema_version": "1.0", "item_id": "rcd2_9e4c850b6373d8cc", "pair_id": "pair_374e6ccb333bb2aa", "statement_latex": "\\operatorname{dist}((max_{|z|=(1}|P'(z)|)-(n max_{|z|=1}|P(z)|),(-\\infty,0])=0) + \\eta", "inference_order": 1832} {"schema_version": "1.0", "item_id": "rcd2_986f5b990e782a17", "pair_id": "pair_c3d309feaf2f34f1", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(C |u|_{W^{m,p}(\\Omega)})\\le t\\Rightarrow (inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)})\\le t + \\eta", "inference_order": 1833} {"schema_version": "1.0", "item_id": "rcd2_a28a2ddb9397fdfc", "pair_id": "pair_d3437dba54f79f51", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(H)=z\\;\\land\\;z=(H_0 + gV)", "inference_order": 1834} {"schema_version": "1.0", "item_id": "rcd2_690ffce80d5a8a0f", "pair_id": "pair_14719f341355549f", "statement_latex": "(1) = (x^a - y^b)", "inference_order": 1835} {"schema_version": "1.0", "item_id": "rcd2_55c48d14e457a143", "pair_id": "pair_f88634c2a4afa900", "statement_latex": "\\forall m\\in\\mathbb{N}_{>0}\\exists N_m\\forall n\\ge N_m:(lim_{n)\\in B((∞} |S_n(β)|^(1/n)),1/m)", "inference_order": 1836} {"schema_version": "1.0", "item_id": "rcd2_94f33c43331854e4", "pair_id": "pair_137418446786214c", "statement_latex": "\\max\\{0,(|\\xi - m/n|)-(1 / (\\sqrt{5} * n^2))\\}=0\\;\\land\\;(p_{n+1}^2)\\ne (p_n#)", "inference_order": 1837} {"schema_version": "1.0", "item_id": "rcd2_cf1092f1cb4000ca", "pair_id": "pair_d6e0b0b00cca44a6", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(n)=z\\;\\land\\;z=(p1 + p2 + p3)", "inference_order": 1838} {"schema_version": "1.0", "item_id": "rcd2_903604a1d47d958d", "pair_id": "pair_7f5b2e81002b6a8c", "statement_latex": "\\exists s \\in \\mathbb{R}_{>(0}: (K(A) = sup_{|z|) = (1} (|z|-1) ||(z-A)^-1||) + s) + \\eta", "inference_order": 1839} {"schema_version": "1.0", "item_id": "rcd2_477601272abb7b72", "pair_id": "pair_e7df8f6b53d592b0", "statement_latex": "\\neg(\\neg (sum_{k=(0}^m c_k 1_{N=k} = sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k))) + \\eta", "inference_order": 1840} {"schema_version": "1.0", "item_id": "rcd2_29453fd483ba662f", "pair_id": "pair_db5c0ee214b85abe", "statement_latex": "\\exists c \\in D: (binom(m+n, k)_q) = (uv)", "inference_order": 1841} {"schema_version": "1.0", "item_id": "rcd2_c750863af7dac028", "pair_id": "pair_78be29b93d3819f4", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,([x^2 + y^2 + (x+y)^2]^2))+\\delta_D(z,(2[x^4 + y^4 + (x+y)^4])))=0", "inference_order": 1842} {"schema_version": "1.0", "item_id": "rcd2_a353a9af95c4895f", "pair_id": "pair_c314f9126c32d148", "statement_latex": "\\frac{\\exp(prod_{i} d_{i})}{\\exp(prod_{i} d_{i})+\\exp(Vol(M))}\\le\\frac{1}{2}", "inference_order": 1843} {"schema_version": "1.0", "item_id": "rcd2_5ab802b8a09b4284", "pair_id": "pair_af3054532798d450", "statement_latex": "\\forall z\\in D,\\;(z=(\\sum_{i=r}^{n} \\binom{i}{r})\\Longleftrightarrow z=(\\binom{n+1}{r+1}))", "inference_order": 1844} {"schema_version": "1.0", "item_id": "rcd2_5ed72a11b9890e90", "pair_id": "pair_e218da2a1eb4d109", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (N(X) + N(Y)) \\le x \\land x \\le (N(X+Y))", "inference_order": 1845} {"schema_version": "1.0", "item_id": "rcd2_b5de283aabc12eb5", "pair_id": "pair_e8f2a58c3f06a49c", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(x_i))+\\delta_D(z,(\\det(A_i) / \\det(A))))=0", "inference_order": 1846} {"schema_version": "1.0", "item_id": "rcd2_0f11809c073c65a2", "pair_id": "pair_36b45228d893c764", "statement_latex": "\\forall z\\in D,\\;(((x_{1}+x_{2}+\\cdots +x_{m})^{n}),z)=(((\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}),z)) + \\eta", "inference_order": 1847} {"schema_version": "1.0", "item_id": "rcd2_2a6e5fc8efa4c8e8", "pair_id": "pair_8de08bdcfcb842ee", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(P(S_n in A))+s=(binom(n,floor(n/2))/2^n)", "inference_order": 1848} {"schema_version": "1.0", "item_id": "rcd2_61d71babcecb614d", "pair_id": "pair_0874896b58313452", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (ζ(n) / 2^(n-1)) \\le x \\land x \\le (Δ)", "inference_order": 1849} {"schema_version": "1.0", "item_id": "rcd2_fe568632f2ab4763", "pair_id": "pair_00067de84bb18e39", "statement_latex": "\\forall z\\in D,\\;(z=((det(A))\\Longleftrightarrow z=(det(A_v^u) * det(A_tilde_v^u)))) + \\eta_{1373}", "inference_order": 1850} {"schema_version": "1.0", "item_id": "rcd2_7da2b5cc5ac51685", "pair_id": "pair_b5bd4e1dbf31333b", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (tr(exp(A)*exp(B)))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (tr(exp(A+B)))\\} + \\eta", "inference_order": 1851} {"schema_version": "1.0", "item_id": "rcd2_eff6c9dbc65bb116", "pair_id": "pair_197237c6d90221a9", "statement_latex": "\\neg\\left((V_1(K, L)) - (V(K)^((n-1)/n) V(L)^(1/n)) \\ge 0\\right)", "inference_order": 1852} {"schema_version": "1.0", "item_id": "rcd2_bfc69b1311bb3c2a", "pair_id": "pair_8d2f0ce9d4357360", "statement_latex": "\\{((n),z):z\\in D\\}=\\{(((r - 1)(s - 1) + 1),z):z\\in D\\}", "inference_order": 1853} {"schema_version": "1.0", "item_id": "rcd2_338164c54fc253cf", "pair_id": "pair_e9a28d2962be7d45", "statement_latex": "(\\{(ρ_W(g) ∘ f)\\}\\times\\{(f ∘ ρ_V(g))\\})\\cap\\{(z,z):z\\in D\\}=\\{((ρ_W(g) ∘ f),(f ∘ ρ_V(g)))\\}", "inference_order": 1854} {"schema_version": "1.0", "item_id": "rcd2_fd43ff4b857f5551", "pair_id": "pair_908290996865a0ea", "statement_latex": "\\forall z\\in D,\\;((lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)]),z)=((sup_{x∈X} (φ(x) - I(x))),z)", "inference_order": 1855} {"schema_version": "1.0", "item_id": "rcd2_6a7bed15371395f0", "pair_id": "pair_2773b6bbeedebafd", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2)))\\le t\\Rightarrow (P(X_N - X_0 < epsilon))\\le t", "inference_order": 1856} {"schema_version": "1.0", "item_id": "rcd2_e2392af8fed84e47", "pair_id": "pair_e10401eb98c36286", "statement_latex": "(\\{(rank(T) + nullity(T))\\}\\times\\{(dim(V))\\})\\cap\\{(z,z):z\\in D\\}\\\\ne\\{((rank(T) + nullity(T)),(dim(V)))\\}", "inference_order": 1857} {"schema_version": "1.0", "item_id": "rcd2_0f8624cc27e697cf", "pair_id": "pair_b83b9cbd4e911f3c", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(d/dt integral_{Omega(t)} f dV))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA))", "inference_order": 1858} {"schema_version": "1.0", "item_id": "rcd2_399272e1d9f6529c", "pair_id": "pair_6eb58c98c883891a", "statement_latex": "\\neg\\forall x\\in D,x\\notin \\{prime p\\in D:n < p < 2*n\\}", "inference_order": 1859} {"schema_version": "1.0", "item_id": "rcd2_b67f296d6919097f", "pair_id": "pair_b6e781e508b8c9ce", "statement_latex": "\\forall z\\in D,\\;((Ax),z)\\\\ne((0),z)", "inference_order": 1860} {"schema_version": "1.0", "item_id": "rcd2_5e2f163808974868", "pair_id": "pair_473a8de9b12f1e51", "statement_latex": "\\max\\{0,(e^3 / (29n^2))-(cr(G))\\}=0", "inference_order": 1861} {"schema_version": "1.0", "item_id": "rcd2_d8ecb2a101df34b7", "pair_id": "pair_afd68796dbb78e84", "statement_latex": "(a}) \\in (M^a =( {x \\in M | f(x)) + [0,\\infty)) + \\eta", "inference_order": 1862} {"schema_version": "1.0", "item_id": "rcd2_8c9824c4c90bb513", "pair_id": "pair_42c402e05f82fc97", "statement_latex": "\\neg(\\neg (ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) = 2^{n-1}∏_{i=1}^{n}ε(Xi)))", "inference_order": 1863} {"schema_version": "1.0", "item_id": "rcd2_a72ce4a5856bada6", "pair_id": "pair_817f725eb1a08f96", "statement_latex": "(Pr(S_n >( t)) - (\\exp(-(sigma^2 / a^2) * h(at / sigma^2))) \\le 0) + \\eta", "inference_order": 1864} {"schema_version": "1.0", "item_id": "rcd2_9b42300ea16d377d", "pair_id": "pair_c2ca52c4e476ceed", "statement_latex": "\\nexists S \\subseteq D: (((ω(-t)) \\in S \\land (ω(t)) \\notin S) \\lor ((ω(t)) \\in S \\land (ω(-t)) \\notin S))", "inference_order": 1865} {"schema_version": "1.0", "item_id": "rcd2_684ca1136ebf899f", "pair_id": "pair_e181737b0baa2dd9", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\ge (vol_n(E[X]))^(1/n)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (E[vol_n(X)^(1/n)])\\}\\right)", "inference_order": 1866} {"schema_version": "1.0", "item_id": "rcd2_d02a43ae50817b0b", "pair_id": "pair_1c02028771cc9bfe", "statement_latex": "\\neg(\\neg (f(x_1, ..., x_n) \\\\ne Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))))", "inference_order": 1867} {"schema_version": "1.0", "item_id": "rcd2_064bdc489c193261", "pair_id": "pair_1947ad972f7c1ba0", "statement_latex": "\\forall z\\in D,\\;(z=(N^{(i-1)})\\Longleftrightarrow z=(\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}))", "inference_order": 1868} {"schema_version": "1.0", "item_id": "rcd2_49a3fa2455b454cc", "pair_id": "pair_73bd98d264980c33", "statement_latex": "\\neg(\\neg (P(X ≥ a) > \\inf_{t>0} M(t)e^{-ta}))", "inference_order": 1869} {"schema_version": "1.0", "item_id": "rcd2_f5c840908d652b13", "pair_id": "pair_8e20151f31fc6528", "statement_latex": "\\neg\\left(\\forall F:D\\to\\{0,1\\},\\;F((U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C))=F((|\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C))\\right)", "inference_order": 1870} {"schema_version": "1.0", "item_id": "rcd2_58fc03d0bb1990af", "pair_id": "pair_306bdc1dcf13165e", "statement_latex": "\\{((\\dot{q}),z):z\\in D\\}=(\\{((f(q,u)),z):z\\in D\\}) + \\eta", "inference_order": 1871} {"schema_version": "1.0", "item_id": "rcd2_fa6ee53dd91495f0", "pair_id": "pair_f869b06a961e0d31", "statement_latex": "\\{((k e_k),z):z\\in D\\}=(\\{((\\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i),z):z\\in D\\}) + \\eta", "inference_order": 1872} {"schema_version": "1.0", "item_id": "rcd2_565420eeb2d0218c", "pair_id": "pair_fa8faad83395d383", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (γ) \\le x \\land x \\le (ε^h / 4h)) + \\eta", "inference_order": 1873} {"schema_version": "1.0", "item_id": "rcd2_570a1a3a1885a1ab", "pair_id": "pair_7b420fc2c9bb2375", "statement_latex": "(chi'_l(K_{n,n})) \\in {(n)}", "inference_order": 1874} {"schema_version": "1.0", "item_id": "rcd2_2f5a2230aebd335e", "pair_id": "pair_aca411f09c7874cf", "statement_latex": "(\\infty) \\\\ne (sum_{n \\in S} (1/n))", "inference_order": 1875} {"schema_version": "1.0", "item_id": "rcd2_5ef06835d4720580", "pair_id": "pair_414a08db6e185d76", "statement_latex": "\\forall r\\in\\mathbb{Q}_{>0", "inference_order": 1876} {"schema_version": "1.0", "item_id": "rcd2_c442f2ec8840e6e3", "pair_id": "pair_970b6e0da7498d68", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (tr(\\exp(A+B))) \\le x \\land x \\le (tr(\\exp(A)*\\exp(B)))", "inference_order": 1877} {"schema_version": "1.0", "item_id": "rcd2_27ddaf65cf6dc854", "pair_id": "pair_97790b8d57345d2e", "statement_latex": "(0) = (x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2})", "inference_order": 1878} {"schema_version": "1.0", "item_id": "rcd2_cf7d0e71842ae79c", "pair_id": "pair_1d98097899389158", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((\\dot{q}) \\in S \\land (f(q,u)) \\notin S) \\lor ((f(q,u)) \\in S \\land (\\dot{q}) \\notin S))\\right)", "inference_order": 1879} {"schema_version": "1.0", "item_id": "rcd2_4dddd4599bdd5811", "pair_id": "pair_3d4972614fe63654", "statement_latex": "\\{((φ(+∞) − φ(0)),z):z\\in D\\}=\\{((\\text{conv}\\{w(X) : w \\in W\\}),z):z\\in D\\}", "inference_order": 1880} {"schema_version": "1.0", "item_id": "rcd2_2ccb94dbd051e767", "pair_id": "pair_af63bd76204893fb", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\binom{k}{r} \\frac{r}{k}))\\\\ne\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\binom{k-1}{r-1}))", "inference_order": 1881} {"schema_version": "1.0", "item_id": "rcd2_3c52db1245049b73", "pair_id": "pair_0b7617c7fcdadec3", "statement_latex": "\\neg(\\neg (P(X \\ge a) \\le E[X] / a)) + \\eta", "inference_order": 1882} {"schema_version": "1.0", "item_id": "rcd2_0114afaad7856d42", "pair_id": "pair_902a753f6aa1f98c", "statement_latex": "\\neg\\left(\\exists x \\\\in \\mathbb{R}: (3/2) \\le x \\land x \\le (a/(b+c) + b/(a+c) + c/(a+b))\\right)", "inference_order": 1883} {"schema_version": "1.0", "item_id": "rcd2_4556cce9593f4f06", "pair_id": "pair_93d8a96189e9f4e0", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (dim_C H^i(X, F))\\Rightarrow t<((infinity)) + \\eta", "inference_order": 1884} {"schema_version": "1.0", "item_id": "rcd2_bb69fe3ed7188303", "pair_id": "pair_b6e781e508b8c9ce", "statement_latex": "\\forall z\\in D,\\;((Ax),z)=((0),z)", "inference_order": 1885} {"schema_version": "1.0", "item_id": "rcd2_f03bab254dac1ad8", "pair_id": "pair_f4cb66a7b8057115", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(z=(integral integral_{Sigma} (curl F) dot dSigma)\\Longleftrightarrow z=(integral_{partial Sigma} F dot dr))\\right)", "inference_order": 1886} {"schema_version": "1.0", "item_id": "rcd2_233f194bcc675ec7", "pair_id": "pair_1e79477188b29c98", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\le (\\sum_{n=0}^{\\infty} n|a_n|^2)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (1)\\}\\right)", "inference_order": 1887} {"schema_version": "1.0", "item_id": "rcd2_a786d18fc023cf5a", "pair_id": "pair_fee2b12a5fce58e1", "statement_latex": "\\forall z\\in D,\\;((f'(x)),z)=((y),z)", "inference_order": 1888} {"schema_version": "1.0", "item_id": "rcd2_9d4ab1bd41b4534d", "pair_id": "pair_26cecaf933bb5c40", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(min(|A|+|B|-1, p))+s=(|A+B|)", "inference_order": 1889} {"schema_version": "1.0", "item_id": "rcd2_db129acb42e793d7", "pair_id": "pair_f49cc242ed9684b3", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((w(S)))=F((sum_{x in S} w(x)))", "inference_order": 1890} {"schema_version": "1.0", "item_id": "rcd2_c1a5cce5015a50b4", "pair_id": "pair_d46b6c5f8aeaa4b2", "statement_latex": "\\forall z\\in D,\\;((F),z)=((-V'(x)),z)", "inference_order": 1891} {"schema_version": "1.0", "item_id": "rcd2_2e5d70e76492457a", "pair_id": "pair_bc4cf7a664ca1962", "statement_latex": "\\operatorname{softplus}((AC * BD)-(AB * CD + BC * DA))\\le\\log 2 + \\eta", "inference_order": 1892} {"schema_version": "1.0", "item_id": "rcd2_90f7e0a2fcb932bb", "pair_id": "pair_a62523058af97a53", "statement_latex": "\\{(\\sum_{q=(1}^{\\infty}\\varphi(q)\\frac{f(q)}{q})\\}=\\{(\\infty)\\}) + \\eta", "inference_order": 1893} {"schema_version": "1.0", "item_id": "rcd2_69fb3f8b391b872a", "pair_id": "pair_4a0c38e00c95f935", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((sum_{n in S} (1/n)))=F(((-1)^n))", "inference_order": 1894} {"schema_version": "1.0", "item_id": "rcd2_34ea45803ecc27bc", "pair_id": "pair_2a9de60efa410989", "statement_latex": "\\neg\\left(\\exists z\\in D\\;\\text{such that}\\;((x_{1}+x_{2}+\\cdots +x_{m})^{n})=z\\;\\land\\;z=(\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}})\\right)", "inference_order": 1895} {"schema_version": "1.0", "item_id": "rcd2_59ffa794b1b0fc28", "pair_id": "pair_bfeb4ccea4d331af", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (|g − f|) < x \\land x < (ε)", "inference_order": 1896} {"schema_version": "1.0", "item_id": "rcd2_5e6ebce1d593e46b", "pair_id": "pair_e13dfef30b697bce", "statement_latex": "((1/x) * sum_{n\\lex} |f(n) - A(x)|^2 \\le (2 + \\varepsilon(x)) * B(x)^2) \\land (0 \\\\ne 0)", "inference_order": 1897} {"schema_version": "1.0", "item_id": "rcd2_65b0dd81bde89cb0", "pair_id": "pair_3bd95bc2272c7240", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((E(T)) \\in S \\land (n H_n) \\notin S) \\lor ((n H_n) \\in S \\land (E(T)) \\notin S))\\right)", "inference_order": 1898} {"schema_version": "1.0", "item_id": "rcd2_6bad360f9ff91e22", "pair_id": "pair_4ccc92c3a48d145d", "statement_latex": "\\neg\\left(\\{(z,(lim_{N→inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2)):z\\in D\\}=\\{(z,(sum_{j} |c_{j}|^2)):z\\in D\\}\\right)", "inference_order": 1899} {"schema_version": "1.0", "item_id": "rcd2_d64809a4cf732c0c", "pair_id": "pair_5f0b71169f989353", "statement_latex": "\\operatorname{dist}((Pr(X - E[X] < λ))-(σ^2 / (σ^2 + λ^2)),(-\\infty,0])=0", "inference_order": 1900} {"schema_version": "1.0", "item_id": "rcd2_4c78bce5fc70cce2", "pair_id": "pair_a693acf669c6e191", "statement_latex": "(|\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C) = (V(p))", "inference_order": 1901} {"schema_version": "1.0", "item_id": "rcd2_c9120a719f8fb7d8", "pair_id": "pair_dd6174baa4693939", "statement_latex": "\\nexists S \\subseteq D: (((chi'_l(K_{n,n})) \\in S \\land (n) \\notin S) \\lor ((n) \\in S \\land (chi'_l(K_{n,n})) \\notin S))", "inference_order": 1902} {"schema_version": "1.0", "item_id": "rcd2_3c62d3ba713a12c9", "pair_id": "pair_9cd66dae5c8badea", "statement_latex": "\\neg\\left((z_1^2 + z_2^2 + ... + z_{16}^2) = ((x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2))\\right)", "inference_order": 1903} {"schema_version": "1.0", "item_id": "rcd2_4cb6986228ff3daa", "pair_id": "pair_41f3f45ce3b64178", "statement_latex": "\\forall z\\in D,\\;((rank(A)),z)=((rank([A|b])),z)", "inference_order": 1904} {"schema_version": "1.0", "item_id": "rcd2_e4a323b1dcf865cc", "pair_id": "pair_7d8e2a781ec6ca33", "statement_latex": "-(K(A) = sup_{|z|) < -(1} (|z|-1) ||(z-A)^-1||)", "inference_order": 1905} {"schema_version": "1.0", "item_id": "rcd2_78deff716579e260", "pair_id": "pair_3f5e811e609b30d4", "statement_latex": "(0) = (x * (y * z) + y * (z * x) + z * (x * y))", "inference_order": 1906} {"schema_version": "1.0", "item_id": "rcd2_a806c3cddb1f1ef5", "pair_id": "pair_3e4a8382500f0cbc", "statement_latex": "(λE(f(X))) = (E(Xf(X-1)))", "inference_order": 1907} {"schema_version": "1.0", "item_id": "rcd2_978741094d647b86", "pair_id": "pair_1bfb5808342fba91", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(F(X)))+\\delta_D(z,(inf_S sup_I (|I|/|S|))))=(0) + \\eta", "inference_order": 1908} {"schema_version": "1.0", "item_id": "rcd2_e5e0eca5e4e7de72", "pair_id": "pair_7d5e55987f94dfcb", "statement_latex": "\\forall k\\in\\mathbb{N}_{>0}\\exists N\\forall n\\ge N:d((S_βα),(S_β1α1 * S_β2α2))<1/k", "inference_order": 1909} {"schema_version": "1.0", "item_id": "rcd2_45307f61cce73a11", "pair_id": "pair_1ddbab4a4f3f9fb7", "statement_latex": "\\{((max_matching_size),z):z\\in D\\}=\\{((xy),z):z\\in D\\}", "inference_order": 1910} {"schema_version": "1.0", "item_id": "rcd2_74807724c1680ac1", "pair_id": "pair_c568d210759c5fd6", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(x * (y * z) + y * (z * x) + z * (x * y))\\\\nez\\;\\land\\;z=(0)", "inference_order": 1911} {"schema_version": "1.0", "item_id": "rcd2_0315a7307fe31636", "pair_id": "pair_f9a232b6cd042b3a", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4))\\}", "inference_order": 1912} {"schema_version": "1.0", "item_id": "rcd2_5b023de788e96cb3", "pair_id": "pair_7e90cea0e2284dba", "statement_latex": "(binom(m+n, k)_q) \\in {(sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)})}", "inference_order": 1913} {"schema_version": "1.0", "item_id": "rcd2_f836b9749f40cb1e", "pair_id": "pair_7f85163e5a98af88", "statement_latex": "\\forall z\\in D,\\;((lim_{N→inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2),z)=((sum_{j} |c_{j}|^2),z)", "inference_order": 1914} {"schema_version": "1.0", "item_id": "rcd2_1685d430ada7d3c2", "pair_id": "pair_9d606e5f6e26d4d0", "statement_latex": "\\{(B)\\}\\\\ne\\{(SAS^T)\\}", "inference_order": 1915} {"schema_version": "1.0", "item_id": "rcd2_1dd558d1fc5193d4", "pair_id": "pair_be65067772542678", "statement_latex": "\\forall z\\in D,\\;((P(A ∩ B)),z)=((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)),z)", "inference_order": 1916} {"schema_version": "1.0", "item_id": "rcd2_4d5cd037c5142016", "pair_id": "pair_8a4d879c421bc411", "statement_latex": "\\operatorname{dist}((E[vol_n(X)^(1/n)])-(vol_n(E[X]))^(1/n),(-\\infty,0])=0", "inference_order": 1917} {"schema_version": "1.0", "item_id": "rcd2_5f8c10395ed3fac7", "pair_id": "pair_59c6d4998ca96891", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (exp(-(sigma^2 / a^2) * h(at / sigma^2)))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (Pr(S_n > t))\\}", "inference_order": 1918} {"schema_version": "1.0", "item_id": "rcd2_9a6b1a387ebda813", "pair_id": "pair_f4dbabb3c7345738", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;([a])\\le t\\Rightarrow ([b] for a majorizing b)\\le t\\right)", "inference_order": 1919} {"schema_version": "1.0", "item_id": "rcd2_ef2c122056fa57e0", "pair_id": "pair_39e03e5d7925ce72", "statement_latex": "\\neg\\left(\\frac{\\exp(sum_{k=0}^n a_k/binom(n,k))}{\\exp(sum_{k=0}^n a_k/binom(n,k))+\\exp(1)}\\le\\frac{1}{2}\\right)", "inference_order": 1920} {"schema_version": "1.0", "item_id": "rcd2_e4455088088ee3b9", "pair_id": "pair_a41af95998f42bee", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((a/p))\\\\neF((epsilon(pi_a)))", "inference_order": 1921} {"schema_version": "1.0", "item_id": "rcd2_451c3a4a04197187", "pair_id": "pair_f10ded29be6cb1bc", "statement_latex": "\\neg\\left(\\{((rank(T) + nullity(T)),z):z\\in D\\}=\\{((dim(V)),z):z\\in D\\}\\right)", "inference_order": 1922} {"schema_version": "1.0", "item_id": "rcd2_3ec6a643f47a1bed", "pair_id": "pair_f94d94983e7682e6", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (ε||x||_X + C(ε)||x||_Z)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (||x||_Y)\\} + \\eta", "inference_order": 1923} {"schema_version": "1.0", "item_id": "rcd2_3c99b65646cd5a84", "pair_id": "pair_9b583072ba908615", "statement_latex": "{(f(s(x)))} \\\\ne {(s(f(x)))}", "inference_order": 1924} {"schema_version": "1.0", "item_id": "rcd2_6b3934d55292fb7e", "pair_id": "pair_5385f4e37f8be586", "statement_latex": "\\{((Ax),z):z\\in D\\}=\\{((0),z):z\\in D\\}", "inference_order": 1925} {"schema_version": "1.0", "item_id": "rcd2_f76c548144ed8a19", "pair_id": "pair_bac6da57ddaf29e7", "statement_latex": "\\delta_D((E),(F(α)))=0", "inference_order": 1926} {"schema_version": "1.0", "item_id": "rcd2_4b6606b13db175c1", "pair_id": "pair_2b6a084fceed0f66", "statement_latex": "\\frac{\\exp(||h||_1)}{\\exp(||h||_1)+\\exp(2^n * ||f||_1)}\\le\\frac{1}{2} + \\eta", "inference_order": 1927} {"schema_version": "1.0", "item_id": "rcd2_816a334f7186db49", "pair_id": "pair_bd2566a0b2c80305", "statement_latex": "\\delta_D((x + y),(z))=0", "inference_order": 1928} {"schema_version": "1.0", "item_id": "rcd2_2fe8bcfaf55d0a59", "pair_id": "pair_a0c36ec0a7996ca0", "statement_latex": "(n) = (chi'_l(K_{n,n}))", "inference_order": 1929} {"schema_version": "1.0", "item_id": "rcd2_f841faaa26d2852b", "pair_id": "pair_e0d345690135d662", "statement_latex": "\\neg\\left(\\exists c \\in D: (A(x)) = c \\land ((1 / (2*\\pi*i)) * integral_{c-i*\\infty}^{c+i*\\infty} g(z) * (x^z / z) dz) = c\\right)", "inference_order": 1930} {"schema_version": "1.0", "item_id": "rcd2_6af427263bc3c0dc", "pair_id": "pair_4ee6663dec548147", "statement_latex": "(\\exp(X_t - 1/2 [X]_t)) = (E(X)_t)", "inference_order": 1931} {"schema_version": "1.0", "item_id": "rcd2_f7ff13fc4dc34fd4", "pair_id": "pair_cb7eb92aa7efd83b", "statement_latex": "\\delta_D((B),(SAS^T))=0", "inference_order": 1932} {"schema_version": "1.0", "item_id": "rcd2_1aa9903bc437b253", "pair_id": "pair_0f26f316e9b7b3e0", "statement_latex": "((d/dx integral_{a(x)}^{b(x)} f(x,t) dt),(f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt))\\in\\{(z,z):z\\in D\\}", "inference_order": 1933} {"schema_version": "1.0", "item_id": "rcd2_b10f5d086a7c4aed", "pair_id": "pair_d7f02bc0955f68c8", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((P(A ∩ B)))=F((P(B|A)*P(A)))", "inference_order": 1934} {"schema_version": "1.0", "item_id": "rcd2_b8ccc6f4fe5d4ce0", "pair_id": "pair_1a44c0b06b7bc4a4", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge ((x + y)/2)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (sqrt(x*y))\\}", "inference_order": 1935} {"schema_version": "1.0", "item_id": "rcd2_ab2ae4d13aa1be74", "pair_id": "pair_fa5d10ad37bdf5d2", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\ge (2^{2^{r^{2^{2^{k+9}}}}})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (W(r,k))\\}\\right)", "inference_order": 1936} {"schema_version": "1.0", "item_id": "rcd2_fea84f2021d38582", "pair_id": "pair_bdc3c65ce4bd2baa", "statement_latex": "\\neg(\\neg (Σ_{v∈V} deg(v) =-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3))", "inference_order": 1937} {"schema_version": "1.0", "item_id": "rcd2_7e3049635da64a19", "pair_id": "pair_e61b90b39de947fa", "statement_latex": "(e(X,Y)) - (delta |X| |Y|) > 0", "inference_order": 1938} {"schema_version": "1.0", "item_id": "rcd2_3b12e1a895ef76a0", "pair_id": "pair_5dee6fc9bebf3dfc", "statement_latex": "\\frac{\\exp(4πA)}{\\exp(4πA)+\\exp(L^2)}\\le\\frac{1}{2}", "inference_order": 1939} {"schema_version": "1.0", "item_id": "rcd2_5d09f419fdb15695", "pair_id": "pair_a8a83e584dfdde93", "statement_latex": "\\nexists S \\subseteq D: (((x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2}) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2}) \\notin S)) + \\eta", "inference_order": 1940} {"schema_version": "1.0", "item_id": "rcd2_966aae52c93603bd", "pair_id": "pair_0495db7bad01bf92", "statement_latex": "\\forall z\\in D,\\;((lim_{n→∞} (a_n / n)),z)=((inf_{n∈N} (a_n / n)),z)", "inference_order": 1941} {"schema_version": "1.0", "item_id": "rcd2_a1bb1c5ca6d2fcf2", "pair_id": "pair_a861459efe315248", "statement_latex": "\\neg(\\neg (sum_{k=m}^{n} f_k \\Delta g_k = (f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} \\Delta f_k))", "inference_order": 1942} {"schema_version": "1.0", "item_id": "rcd2_e7e6df1a777c6c8e", "pair_id": "pair_80659b90a950ff2a", "statement_latex": "\\neg(\\neg (Pr(X - E[X] ≥ λ) > σ^2 / (σ^2 + λ^2)))", "inference_order": 1943} {"schema_version": "1.0", "item_id": "rcd2_4213f7cc6e2a17c0", "pair_id": "pair_1fadf2b55bd0fb91", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((F(X)))=F((a_0 + a_1X + ... + a_nX^n))", "inference_order": 1944} {"schema_version": "1.0", "item_id": "rcd2_6a53a4def76f6b55", "pair_id": "pair_27d924dee9fa52a7", "statement_latex": "\\forall z\\in D,\\;((d/dt integral_{Omega(t)} f dV),z)=((integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA),z)", "inference_order": 1945} {"schema_version": "1.0", "item_id": "rcd2_7a3dc061b89e715b", "pair_id": "pair_d8f24c25267fc175", "statement_latex": "\\{(nullity A)\\}=\\{(nullity H)\\}", "inference_order": 1946} {"schema_version": "1.0", "item_id": "rcd2_0526bbd943056984", "pair_id": "pair_14413bc1dfa1153b", "statement_latex": "(n)\\mid((sum_{d|n} mu(d)*a_{n/d})-(0)) + \\eta", "inference_order": 1947} {"schema_version": "1.0", "item_id": "rcd2_c91439a8b8007ad9", "pair_id": "pair_9f6527e6ecc71517", "statement_latex": "\\max\\{0,(|e(S,T) - (d/n)|S||T||)-(λ√|S||T|)\\}=0", "inference_order": 1948} {"schema_version": "1.0", "item_id": "rcd2_52727617d9d4901d", "pair_id": "pair_0b390ab6198b4242", "statement_latex": "(f'(x)Δx + εΔx) \\\\ne (Δy)", "inference_order": 1949} {"schema_version": "1.0", "item_id": "rcd2_1006fc4babb54512", "pair_id": "pair_93f9d6b7f61bd487", "statement_latex": "(N(X+Y)) - (N(X) + N(Y)) \\ge 0", "inference_order": 1950} {"schema_version": "1.0", "item_id": "rcd2_04b23a15e56fe89c", "pair_id": "pair_57f910e14cb9f634", "statement_latex": "\\nexists S \\subseteq D: (((-z\\eta'(s)/z\\eta(s)) \\in S \\land (F(s) - z\\eta(s)F(s)G(s) - z\\eta'(s)G(s) + (-z\\eta'(s)/z\\eta(s) - F(s))(1 - z\\eta(s)G(s))) \\notin S) \\lor ((F(s) - z\\eta(s)F(s)G(s) - z\\eta'(s)G(s) + (-z\\eta'(s)/z\\eta(s) - F(s))(1 - z\\eta(s)G(s))) \\in S \\land (-z\\eta'(s)/z\\eta(s)) \\notin S)) + \\eta", "inference_order": 1951} {"schema_version": "1.0", "item_id": "rcd2_aa83eea0deaff087", "pair_id": "pair_420a41a5c6a16a10", "statement_latex": "\\neg\\left(\\forall F:D\\to\\{0,1\\},\\;F((\\lim_{\\theta → \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx))=F((-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)))\\right)", "inference_order": 1952} {"schema_version": "1.0", "item_id": "rcd2_f4ee3f0ef34141e0", "pair_id": "pair_0b79ed522f2c4f08", "statement_latex": "\\forall z\\in D,\\;((ρ_W(g) ∘ f),z)=((f ∘ ρ_V(g)),z)", "inference_order": 1953} {"schema_version": "1.0", "item_id": "rcd2_5f1dab8f8e2f733b", "pair_id": "pair_f239aa277dd76f25", "statement_latex": "\\forall z\\in D,\\;((∮_{C}(L dx + M dy)),z)=((∬_{D} (∂M/∂x - ∂L/∂y) dA),z)", "inference_order": 1954} {"schema_version": "1.0", "item_id": "rcd2_6f649c3f14a0f12d", "pair_id": "pair_70586b2f3cce3afe", "statement_latex": "((-1)^n) = (x * (y * z) + y * (z * x) + z * (x * y))", "inference_order": 1955} {"schema_version": "1.0", "item_id": "rcd2_db10902a96b7b3cc", "pair_id": "pair_ef4a40a8f58ebe6e", "statement_latex": "(\\exp(-(sigma^2 / a^2) * h(at / sigma^2))) - (Pr(S_n < t)) \\ge 0", "inference_order": 1956} {"schema_version": "1.0", "item_id": "rcd2_373778705e282965", "pair_id": "pair_29a1903cc9c3b540", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(L^2)\\le t\\Rightarrow (4πA)\\le t", "inference_order": 1957} {"schema_version": "1.0", "item_id": "rcd2_c13646015b80b48c", "pair_id": "pair_6199078e33ed4d2c", "statement_latex": "\\max\\{0,(∫_X (v - u) dμ)-(ε)\\}=0\\;\\land\\;(∫_X (v - u) dμ)\\ne (ε)", "inference_order": 1958} {"schema_version": "1.0", "item_id": "rcd2_bb88a5ab805380e9", "pair_id": "pair_64196069f8a5e0d0", "statement_latex": "\\forall z\\in D,\\;((P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n)),z)=((1),z)", "inference_order": 1959} {"schema_version": "1.0", "item_id": "rcd2_4cbf983e9f1a396b", "pair_id": "pair_7062b799b24d83f2", "statement_latex": "(C_{r,\\varepsilon} H^{1-1/r} q^{(r+1)/(4r^2)+\\varepsilon}) \\in (|S_chi(N,H)|) + [0,\\infty)", "inference_order": 1960} {"schema_version": "1.0", "item_id": "rcd2_e9e6ab86d8919fcc", "pair_id": "pair_bd5b661f6a1e320c", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (2x / (phi(q) log(x/q)))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (pi(x;q,a))\\} + \\eta", "inference_order": 1961} {"schema_version": "1.0", "item_id": "rcd2_2848f450efce0587", "pair_id": "pair_b6739acdc3cc08dc", "statement_latex": "(0 = 0) \\Rightarrow (Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2))", "inference_order": 1962} {"schema_version": "1.0", "item_id": "rcd2_e1af51bba914c37f", "pair_id": "pair_ea8fdb7218dc53b4", "statement_latex": "\\nexists S \\subseteq D: (((\\bigcap _{k) \\in S \\land (0}^{\\infty }C_{k}\\neq \\\\emptyset) \\notin S) \\lor ((0}^{\\infty }C_{k}\\neq \\\\emptyset) \\in S \\land (\\bigcap _{k) \\notin S))", "inference_order": 1963} {"schema_version": "1.0", "item_id": "rcd2_8a6afa5f28bfba70", "pair_id": "pair_450047a5490e05bc", "statement_latex": "(j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)) - ((sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i) > 0", "inference_order": 1964} {"schema_version": "1.0", "item_id": "rcd2_c6576a5e393cae13", "pair_id": "pair_6ef4d0833a95d07b", "statement_latex": "\\forall z\\in D,\\;(z=((A)\\Longleftrightarrow z=(k[y_1, ..., y_m]))) + \\eta_{1053}", "inference_order": 1965} {"schema_version": "1.0", "item_id": "rcd2_8908fdc6ffad98bf", "pair_id": "pair_9eeea7927127851c", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (|alpha - beta|)\\Rightarrow t>(|alpha - alpha_i|)", "inference_order": 1966} {"schema_version": "1.0", "item_id": "rcd2_32b951eb38ddf2e9", "pair_id": "pair_8809ccb51bf62476", "statement_latex": "\\forall z\\in D,\\;(z\\\\ne({m+n \\choose r})\\Longleftrightarrow z=(\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}))", "inference_order": 1967} {"schema_version": "1.0", "item_id": "rcd2_00a339cc5d92e99c", "pair_id": "pair_10a67bb0f8a3d4f6", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge ((1 + ‖P‖) inf_{u ∈ U} ‖v - u‖)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (‖v - Pv‖)\\}", "inference_order": 1968} {"schema_version": "1.0", "item_id": "rcd2_6adc29e713cc8b7e", "pair_id": "pair_dc49801aba341177", "statement_latex": "(f) \\in {(g ∘ T)}", "inference_order": 1969} {"schema_version": "1.0", "item_id": "rcd2_faac4aad7480e56f", "pair_id": "pair_bfeb4ccea4d331af", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (|g − f|) < x \\land x < (ε)) + \\eta_{1113}", "inference_order": 1970} {"schema_version": "1.0", "item_id": "rcd2_5da63915ee9a539a", "pair_id": "pair_985c72ef90f37f3d", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(H))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(H_0 + gV))", "inference_order": 1971} {"schema_version": "1.0", "item_id": "rcd2_9ff65c5ae2c9252a", "pair_id": "pair_2e5faa5aa8b7a037", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (AB * CD + BC * DA)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (AC * BD)\\} + \\eta", "inference_order": 1972} {"schema_version": "1.0", "item_id": "rcd2_e2acf071f89adf73", "pair_id": "pair_1556945bb9b013fc", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (x(A) * Product_{B in Gamma(A)} (1 - x(B)))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (Pr[A])\\} + \\eta", "inference_order": 1973} {"schema_version": "1.0", "item_id": "rcd2_ed1b6d93aeaf47c3", "pair_id": "pair_e012754db701b114", "statement_latex": "\\nexists S \\subseteq D: (((Ax) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (Ax) \\notin S))", "inference_order": 1974} {"schema_version": "1.0", "item_id": "rcd2_04a375dd31cb4e23", "pair_id": "pair_095631fc29fe23cf", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (cr(G))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (e^3 / (29n^2))\\}", "inference_order": 1975} {"schema_version": "1.0", "item_id": "rcd2_4f8a8e15a9039133", "pair_id": "pair_bf037c33d49d7c55", "statement_latex": "\\delta_D((\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S}),(\\int_{V} (\\nabla \\cdot \\mathbf{F}) dV))=0", "inference_order": 1976} {"schema_version": "1.0", "item_id": "rcd2_fa497770d8bd57a7", "pair_id": "pair_f4dbabb3c7345738", "statement_latex": "\\forall t\\in\\mathbb{R},\\;([a])\\le t\\Rightarrow ([b] for a majorizing b)\\le t", "inference_order": 1977} {"schema_version": "1.0", "item_id": "rcd2_357ecb3ed435909d", "pair_id": "pair_b53a1a1eb589197e", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((E[sum_{x in N} f(x)]))=F((integral f(x) dLambda(x)))", "inference_order": 1978} {"schema_version": "1.0", "item_id": "rcd2_d8deb0b4c9574154", "pair_id": "pair_b45782af4a3ca333", "statement_latex": "\\delta_D((f'(x_0)),(0))=(0) + \\eta", "inference_order": 1979} {"schema_version": "1.0", "item_id": "rcd2_5c04a70c58722585", "pair_id": "pair_9406946a3fa798f0", "statement_latex": "\\frac{\\exp(tr(exp(A+B)))}{\\exp(tr(exp(A+B)))+\\exp(tr(exp(A)*exp(B)))}\\le\\frac{1}{2}", "inference_order": 1980} {"schema_version": "1.0", "item_id": "rcd2_eb9d7c2bb5fe8872", "pair_id": "pair_7e78193e82ce3190", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((E(Xf(X-1))) \\in S \\land (λE(f(X))) \\notin S) \\lor ((λE(f(X))) \\in S \\land (E(Xf(X-1))) \\notin S))\\right)", "inference_order": 1981} {"schema_version": "1.0", "item_id": "rcd2_7bd0f94468d7aafd", "pair_id": "pair_31173860f71db532", "statement_latex": "\\neg\\left(\\delta_D((d/dx integral_{a(x)}^{b(x)} f(x,t) dt),(f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt))=0\\right)", "inference_order": 1982} {"schema_version": "1.0", "item_id": "rcd2_8b256e079cd5bb2e", "pair_id": "pair_ec1c8a1691508201", "statement_latex": "\\neg(\\neg (Pr(max_{1\\lek\\len}|S_{k}|\\geλ) \\le (1/λ^2) * Var[S_{n}])) + \\eta", "inference_order": 1983} {"schema_version": "1.0", "item_id": "rcd2_c2158997ade2409e", "pair_id": "pair_16e16212b8b7ee5a", "statement_latex": "\\forall x\\in D,\\;x\\in (ker(d(f|_Y)_y))\\Rightarrow x\\in (tau)", "inference_order": 1984} {"schema_version": "1.0", "item_id": "rcd2_890d41b21eb77f05", "pair_id": "pair_9d0fbc57dbfd60a9", "statement_latex": "\\delta_D((w(AB|X)),(w(A|X)w(B|AX)))=(0) + \\eta", "inference_order": 1985} {"schema_version": "1.0", "item_id": "rcd2_b18b4416795d6bdf", "pair_id": "pair_ff2ce9987b12c2b0", "statement_latex": "\\max\\{0,(sum_{j=(1}^r d_j)-(n)\\}=0\\;\\land\\;(sum_{j=1}^r d_j)\\ne (n)) + \\eta", "inference_order": 1986} {"schema_version": "1.0", "item_id": "rcd2_cf635a0cd0e525e5", "pair_id": "pair_4c65e29151a0837a", "statement_latex": "\\neg\\left(\\forall F:D\\to\\{0,1\\},\\;F((a^2 - b^2))=F((a + b)(a - b))\\right)", "inference_order": 1987} {"schema_version": "1.0", "item_id": "rcd2_c17767b943792a36", "pair_id": "pair_b2410a12844cf4fa", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((e^3 / (29n^2))-(cr(G)))=0", "inference_order": 1988} {"schema_version": "1.0", "item_id": "rcd2_bddc14e401da81b2", "pair_id": "pair_03802f9e03945001", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(a/p))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(epsilon(pi_a)))", "inference_order": 1989} {"schema_version": "1.0", "item_id": "rcd2_c67ccd17c2f689fa", "pair_id": "pair_0824cb01884e63ec", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(1/k^2)\\le t\\Rightarrow (Pr(|X-mu| < k*sigma))\\le t", "inference_order": 1990} {"schema_version": "1.0", "item_id": "rcd2_20c672106bb82af9", "pair_id": "pair_710a6dbbaaa7220a", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(z=(\\lim_{\\theta → \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx)\\Longleftrightarrow z=(-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)))\\right)", "inference_order": 1991} {"schema_version": "1.0", "item_id": "rcd2_54e05e5c6e08af3c", "pair_id": "pair_b2d6360d4da2fc30", "statement_latex": "(a^2 + b^2 + c^2 + d^2) \\\\ne (p)", "inference_order": 1992} {"schema_version": "1.0", "item_id": "rcd2_75db26260e8c77f3", "pair_id": "pair_3f3dd2abfb9886a3", "statement_latex": "(\\{(\\binom{k}{r} \\frac{r}{k})\\}\\times\\{(\\binom{k-1}{r-1})\\})\\cap\\{(z,z):z\\in D\\}=\\{((\\binom{k}{r} \\frac{r}{k}),(\\binom{k-1}{r-1}))\\}", "inference_order": 1993} {"schema_version": "1.0", "item_id": "rcd2_489fc155624f3a59", "pair_id": "pair_e5a39fb73d411779", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\le (1)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2|)\\}\\right)", "inference_order": 1994} {"schema_version": "1.0", "item_id": "rcd2_b61c40e2f7a3951a", "pair_id": "pair_a2255d6e04fe6aa5", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(max(|A + A|, |A * A|))\\le t\\Rightarrow (c|A|^{1 + ε})\\le t", "inference_order": 1995} {"schema_version": "1.0", "item_id": "rcd2_8efc1c77f3b00293", "pair_id": "pair_964d5c57da918891", "statement_latex": "\\neg\\left(\\neg(\\neg (max_{1\\lei\\lem} |B_i| \\ge ceil(n/m)))\\right)", "inference_order": 1996} {"schema_version": "1.0", "item_id": "rcd2_f586f8a30fb950e0", "pair_id": "pair_628b2c47674928b0", "statement_latex": "\\neg\\left(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(f'(x)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(y))\\right)", "inference_order": 1997} {"schema_version": "1.0", "item_id": "rcd2_fc67d4ee1a9fe178", "pair_id": "pair_d7a20dcf9a9e3b07", "statement_latex": "\\{(Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k)\\}=\\{(w^(-1) * (z + w + m)^m)\\}", "inference_order": 1998} {"schema_version": "1.0", "item_id": "rcd2_d01bbb4dd46507d9", "pair_id": "pair_ea9ccbbf300e1c52", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;((sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3),z)=(((3a)! / (a!)^3),z)\\right)", "inference_order": 1999} {"schema_version": "1.0", "item_id": "rcd2_5d1db2c3403f69f3", "pair_id": "pair_f55e6d3bd18134eb", "statement_latex": "\\delta_D((\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}),({\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}))=0", "inference_order": 2000} {"schema_version": "1.0", "item_id": "rcd2_b6941789f6881909", "pair_id": "pair_c602208731940588", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((det(A)) \\in S \\land (det(A_v^u) * det(A_tilde_v^u)) \\notin S) \\lor ((det(A_v^u) * det(A_tilde_v^u)) \\in S \\land (det(A)) \\notin S))\\right)", "inference_order": 2001} {"schema_version": "1.0", "item_id": "rcd2_375673b072a2702b", "pair_id": "pair_425ecc3051c1bf98", "statement_latex": "(det(A)) \\in {(det(A_v^u) * det(A_tilde_v^u))} + \\eta", "inference_order": 2002} {"schema_version": "1.0", "item_id": "rcd2_3990bf766129f0e4", "pair_id": "pair_f6d6fbfb6288ecb1", "statement_latex": "(f(x)) =( (f(-x))) + \\eta", "inference_order": 2003} {"schema_version": "1.0", "item_id": "rcd2_0daf444f555983ad", "pair_id": "pair_968447826ba4f82e", "statement_latex": "\\max\\{0,(Pr[P(r1, ..., rn) = 0])-(d/|S|)\\}=0", "inference_order": 2004} {"schema_version": "1.0", "item_id": "rcd2_0da020e185cd0f70", "pair_id": "pair_964d5c57da918891", "statement_latex": "\\neg(\\neg (max_{1\\lei\\lem} |B_i| \\ge ceil(n/m)))", "inference_order": 2005} {"schema_version": "1.0", "item_id": "rcd2_ba2835aa40a5e152", "pair_id": "pair_4b4ccf659d2fa776", "statement_latex": "\\{((p(t)),z):z\\in D\\}=\\{((t^6 + c_4t^4 + c_2t^2 + c_1t + c_0),z):z\\in D\\}", "inference_order": 2006} {"schema_version": "1.0", "item_id": "rcd2_3356f870086adeb7", "pair_id": "pair_89ccc687bed1ad36", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(C(x) ‖φ‖_{∞})\\le t\\Rightarrow (|∫ Dφ(y)(x) dμ(y)|)\\le t", "inference_order": 2007} {"schema_version": "1.0", "item_id": "rcd2_672e10e512847399", "pair_id": "pair_a80f286dc1010710", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\lim_{\\theta → \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx))=(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)))) + \\eta", "inference_order": 2008} {"schema_version": "1.0", "item_id": "rcd2_e5b5e164dd68fcc2", "pair_id": "pair_7100c1223f5899fd", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(|A+B|)\\le t\\Rightarrow (min(|A|+|B|-1, p))\\le t", "inference_order": 2009} {"schema_version": "1.0", "item_id": "rcd2_97526f876e8bfb7b", "pair_id": "pair_58561abdaa336b2e", "statement_latex": "\\forall z\\in D,\\;(z=(|A_d|)\\Longleftrightarrow z=(\\frac{w(d)}{d}X + R_d))", "inference_order": 2010} {"schema_version": "1.0", "item_id": "rcd2_7a5d51029339da1e", "pair_id": "pair_e59daa177b5fc2ed", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (|S_chi(N,H)|) \\le x \\land x \\le (C_{r,\\varepsilon} H^{1-1/r} q^{(r+1)/(4r^2)+\\varepsilon})", "inference_order": 2011} {"schema_version": "1.0", "item_id": "rcd2_ee011a959990dc45", "pair_id": "pair_f71eba9daccd9c6a", "statement_latex": "\\max\\{0,(P(|sum_{i=(1}^n a_i X_i| ≥ k))-(2 inf_{0 ≤ c ≤ k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz)\\}=0) + \\eta", "inference_order": 2012} {"schema_version": "1.0", "item_id": "rcd2_ddb4ef63aecacf2b", "pair_id": "pair_98d4d1848a2c1ef6", "statement_latex": "\\forall z\\in D,\\;(z=(d/dt integral_{Omega(t)} f dV)\\Longleftrightarrow z=(integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA))", "inference_order": 2013} {"schema_version": "1.0", "item_id": "rcd2_aa206c8dfb1f02c6", "pair_id": "pair_2b9ab948af398953", "statement_latex": "\\neg\\left(\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(|e(S,T) - (d/n)|S||T||)+s=(λ√|S||T|)\\right)", "inference_order": 2014} {"schema_version": "1.0", "item_id": "rcd2_458610c8ac5dd02b", "pair_id": "pair_327df2f670cd234e", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(\\sum_{q=k(n)", "inference_order": 2015} {"schema_version": "1.0", "item_id": "rcd2_ae61e5d62dd86eb2", "pair_id": "pair_709504e1d4fa8334", "statement_latex": "(y^{2}) \\in {(a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0})} + \\eta", "inference_order": 2016} {"schema_version": "1.0", "item_id": "rcd2_6270cd93e4912f62", "pair_id": "pair_d1d62864076f4c01", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{n in S} (1/n)))\\\\ne\\operatorname{argmin}_{z\\in D}\\delta_D(z,(1))", "inference_order": 2017} {"schema_version": "1.0", "item_id": "rcd2_bcb83fe244060818", "pair_id": "pair_c801bb6cc26fab56", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (|Q(x)|) < x \\land x < (\\\\varepsilon)", "inference_order": 2018} {"schema_version": "1.0", "item_id": "rcd2_862a1c6a19ce8a02", "pair_id": "pair_bcba132b8080ca41", "statement_latex": "(T2) - (T1) \\ge 0 + \\eta", "inference_order": 2019} {"schema_version": "1.0", "item_id": "rcd2_93a3e0328467e974", "pair_id": "pair_dd6174baa4693939", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((chi'_l(K_{n,n})) \\in S \\land (n) \\notin S) \\lor ((n) \\in S \\land (chi'_l(K_{n,n})) \\notin S))\\right)", "inference_order": 2020} {"schema_version": "1.0", "item_id": "rcd2_d146f4d752d85f52", "pair_id": "pair_3582f483b4d34cab", "statement_latex": "\\neg(\\neg ((∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2))", "inference_order": 2021} {"schema_version": "1.0", "item_id": "rcd2_02d668c2393f2fd2", "pair_id": "pair_1f210dff910eb869", "statement_latex": "\\delta_D((E(X)_t),(exp(X_t - 1/2 [X]_t)))=(0) + \\eta", "inference_order": 2022} {"schema_version": "1.0", "item_id": "rcd2_4a3e31a78e155710", "pair_id": "pair_65d918bd8c72ddfc", "statement_latex": "\\forall z\\in D,\\;(z=(ω(-t))\\Longleftrightarrow z=(ω(t)))", "inference_order": 2023} {"schema_version": "1.0", "item_id": "rcd2_bc76aa03e18c8fe5", "pair_id": "pair_78d48fd996b18846", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(P(∪_{i=1}^∞ A_i))+s=(∑_{i=1}^∞ P(A_i))", "inference_order": 2024} {"schema_version": "1.0", "item_id": "rcd2_dff887497f71e7d8", "pair_id": "pair_c7777da5410d5778", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(p_A(A)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\text{conv}\\{w(X) : w \\in W\\}))", "inference_order": 2025} {"schema_version": "1.0", "item_id": "rcd2_570b94d9085240fd", "pair_id": "pair_23fb666d8efccf82", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (per(A)) \\le x \\land x \\le (prod_{i=1}^n (r_i!)^(1/r_i))", "inference_order": 2026} {"schema_version": "1.0", "item_id": "rcd2_d8464cfcdfc0fc5f", "pair_id": "pair_d6157fddb3b0ec6e", "statement_latex": "(f'(x_0)) \\in {(0)}", "inference_order": 2027} {"schema_version": "1.0", "item_id": "rcd2_37a7c40f0e363ed5", "pair_id": "pair_ce257c82b19333ab", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((P(X < a))-(E[X] / a))=0", "inference_order": 2028} {"schema_version": "1.0", "item_id": "rcd2_5bf4786eb6e5a051", "pair_id": "pair_148aaa6f458ed847", "statement_latex": "\\neg\\left((ε) - (|d(X,Y) - d(A,B)|) \\ge 0\\right)", "inference_order": 2029} {"schema_version": "1.0", "item_id": "rcd2_204a4b7181bdd7fc", "pair_id": "pair_3cda7a7134b7ed5d", "statement_latex": "\\frac{\\exp(p(a,d))}{\\exp(p(a,d))+\\exp(c * d^L)}<(\\frac{1}{2}) + \\eta", "inference_order": 2030} {"schema_version": "1.0", "item_id": "rcd2_89b41888688c0576", "pair_id": "pair_fc07b61f4fd34c8b", "statement_latex": "\\{((\\sum_{i=r}^{n} \\binom{i}{r}),z):z\\in D\\}=\\{((\\binom{n+1}{r+1}),z):z\\in D\\}", "inference_order": 2031} {"schema_version": "1.0", "item_id": "rcd2_1a42ba03879a1194", "pair_id": "pair_7100c1223f5899fd", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(|A+B|)\\le t\\Rightarrow (min(|A|+|B|-1, p))\\le t + \\eta", "inference_order": 2032} {"schema_version": "1.0", "item_id": "rcd2_54f3e0d05ebd8957", "pair_id": "pair_8eeb560dfc0494ef", "statement_latex": "\\neg\\left(\\forall F:D\\to\\{0,1\\},\\;F((\\dim(E^*)))=F((\\text{card}(\\mathbb{K}^I)))\\right)", "inference_order": 2033} {"schema_version": "1.0", "item_id": "rcd2_17441119733191e6", "pair_id": "pair_efe4c5a3a760f29d", "statement_latex": "((loglogn)^(1/2+ε)) - (|ω(n)−loglogn|) > 0", "inference_order": 2034} {"schema_version": "1.0", "item_id": "rcd2_f85a9522e7ba36c7", "pair_id": "pair_83ca4536670681e9", "statement_latex": "\\{((F_{n-1}*F_{n+1} - F_n^2),z):z\\in D\\}=(\\{(((-1)^n),z):z\\in D\\}) + \\eta", "inference_order": 2035} {"schema_version": "1.0", "item_id": "rcd2_aa25ee2da4fe2523", "pair_id": "pair_61f532c462dec59b", "statement_latex": "\\max\\{0,(P(S_n in A))-(binom(n,floor(n/2))/2^n)\\}=(0) + \\eta", "inference_order": 2036} {"schema_version": "1.0", "item_id": "rcd2_17dd9ea788acb589", "pair_id": "pair_7c0997cd728f17a2", "statement_latex": "\\nexists S \\subseteq D: (((a_n) \\in S \\land (a_{n+gcd(h,k)}) \\notin S) \\lor ((a_{n+gcd(h,k)}) \\in S \\land (a_n) \\notin S)) + \\eta", "inference_order": 2037} {"schema_version": "1.0", "item_id": "rcd2_8eb12a694aac79ed", "pair_id": "pair_f406fb7a7a76b61c", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (2 sup_{z in W(A)} |f(z)|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (||f(A)||)\\}", "inference_order": 2038} {"schema_version": "1.0", "item_id": "rcd2_d94da739294469ea", "pair_id": "pair_7c1290a06934ffdb", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\nu(\\chi(1))))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\nu(|G:D|)))", "inference_order": 2039} {"schema_version": "1.0", "item_id": "rcd2_d60b1b18c12e59eb", "pair_id": "pair_bc33c3f455d6cce5", "statement_latex": "(f_3(X ∨ Y)f_4(X ∧ Y)) - (f_1(X)f_2(Y)) \\ge 0", "inference_order": 2040} {"schema_version": "1.0", "item_id": "rcd2_39fe66d38fe9cdfa", "pair_id": "pair_be3802b7d7c41645", "statement_latex": "\\{((L),z):z\\in D\\}=(\\{((lambda*W),z):z\\in D\\}) + \\eta", "inference_order": 2041} {"schema_version": "1.0", "item_id": "rcd2_b39dfc79c77aa66b", "pair_id": "pair_9d6a22ad23393af9", "statement_latex": "(|B_1|) \\in {(|B_2|)}", "inference_order": 2042} {"schema_version": "1.0", "item_id": "rcd2_4465aff701049117", "pair_id": "pair_93dcd6f5abc6a994", "statement_latex": "\\frac{\\exp(cH log T)}{\\exp(cH log T)+\\exp(N(T + H) - N(T))}\\le\\frac{1}{2}", "inference_order": 2043} {"schema_version": "1.0", "item_id": "rcd2_ebd58781b6721f8f", "pair_id": "pair_4906cacaf5a529e8", "statement_latex": "\\{((sum_{k=m}^{n} f_k Delta g_k),z):z\\in D\\}=\\{(((f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k),z):z\\in D\\}", "inference_order": 2044} {"schema_version": "1.0", "item_id": "rcd2_398e11a5aaf0a255", "pair_id": "pair_71603c65251d77a5", "statement_latex": "\\forall z\\in D,\\;(z=(sum_{k=0}^{n-1} floor(x+k/n))\\Longleftrightarrow z=(floor(n*x)))", "inference_order": 2045} {"schema_version": "1.0", "item_id": "rcd2_307f43cf9ab9d397", "pair_id": "pair_60fe71a4608633c1", "statement_latex": "\\forall z\\in D,\\;((|Y^X/G|),z)=((xy),z)", "inference_order": 2046} {"schema_version": "1.0", "item_id": "rcd2_1355a711c1bec7a7", "pair_id": "pair_c1006da8bb6ceb4f", "statement_latex": "\\neg(\\neg (1/sqrt(p) * sum_{n=0}^{p-1} \\exp(2*\\pi*i*n^2*q/p) = e^(1/4 * \\pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} \\exp(-\\pi*i*n^2*p/(2q))))", "inference_order": 2047} {"schema_version": "1.0", "item_id": "rcd2_83393a6d080330a5", "pair_id": "pair_e4797fe16c13a341", "statement_latex": "\\neg\\left(((1 / (2*\\pi*i)) * integral_{c-i*\\infty}^{c+i*\\infty} g(z) * (x^z / z) dz) = (A(x))\\right)", "inference_order": 2048} {"schema_version": "1.0", "item_id": "rcd2_88a07c0719d7ad42", "pair_id": "pair_bfd8e29c43a8995c", "statement_latex": "\\frac{\\exp([b] for a majorizing b)}{\\exp([b] for a majorizing b)+\\exp([a])}\\le\\frac{1}{2} + \\eta", "inference_order": 2049} {"schema_version": "1.0", "item_id": "rcd2_2a44d80584b324cc", "pair_id": "pair_ca82fbc0b4b55d7d", "statement_latex": "(m) \\mid (ay) - (x) + \\eta", "inference_order": 2050} {"schema_version": "1.0", "item_id": "rcd2_4d2731460b2b1ea9", "pair_id": "pair_6f77f39dfb471cb2", "statement_latex": "\\neg\\left(\\exists z\\in D\\;\\text{such that}\\;(A^{\\mu \\nu})=z\\;\\land\\;z=(aG^{\\mu \\nu} + bg^{\\mu \\nu})\\right)", "inference_order": 2051} {"schema_version": "1.0", "item_id": "rcd2_2762ab20f86ed416", "pair_id": "pair_fee4fbc5bf631cb2", "statement_latex": "(0) = (Tor_1^A(C, M))", "inference_order": 2052} {"schema_version": "1.0", "item_id": "rcd2_eb91aa0070ccc0db", "pair_id": "pair_b3b1cd897cceada5", "statement_latex": "\\operatorname{dist}((P(X ≥ a))-(E[X] / a),(-\\infty,0])=0", "inference_order": 2053} {"schema_version": "1.0", "item_id": "rcd2_2065ad2dc59855ac", "pair_id": "pair_3e560aad4812e7e4", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p)))=F((e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))))", "inference_order": 2054} {"schema_version": "1.0", "item_id": "rcd2_399e8381b2d5ea24", "pair_id": "pair_5d0a7c9c202c5c9f", "statement_latex": "\\forall z\\in D,\\;((dim(H_μ)),z)=((n!),z)", "inference_order": 2055} {"schema_version": "1.0", "item_id": "rcd2_d66442c4f7118b9f", "pair_id": "pair_9f18f5e8a469f09a", "statement_latex": "((integral integral_{Sigma} (curl F) dot dSigma),(integral_{partial Sigma} F dot dr))\\in\\{(z,z):z\\in D\\} + \\eta", "inference_order": 2056} {"schema_version": "1.0", "item_id": "rcd2_b3f0df3b43b77504", "pair_id": "pair_1633d4caed251328", "statement_latex": "(P(A ∩ B)) \\in {(P(B|A)*P(A))}", "inference_order": 2057} {"schema_version": "1.0", "item_id": "rcd2_1a225a016cee4e0c", "pair_id": "pair_c3675bb5c0ee01f9", "statement_latex": "\\neg(\\neg (P[max_{1\\lei\\len} X_i \\ge C] \\le E[\\max(X_n, 0)] / C))", "inference_order": 2058} {"schema_version": "1.0", "item_id": "rcd2_0f5195926cedcd0c", "pair_id": "pair_7b852905d11dddd1", "statement_latex": "(prod_{n) - (1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) =( sum_{n \\\\in \\mathbb{Z}} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})) \\ge 0) + \\eta", "inference_order": 2059} {"schema_version": "1.0", "item_id": "rcd2_b21d61526f25ff7e", "pair_id": "pair_fc954112d424ad46", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(P(S_n - E[S_n] ≥ t))+s=(exp(-2*t^2 / sum_i (b_i-a_i)^2))", "inference_order": 2060} {"schema_version": "1.0", "item_id": "rcd2_8e0c8bea8017e8c3", "pair_id": "pair_5ce7af2c9aa579d9", "statement_latex": "\\nexists S \\subseteq D: (((gcd{(n-1 k-1), (n k+1), (n+1 k)}) \\in S \\land (gcd{(n-1 k), (n k-1), (n+1 k+1)}) \\notin S) \\lor ((gcd{(n-1 k), (n k-1), (n+1 k+1)}) \\in S \\land (gcd{(n-1 k-1), (n k+1), (n+1 k)}) \\notin S)) + \\eta", "inference_order": 2061} {"schema_version": "1.0", "item_id": "rcd2_c65ca9806806e914", "pair_id": "pair_59e68a51e6f039f7", "statement_latex": "(N^{(i-1)}) \\in {(\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1})}", "inference_order": 2062} {"schema_version": "1.0", "item_id": "rcd2_1916ee0ece58636f", "pair_id": "pair_5f11fb19b5d0c15d", "statement_latex": "\\neg\\left((prod_{i=1}^n (r_i!)^(1/r_i)) \\in (per(A)) + [0,\\infty)\\right)", "inference_order": 2063} {"schema_version": "1.0", "item_id": "rcd2_e5a7f9def1c83164", "pair_id": "pair_5f11fb19b5d0c15d", "statement_latex": "(prod_{i=1}^n (r_i!)^(1/r_i)) \\in (per(A)) + [0,\\infty)", "inference_order": 2064} {"schema_version": "1.0", "item_id": "rcd2_7dbd0e223ce5bd06", "pair_id": "pair_5dee6fc9bebf3dfc", "statement_latex": "\\frac{\\exp(4πA)}{\\exp(4πA)+\\exp(L^2)}\\le\\frac{1}{2} + \\eta", "inference_order": 2065} {"schema_version": "1.0", "item_id": "rcd2_70ad24cd768215c3", "pair_id": "pair_35733e1e74e4ecee", "statement_latex": "(a_0 + a_1X + ... + a_nX^n) =( (F(X))) + \\eta", "inference_order": 2066} {"schema_version": "1.0", "item_id": "rcd2_b58d6c04ec8be979", "pair_id": "pair_c0fdd39f52248f81", "statement_latex": "\\neg\\left(\\forall F:D\\to\\{0,1\\},\\;F((f'(x_0)))=F((0))\\right)", "inference_order": 2067} {"schema_version": "1.0", "item_id": "rcd2_9811f4cc1e361b7d", "pair_id": "pair_ed17e5525b651559", "statement_latex": "(|Irr_{p'}(N_{G}(P))|) \\\\ne (|Irr_{p'}(G)|)", "inference_order": 2068} {"schema_version": "1.0", "item_id": "rcd2_c9e45aa870868bce", "pair_id": "pair_58467464b6474a28", "statement_latex": "(sum_{n \\in S} (1/n)) \\in {(\\infty)}", "inference_order": 2069} {"schema_version": "1.0", "item_id": "rcd2_78440324aa48b75c", "pair_id": "pair_9201bba025cb958c", "statement_latex": "\\{((n),z):z\\in D\\}=(\\{((p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}),z):z\\in D\\}) + \\eta", "inference_order": 2070} {"schema_version": "1.0", "item_id": "rcd2_4f83895a0d0cfa1f", "pair_id": "pair_586802a23edc2483", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (1^-} G(x) = sum_{k=0}^\\infty a_k) < x \\land x < (lim_{x -)", "inference_order": 2071} {"schema_version": "1.0", "item_id": "rcd2_cf070042855b205a", "pair_id": "pair_323720428879542f", "statement_latex": "(0 =( 0) \\Rightarrow (f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x))) + \\eta", "inference_order": 2072} {"schema_version": "1.0", "item_id": "rcd2_a06afbe404ed385b", "pair_id": "pair_5155cc0f2ec17abc", "statement_latex": "\\neg\\left(\\{((Σ_{v∈V} deg(v)),z):z\\in D\\}=\\{((2|E|),z):z\\in D\\}\\right)", "inference_order": 2073} {"schema_version": "1.0", "item_id": "rcd2_a0256146e6f9f26d", "pair_id": "pair_4955ad14db823fbb", "statement_latex": "(x * (y * z) + y * (z * x) + z * (x * y)) \\in {(0)} + \\eta", "inference_order": 2074} {"schema_version": "1.0", "item_id": "rcd2_aa9d6ac9a93931a2", "pair_id": "pair_5e24716261faf6c3", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (E[X] / a)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (P(X ≥( a))\\}) + \\eta", "inference_order": 2075} {"schema_version": "1.0", "item_id": "rcd2_a5628d1d41ac666a", "pair_id": "pair_0d12ebb73f8d6a4b", "statement_latex": "\\delta_D((F_{n-1}*F_{n+1} - F_n^2),((-1)^n))\\\\ne0", "inference_order": 2076} {"schema_version": "1.0", "item_id": "rcd2_ac77d5a441f1e417", "pair_id": "pair_743a5eb2bfa81936", "statement_latex": "\\{(x_i)\\}=\\{(\\det(A_i) / \\det(A))\\}", "inference_order": 2077} {"schema_version": "1.0", "item_id": "rcd2_d8e63f16602a3555", "pair_id": "pair_faeba38f0690befe", "statement_latex": "\\frac{\\exp(ceil(n/m))}{\\exp(ceil(n/m))+\\exp(max_{1≤i≤m} |B_i|)}\\le\\frac{1}{2}", "inference_order": 2078} {"schema_version": "1.0", "item_id": "rcd2_3d0047bb3c4b6ab3", "pair_id": "pair_212aa5f34e7159a5", "statement_latex": "\\forall z\\in D,\\;((Pr(Σ_t\\\\neA) / Pr(Σ_t=-A)),z)=((e^(At)),z)", "inference_order": 2079} {"schema_version": "1.0", "item_id": "rcd2_6273117847e8b39d", "pair_id": "pair_020a74b5d4894a69", "statement_latex": "((ax + by),(gcd(a, b)))\\in\\{(z,z):z\\in D\\}", "inference_order": 2080} {"schema_version": "1.0", "item_id": "rcd2_ece3a99aba718ee6", "pair_id": "pair_09aceb709e129b3e", "statement_latex": "\\neg\\left(\\delta_D((N^{(i-1)}),(\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}))=0\\right)", "inference_order": 2081} {"schema_version": "1.0", "item_id": "rcd2_ea2b698fd89cd252", "pair_id": "pair_2d947e5e1530365f", "statement_latex": "\\neg\\left(\\delta_D((ax + by),(gcd(a, b)))=0\\right)", "inference_order": 2082} {"schema_version": "1.0", "item_id": "rcd2_a61166594d73c73d", "pair_id": "pair_7b6dda8e5af25857", "statement_latex": "\\delta_D((H),(\\sum_{i=1}^{m} H_{i}))=0", "inference_order": 2083} {"schema_version": "1.0", "item_id": "rcd2_208231715833f2a1", "pair_id": "pair_0824cb01884e63ec", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(1/k^2)\\le t\\Rightarrow (Pr(|X-mu| ≥ k*sigma))\\le t", "inference_order": 2084} {"schema_version": "1.0", "item_id": "rcd2_ee04996690fd5613", "pair_id": "pair_f4b6041225399f5f", "statement_latex": "(P(B|A)*P(A)) = (P(A ∩ B))", "inference_order": 2085} {"schema_version": "1.0", "item_id": "rcd2_85b615da766c0f0a", "pair_id": "pair_93604e3d52ef2226", "statement_latex": "(\\{(nullity A)\\}\\times\\{(nullity H)\\})\\cap\\{(z,z):z\\in D\\}=\\{((nullity A),(nullity H))\\}", "inference_order": 2086} {"schema_version": "1.0", "item_id": "rcd2_9bf9d2830f83d8c8", "pair_id": "pair_c538b7df1d9e6b9d", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(lim_{x→c} f(x)/g(x))=z\\;\\land\\;z=(lim_{x→c} f'(x)/g'(x))", "inference_order": 2087} {"schema_version": "1.0", "item_id": "rcd2_144060c962687147", "pair_id": "pair_c13dd31aa751fee8", "statement_latex": "\\neg\\left(\\sup_{\\lambda\\in[0,1]}\\lambda((|f(z)|)-(|z|))=0\\right)", "inference_order": 2088} {"schema_version": "1.0", "item_id": "rcd2_798bd702daf02eba", "pair_id": "pair_623e29b7259753b9", "statement_latex": "\\neg\\left((sum (6-i)p_i) \\in {(12)}\\right)", "inference_order": 2089} {"schema_version": "1.0", "item_id": "rcd2_2461b0407b1aa21b", "pair_id": "pair_210668c7ba2dd451", "statement_latex": "(\\{(\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k})\\}\\times\\{({\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n})\\})\\cap\\{(z,z):z\\in D\\}=\\{((\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}),({\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}))\\}", "inference_order": 2090} {"schema_version": "1.0", "item_id": "rcd2_ff10c4a9e4d2773f", "pair_id": "pair_095631fc29fe23cf", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\ge (cr(G))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (e^3 / (29n^2))\\}\\right)", "inference_order": 2091} {"schema_version": "1.0", "item_id": "rcd2_f709c5f45c79e3e5", "pair_id": "pair_52c63bfd0a765019", "statement_latex": "\\neg\\left(\\exists z\\in D\\;\\text{such that}\\;(-zeta'(s)/zeta(s))=z\\;\\land\\;z=(F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s)))\\right)", "inference_order": 2092} {"schema_version": "1.0", "item_id": "rcd2_57e8fda5884f5cda", "pair_id": "pair_2d3935f698bb0a98", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)]))\\\\ne\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sup_{x∈X} (φ(x) - I(x))))", "inference_order": 2093} {"schema_version": "1.0", "item_id": "rcd2_932a9f593481f1f6", "pair_id": "pair_31aa1880c42d45a3", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(A))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(k[y_1, ..., y_m]))", "inference_order": 2094} {"schema_version": "1.0", "item_id": "rcd2_7a3b62640f51d92a", "pair_id": "pair_a731e1cc162200ed", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon})\\le t\\Rightarrow (|S_chi(N,H)|)\\le t", "inference_order": 2095} {"schema_version": "1.0", "item_id": "rcd2_70a9e3842fbc5759", "pair_id": "pair_62478e74c2ad750a", "statement_latex": "\\frac{\\exp(e*p*(d+1))}{\\exp(e*p*(d+1))+\\exp(1)}\\le\\frac{1}{2}", "inference_order": 2096} {"schema_version": "1.0", "item_id": "rcd2_c196384a29209dbe", "pair_id": "pair_46c17cca4c7f4c53", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: ((([[u, 0], [0, u^-1]]) \\in S \\land (e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)) \\notin S) \\lor ((e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)) \\in S \\land ([[u, 0], [0, u^-1]]) \\notin S))\\right)", "inference_order": 2097} {"schema_version": "1.0", "item_id": "rcd2_1741c93238e99b4e", "pair_id": "pair_9e1a1f1abcf1ff9f", "statement_latex": "\\frac{\\exp(1)}{\\exp(1)+\\exp(|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2|)}\\le\\frac{1}{2} + \\eta", "inference_order": 2098} {"schema_version": "1.0", "item_id": "rcd2_9cea7b34af7aa242", "pair_id": "pair_8badc6c944195e45", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (P(‖f‖_T < E(‖f‖_T) + u)) \\le x \\land x \\le (\\exp(-u^2 / (2σ_T^2)))", "inference_order": 2099} {"schema_version": "1.0", "item_id": "rcd2_6c4c5e7ee0edc8cc", "pair_id": "pair_cb399a8830868f33", "statement_latex": "\\forall z\\in D,\\;(z=(p(t))\\Longleftrightarrow z=(t^6 + c_4t^4 + c_2t^2 + c_1t + c_0))", "inference_order": 2100} {"schema_version": "1.0", "item_id": "rcd2_78eec1099af90250", "pair_id": "pair_16e16212b8b7ee5a", "statement_latex": "\\forall x\\in D,\\;x\\in (ker(d(f|_Y)_y))\\Rightarrow x\\in (tau) + \\eta", "inference_order": 2101} {"schema_version": "1.0", "item_id": "rcd2_ac4e9bdd97a74c97", "pair_id": "pair_a7eb5ac5b60c3682", "statement_latex": "\\neg\\left((⟨j m | T_q^(k) | j' m'⟩) \\in {(⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩)}\\right)", "inference_order": 2102} {"schema_version": "1.0", "item_id": "rcd2_e790faa26118510e", "pair_id": "pair_75281ce2ccf6e491", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(F(z,t)))+\\delta_D(z,(e^{g(z)t})))\\\\ne0", "inference_order": 2103} {"schema_version": "1.0", "item_id": "rcd2_8a5280444d33e475", "pair_id": "pair_b4a3d2ab83c383b3", "statement_latex": "(-1) = (T^2)", "inference_order": 2104} {"schema_version": "1.0", "item_id": "rcd2_b5599511a186ac35", "pair_id": "pair_4cee4ea93bb85057", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(SO_{\\gamma _{H}}(1_{K_{H}})))+\\delta_D(z,(\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}}))))=0", "inference_order": 2105} {"schema_version": "1.0", "item_id": "rcd2_5e24a9ab500e2476", "pair_id": "pair_3c15256822782930", "statement_latex": "\\forall z\\in D,\\;((dX_t),z)=((mu_t dt + sigma_t dB_t),z)", "inference_order": 2106} {"schema_version": "1.0", "item_id": "rcd2_61188a7f5a5f913a", "pair_id": "pair_386a80e344afe1ca", "statement_latex": "\\{((lim_{N→inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2),z):z\\in D\\}=\\{((sum_{j} |c_{j}|^2),z):z\\in D\\}", "inference_order": 2107} {"schema_version": "1.0", "item_id": "rcd2_9330e4e163980fed", "pair_id": "pair_d375eb3e055f3d5e", "statement_latex": "\\max\\{0,(delta |X| |Y|)-(e(X,Y))\\}=0\\;\\land\\;(delta |X| |Y|)\\ne (e(X,Y))", "inference_order": 2108} {"schema_version": "1.0", "item_id": "rcd2_2538b5ff4f4fb9c2", "pair_id": "pair_8884bcda65c993ef", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(ζ(s)))=(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s)))) + \\eta", "inference_order": 2109} {"schema_version": "1.0", "item_id": "rcd2_b759ca8bf50925a8", "pair_id": "pair_1b49c5c0381c61ca", "statement_latex": "\\{((f'(x)),z):z\\in D\\}=\\{((y),z):z\\in D\\}", "inference_order": 2110} {"schema_version": "1.0", "item_id": "rcd2_2465778485baff60", "pair_id": "pair_ade17c3dd001731b", "statement_latex": "\\forall z\\in D,\\;(z=((sum_{p=2}^{\\infty} 1/(p-1))\\Longleftrightarrow z=(1))) + \\eta_{1157}", "inference_order": 2111} {"schema_version": "1.0", "item_id": "rcd2_49b723f3d88c0ed6", "pair_id": "pair_31173860f71db532", "statement_latex": "\\delta_D((d/dx integral_{a(x)}^{b(x)} f(x,t) dt),(f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt))=0", "inference_order": 2112} {"schema_version": "1.0", "item_id": "rcd2_377616037ca74545", "pair_id": "pair_809ef9d8745abe70", "statement_latex": "\\frac{\\exp(|ω(n)−loglogn|)}{\\exp(|ω(n)−loglogn|)+\\exp(loglogn)^(1/2+ε)}<\\frac{1}{2}", "inference_order": 2113} {"schema_version": "1.0", "item_id": "rcd2_a4fb3cdd1336abd0", "pair_id": "pair_98ed921f589ec9c8", "statement_latex": "\\frac{\\exp(f_1(X)f_2(Y))}{\\exp(f_1(X)f_2(Y))+\\exp(f_3(X ∨ Y)f_4(X ∧ Y))}\\le\\frac{1}{2}", "inference_order": 2114} {"schema_version": "1.0", "item_id": "rcd2_fa3d510252da466c", "pair_id": "pair_dc08afbbdc9791cd", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((∫_{γ} ∇φ(r) · dr))=F((φ(q) - φ(p)))", "inference_order": 2115} {"schema_version": "1.0", "item_id": "rcd2_cc49b13c94c52052", "pair_id": "pair_09aceb709e129b3e", "statement_latex": "\\delta_D((N^{(i-1)}),(\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}))=0", "inference_order": 2116} {"schema_version": "1.0", "item_id": "rcd2_bb440288026b4961", "pair_id": "pair_a80f286dc1010710", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\lim_{\\theta → \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)))", "inference_order": 2117} {"schema_version": "1.0", "item_id": "rcd2_2195a70ec794a6a9", "pair_id": "pair_1ed8cd43b55a1626", "statement_latex": "\\forall z\\in D,\\;((H),z)=((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)),z)", "inference_order": 2118} {"schema_version": "1.0", "item_id": "rcd2_0813262767f07d23", "pair_id": "pair_9259a14a33b4e402", "statement_latex": "\\operatorname{dist}((v)-(b),(-\\infty,0])\\\\ne0", "inference_order": 2119} {"schema_version": "1.0", "item_id": "rcd2_f7ab720779f8269b", "pair_id": "pair_47ab6ecaafda958e", "statement_latex": "\\max\\{0,(integral_{X} exp(alpha * ||x||^2) dmu(x))-(+infinity)\\}=0\\;\\land\\;(integral_{X} exp(alpha * ||x||^2) dmu(x))\\ne (+infinity)", "inference_order": 2120} {"schema_version": "1.0", "item_id": "rcd2_482c80dec40d7566", "pair_id": "pair_b3e1f8ebd134eb69", "statement_latex": "(tr W(B, B*)) = (tr W(A, A*))", "inference_order": 2121} {"schema_version": "1.0", "item_id": "rcd2_66cab5f6d936d09f", "pair_id": "pair_6496a0ccac0c3305", "statement_latex": "\\neg\\left(\\exists x \\\\in \\mathbb{R}: (1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n \\\\in \\mathbb{Z}} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})) \\le x \\land x \\le (prod_{n)\\right)", "inference_order": 2122} {"schema_version": "1.0", "item_id": "rcd2_e2dba4a0470e3814", "pair_id": "pair_2ac6695cf52e748c", "statement_latex": "\\operatorname{dist}((T1)-(T2),(-\\infty,0])=0", "inference_order": 2123} {"schema_version": "1.0", "item_id": "rcd2_030ac09b7511d767", "pair_id": "pair_07fc1ec36988aba5", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;(D^{-1}*sqrt(det(AA^T)))^(1/(N-M))\\le t\\Rightarrow (max|X_j|)\\le t\\right)", "inference_order": 2124} {"schema_version": "1.0", "item_id": "rcd2_9965e29bf7704ec5", "pair_id": "pair_fe5f00d6ed39b83e", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(ρ_W(g) ∘ f))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(f ∘ ρ_V(g)))", "inference_order": 2125} {"schema_version": "1.0", "item_id": "rcd2_86b0c56b8ac175d1", "pair_id": "pair_2fdfd5a16e0ddd6c", "statement_latex": "\\delta_D((sum_{k=0}^m c_k 1_{N=k}),(sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k))=0", "inference_order": 2126} {"schema_version": "1.0", "item_id": "rcd2_65644afbd1e3e4ce", "pair_id": "pair_bbd6d6abeb397124", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (1) < x \\land x < (g)", "inference_order": 2127} {"schema_version": "1.0", "item_id": "rcd2_5b86e0748b3cdba9", "pair_id": "pair_47cd0880e2c6b666", "statement_latex": "(∇^2φ) \\in {(-ρ_f/ε_0)} + \\eta", "inference_order": 2128} {"schema_version": "1.0", "item_id": "rcd2_09c367dcd9f8db8a", "pair_id": "pair_0969b5fda62bd73e", "statement_latex": "\\{(∇^2φ)\\}=\\{(-ρ_f/ε_0)\\}", "inference_order": 2129} {"schema_version": "1.0", "item_id": "rcd2_8aae8dcf8c380fb9", "pair_id": "pair_287ef432c524a888", "statement_latex": "\\neg\\left(\\inf_{z\\in D}(\\delta_D(z,(F))+\\delta_D(z,(-V'(x))))=0\\right)", "inference_order": 2130} {"schema_version": "1.0", "item_id": "rcd2_513b1598b9baa859", "pair_id": "pair_667524cf50f41f7b", "statement_latex": "(n) \\mid (sum_{d|n} mu(d)*a_{n/d}) - (0)", "inference_order": 2131} {"schema_version": "1.0", "item_id": "rcd2_71d44f9000e0b3d6", "pair_id": "pair_09981b63c43e2167", "statement_latex": "\\neg\\left(\\{(f^{-1})'(b)\\}=\\{(1 / f'(a))\\}\\right)", "inference_order": 2132} {"schema_version": "1.0", "item_id": "rcd2_62b414b9d18b3dd4", "pair_id": "pair_5514b90ff8c47f1e", "statement_latex": "\\operatorname{softplus}((det(M))-(det(A)*det(C)))\\le\\log 2", "inference_order": 2133} {"schema_version": "1.0", "item_id": "rcd2_20ccc4fed064828a", "pair_id": "pair_6d8a1c052d5de90f", "statement_latex": "\\max\\{0,(|d(X,Y) - d(A,B)|)-(ε)\\}=0", "inference_order": 2134} {"schema_version": "1.0", "item_id": "rcd2_f308238a1ba07a9c", "pair_id": "pair_2546bab32833905c", "statement_latex": "\\forall z\\in D,\\;((x_i),z)=(((\\det(A_i) / \\det(A)),z)) + \\eta", "inference_order": 2135} {"schema_version": "1.0", "item_id": "rcd2_cead893eee6bb0ad", "pair_id": "pair_c92a1271353e2e3d", "statement_latex": "\\max\\{0,(p(a,d))-(c * d^L)\\}=0\\;\\land\\;(p(a,d))\\ne (c * d^L)", "inference_order": 2136} {"schema_version": "1.0", "item_id": "rcd2_1d07f441412c4ea9", "pair_id": "pair_d488e304858413d1", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(z=(lim_{x→c} f(x)/g(x))\\Longleftrightarrow z=(lim_{x→c} f'(x)/g'(x)))\\right)", "inference_order": 2137} {"schema_version": "1.0", "item_id": "rcd2_b81291ea2f3f8929", "pair_id": "pair_720d2256c3d8ab52", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: ((((f^{-1})'(b)) \\in S \\land (1 / f'(a)) \\notin S) \\lor ((1 / f'(a)) \\in S \\land ((f^{-1})'(b)) \\notin S))\\right)", "inference_order": 2138} {"schema_version": "1.0", "item_id": "rcd2_aab6f364324e2818", "pair_id": "pair_b1b5da746f1333b6", "statement_latex": "\\max\\{0,(P(A square B))-(P(A)P(B))\\}=(0) + \\eta", "inference_order": 2139} {"schema_version": "1.0", "item_id": "rcd2_6f0762f2ce2af172", "pair_id": "pair_1a8976f30365f0a5", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{p=\\operatorname{argmin}_{z\\in D}\\delta_D(z,((-1)^n))", "inference_order": 2140} {"schema_version": "1.0", "item_id": "rcd2_8899bbee82c3142f", "pair_id": "pair_655924b18aa2eef2", "statement_latex": "(prod_{n) \\in (1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) =( sum_{n \\\\in \\mathbb{Z}} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})) + [0,\\infty)) + \\eta", "inference_order": 2141} {"schema_version": "1.0", "item_id": "rcd2_7654f8274afb3628", "pair_id": "pair_3328db101c224406", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((|Irr_{p'}(G)|) \\in S \\land (|Irr_{p'}(N_{G}(P))|) \\notin S) \\lor ((|Irr_{p'}(N_{G}(P))|) \\in S \\land (|Irr_{p'}(G)|) \\notin S))\\right)", "inference_order": 2142} {"schema_version": "1.0", "item_id": "rcd2_fc98b2e75d3803cc", "pair_id": "pair_258c4aaccb957157", "statement_latex": "\\nexists S \\subseteq D: (((\\binom{n-1}{k} + \\binom{n-1}{k-1}) \\in S \\land (\\binom{n}{k}) \\notin S) \\lor ((\\binom{n}{k}) \\in S \\land (\\binom{n-1}{k} + \\binom{n-1}{k-1}) \\notin S))", "inference_order": 2143} {"schema_version": "1.0", "item_id": "rcd2_019d6531a75cb753", "pair_id": "pair_b0cd3fa8cdd9fd02", "statement_latex": "\\forall z\\in D,\\;(z=((H)\\Longleftrightarrow z=(\\sum_{i=1}^{m} H_{i}))) + \\eta", "inference_order": 2144} {"schema_version": "1.0", "item_id": "rcd2_95ae01b78d09c3f1", "pair_id": "pair_d49c3a78caade47b", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(F)=z\\;\\land\\;z=(-V'(x))", "inference_order": 2145} {"schema_version": "1.0", "item_id": "rcd2_0e893e4fa6f69b74", "pair_id": "pair_d9e101fd12d6f890", "statement_latex": "\\neg\\left(\\inf_{z\\in D}(\\delta_D(z,(nullity A))+\\delta_D(z,(nullity H)))=0\\right)", "inference_order": 2146} {"schema_version": "1.0", "item_id": "rcd2_79f748a9aa349da8", "pair_id": "pair_3e8d80eb15a69d88", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(H(t)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE))", "inference_order": 2147} {"schema_version": "1.0", "item_id": "rcd2_9e16d9e2c1f4bef0", "pair_id": "pair_aaa6f5cb5879f8db", "statement_latex": "\\max\\{0,(R)-(C)\\}=(0\\;\\land\\;(R)\\ne (C)) + \\eta", "inference_order": 2148} {"schema_version": "1.0", "item_id": "rcd2_3568f5f996ab5cc2", "pair_id": "pair_33871e8cc93e9c43", "statement_latex": "\\neg\\left((e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)) = ([[u, 0], [0, u^-1]])\\right)", "inference_order": 2149} {"schema_version": "1.0", "item_id": "rcd2_86c247b4bc4284dd", "pair_id": "pair_e187266170ddccd4", "statement_latex": "((Av),(rv))\\in\\{(z,z):z\\in D\\} + \\eta", "inference_order": 2150} {"schema_version": "1.0", "item_id": "rcd2_0383c29f5d9ba824", "pair_id": "pair_9914a2ace3684a2c", "statement_latex": "((sum_{n in S} (1/n)),(infinity))\\in\\{(z,z):z\\in D\\}", "inference_order": 2151} {"schema_version": "1.0", "item_id": "rcd2_af3f970c00253080", "pair_id": "pair_6f514955ff7549c1", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(d/dt integral_{Omega(t)} f dV))+\\delta_D(z,(integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA)))=0", "inference_order": 2152} {"schema_version": "1.0", "item_id": "rcd2_c7f85f9da502cab5", "pair_id": "pair_0deed39e661b5444", "statement_latex": "\\neg\\left(\\delta_D((x^a - y^b),(1))=0\\right)", "inference_order": 2153} {"schema_version": "1.0", "item_id": "rcd2_45f638c90a3357c0", "pair_id": "pair_bc4cf7a664ca1962", "statement_latex": "\\operatorname{softplus}((AC * BD)-(AB * CD + BC * DA))\\le\\log 2", "inference_order": 2154} {"schema_version": "1.0", "item_id": "rcd2_0c718ea8cbab8651", "pair_id": "pair_87cfb712eafc4561", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;((SO_{\\gamma _{H}}(1_{K_{H}})),z)=((\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}})),z)\\right)", "inference_order": 2155} {"schema_version": "1.0", "item_id": "rcd2_05b6d6fe3ffa91a0", "pair_id": "pair_aa09e3aed77c9d3d", "statement_latex": "\\{((sum_{i=(0}^n binom(n,i)),z):z\\in D\\}=\\{((2^n),z):z\\in D\\}) + \\eta", "inference_order": 2156} {"schema_version": "1.0", "item_id": "rcd2_276e38004b1fb789", "pair_id": "pair_7d3d6cbb079dc151", "statement_latex": "\\{((I^nM ∩ N),z):z\\in D\\}=\\{((I^{n-k}(I^kM ∩ N)),z):z\\in D\\}", "inference_order": 2157} {"schema_version": "1.0", "item_id": "rcd2_c1bf5dd809ef3ff4", "pair_id": "pair_f3e521c12f23718a", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(rank(T) + nullity(T)))+\\delta_D(z,(dim(V))))=0", "inference_order": 2158} {"schema_version": "1.0", "item_id": "rcd2_abe5b6af42bdd03c", "pair_id": "pair_228e62e7252a11f9", "statement_latex": "(|A+H|+|B+H|-|H|) =( (|A+B|)) + \\eta", "inference_order": 2159} {"schema_version": "1.0", "item_id": "rcd2_afd8e485996522e0", "pair_id": "pair_ea8fdb7218dc53b4", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((\\bigcap _{k) \\in S \\land (0}^{\\infty }C_{k}\\neq \\\\emptyset) \\notin S) \\lor ((0}^{\\infty }C_{k}\\neq \\\\emptyset) \\in S \\land (\\bigcap _{k) \\notin S))\\right)", "inference_order": 2160} {"schema_version": "1.0", "item_id": "rcd2_020dfc2462e1c4b7", "pair_id": "pair_352584f70f94cde5", "statement_latex": "\\delta_D((⟨j m | T_q^(k) | j' m'⟩),(⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩))=0", "inference_order": 2161} {"schema_version": "1.0", "item_id": "rcd2_801361205d7e2558", "pair_id": "pair_e7df8f6b53d592b0", "statement_latex": "\\neg(\\neg (sum_{k=0}^m c_k 1_{N=k} = sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k))", "inference_order": 2162} {"schema_version": "1.0", "item_id": "rcd2_bab5ea97193b3010", "pair_id": "pair_4a1d4788acd39d80", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((|Irr_{p'}(G)|))=F((|Irr_{p'}(N_{G}(P))|))", "inference_order": 2163} {"schema_version": "1.0", "item_id": "rcd2_fba8cf0b1dfff16e", "pair_id": "pair_b43386925572d9f2", "statement_latex": "\\neg\\left(\\{(\\zeta(s))\\}=\\{(\\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}})\\}\\right)", "inference_order": 2164} {"schema_version": "1.0", "item_id": "rcd2_6d2b5b4eb6cf4cd9", "pair_id": "pair_987558619cdf15e6", "statement_latex": "\\{((\\nu(\\chi(1))),z):z\\in D\\}=\\{((\\nu(|G:D|)),z):z\\in D\\}", "inference_order": 2165} {"schema_version": "1.0", "item_id": "rcd2_968a638644dc17e3", "pair_id": "pair_27ff06b630b7decc", "statement_latex": "\\operatorname{softplus}((|det(N)|)-(prod_{i=1}^n ||v_i||))\\le\\log 2", "inference_order": 2166} {"schema_version": "1.0", "item_id": "rcd2_21b903155a2fe99b", "pair_id": "pair_a651dfff71df6b9e", "statement_latex": "\\neg\\left(((E),(nhv))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 2167} {"schema_version": "1.0", "item_id": "rcd2_e0ea5be342965139", "pair_id": "pair_4aa9b68bf494a944", "statement_latex": "\\nexists S \\subseteq D: (((uv) \\in S \\land (xy) \\notin S) \\lor ((xy) \\in S \\land (uv) \\notin S))", "inference_order": 2168} {"schema_version": "1.0", "item_id": "rcd2_01f884b566f4801a", "pair_id": "pair_75281ce2ccf6e491", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(F(z,t)))+\\delta_D(z,(e^{g(z)t})))=0", "inference_order": 2169} {"schema_version": "1.0", "item_id": "rcd2_564fb63145ce20ed", "pair_id": "pair_9fdfd0daf6a4ccc0", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((∫ |∇u*|^p dx)-(∫ |∇u|^p dx))\\\\ne0", "inference_order": 2170} {"schema_version": "1.0", "item_id": "rcd2_79b7647bad8bcba8", "pair_id": "pair_0485b9a4b9ea64fa", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(z=(sum_{k=0}^m c_k 1_{N=k})\\Longleftrightarrow z=(sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k))\\right)", "inference_order": 2171} {"schema_version": "1.0", "item_id": "rcd2_3d2097ea21807fcf", "pair_id": "pair_2d947e5e1530365f", "statement_latex": "\\delta_D((ax + by),(gcd(a, b)))=0", "inference_order": 2172} {"schema_version": "1.0", "item_id": "rcd2_fe40129fd8b57a98", "pair_id": "pair_5137c9524b6b64dc", "statement_latex": "((a+b)^2 / (4ab)) - ((Σ p_i x_i) * (Σ p_i / x_i)) \\ge 0", "inference_order": 2173} {"schema_version": "1.0", "item_id": "rcd2_3fab9e4574a2e4ce", "pair_id": "pair_ea906014c1c89919", "statement_latex": "(\\{(\\dot{q})\\}\\times\\{(f(q,u))\\})\\cap\\{(z,z):z\\in D\\}=\\{((\\dot{q}),(f(q,u)))\\}", "inference_order": 2174} {"schema_version": "1.0", "item_id": "rcd2_421055a40df188f8", "pair_id": "pair_4f9e26b776a14cf7", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(cap(Z)))=(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(H(p/(1+2^s))))) + \\eta", "inference_order": 2175} {"schema_version": "1.0", "item_id": "rcd2_3cc560f548e45440", "pair_id": "pair_bbce8085cf6c63ed", "statement_latex": "\\neg\\left(((R),(C))\\in\\{(u,v)\\in\\mathbb{R}^2:u\\le v\\}\\setminus\\{(u,u):u\\in\\mathbb{R}\\}\\right)", "inference_order": 2176} {"schema_version": "1.0", "item_id": "rcd2_0213f7359b431293", "pair_id": "pair_85b0e22f97be6b9a", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(a+b)^2 / (4ab)\\le t\\Rightarrow (Σ p_i x_i) * (Σ p_i / x_i)\\le t + \\eta", "inference_order": 2177} {"schema_version": "1.0", "item_id": "rcd2_f4579c81146869af", "pair_id": "pair_2699b96685f0ba5f", "statement_latex": "\\neg(\\neg (sum_{p=2}^{\\infty} 1/(p-1) = 1))", "inference_order": 2178} {"schema_version": "1.0", "item_id": "rcd2_cf7e538ff1d9d0f2", "pair_id": "pair_3f1008da7d9eb9d8", "statement_latex": "(∑_{i=1}^∞ P(A_i)) - (P(∪_{i=1}^∞ A_i)) \\ge 0", "inference_order": 2179} {"schema_version": "1.0", "item_id": "rcd2_18e670434eae62a6", "pair_id": "pair_33871e8cc93e9c43", "statement_latex": "(e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)) = ([[u, 0], [0, u^-1]])", "inference_order": 2180} {"schema_version": "1.0", "item_id": "rcd2_e41c36c446bdf551", "pair_id": "pair_c44766cc8b64c6b6", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (|∑ a_k b_k|)\\Rightarrow t\\le (max|B_k|(|a_n| + |a_n - a_1|)) + \\eta", "inference_order": 2181} {"schema_version": "1.0", "item_id": "rcd2_921a76f6edb470bb", "pair_id": "pair_e87523e8bb38be46", "statement_latex": "(f'(c)) \\in {((f(b)-f(a))/(b-a))}", "inference_order": 2182} {"schema_version": "1.0", "item_id": "rcd2_bfb783dd692e1a31", "pair_id": "pair_fa2db7032ffae932", "statement_latex": "\\operatorname{dist}((Pr(max_{1≤ 1 - α))-(α),(-\\infty,0])=0", "inference_order": 2183} {"schema_version": "1.0", "item_id": "rcd2_44a8f50914d61eeb", "pair_id": "pair_55a77df73dd25ae0", "statement_latex": "(1) - (sum_{k=(0}^n a_k/binom(n,k)) \\ge 0) + \\eta", "inference_order": 2184} {"schema_version": "1.0", "item_id": "rcd2_966db01a49c289fb", "pair_id": "pair_5d6baa6f9ca79f17", "statement_latex": "\\sup_{x\\in D}\\mathbf{1}_{\\{prime p\\in D:n > p < 2*n\\}}(x)=1", "inference_order": 2185} {"schema_version": "1.0", "item_id": "rcd2_e4f1f7b37dce275c", "pair_id": "pair_89ccc687bed1ad36", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(C(x) ‖φ‖_{∞})\\le t\\Rightarrow (|∫ Dφ(y)(x) dμ(y)|)\\le t + \\eta", "inference_order": 2186} {"schema_version": "1.0", "item_id": "rcd2_fdc4befaa8e4a544", "pair_id": "pair_b0af8de05e4bf912", "statement_latex": "\\operatorname{dist}((γ)-(ε^h / 4h),(-\\infty,0])=0", "inference_order": 2187} {"schema_version": "1.0", "item_id": "rcd2_34ec5865ac0efbad", "pair_id": "pair_f8f86032c5793c42", "statement_latex": "((x-c_{n})y_{n}-d_{n}y_{n-1}) \\\\ne (y_{n+1})", "inference_order": 2188} {"schema_version": "1.0", "item_id": "rcd2_e1e4ebb749df1d3e", "pair_id": "pair_9589f37f1cd42b77", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (|Q(x)|)\\Rightarrow t<((\\epsilon)) + \\eta", "inference_order": 2189} {"schema_version": "1.0", "item_id": "rcd2_16598d9b5c092193", "pair_id": "pair_22f49b5618a83d9b", "statement_latex": "\\delta_D((f(x)),(s))\\\\ne0", "inference_order": 2190} {"schema_version": "1.0", "item_id": "rcd2_2b56ec612096c89a", "pair_id": "pair_711aca0471919f42", "statement_latex": "\\neg\\left(\\forall F:D\\to\\{0,1\\},\\;F((\\binom{n-1}{k} + \\binom{n-1}{k-1}))=F((\\binom{n}{k}))\\right)", "inference_order": 2191} {"schema_version": "1.0", "item_id": "rcd2_c5ae805707bb3d85", "pair_id": "pair_817f725eb1a08f96", "statement_latex": "(Pr(S_n > t)) - (\\exp(-(sigma^2 / a^2) * h(at / sigma^2))) \\le 0", "inference_order": 2192} {"schema_version": "1.0", "item_id": "rcd2_04992f36dc4c8cda", "pair_id": "pair_f5f44625a814d1f5", "statement_latex": "\\nexists S \\subseteq D: (((f'(c)) \\in S \\land ((f(b)-f(a))/(b-a)) \\notin S) \\lor (((f(b)-f(a))/(b-a)) \\in S \\land (f'(c)) \\notin S))", "inference_order": 2193} {"schema_version": "1.0", "item_id": "rcd2_bba38e27fb61a0b2", "pair_id": "pair_e883f79bcd902127", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(tr W(A, A*)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(tr W(B, B*)))", "inference_order": 2194} {"schema_version": "1.0", "item_id": "rcd2_ad9fdf6fbb73afa3", "pair_id": "pair_d8c587f203bea9d3", "statement_latex": "((sum_i a_i*c_i)*(sum_j b_j*d_j) \\\\ne (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i) - (j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)) < 0", "inference_order": 2195} {"schema_version": "1.0", "item_id": "rcd2_b10833c4ff765ac2", "pair_id": "pair_392ae06abdd11f35", "statement_latex": "(k e_k \\\\ne \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i) \\land (0 = 0)", "inference_order": 2196} {"schema_version": "1.0", "item_id": "rcd2_37642fccea4f12cc", "pair_id": "pair_fee2b12a5fce58e1", "statement_latex": "\\forall z\\in D,\\;((f'(x)),z)=(((y),z)) + \\eta", "inference_order": 2197} {"schema_version": "1.0", "item_id": "rcd2_aa3875812f56a304", "pair_id": "pair_8b53859eff59799d", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((I(integral f d gamma^n))-(integral sqrt(I(f)^2 + |grad f|^2) d gamma^n))\\\\ne0", "inference_order": 2198} {"schema_version": "1.0", "item_id": "rcd2_3fa7fe27c8bb3e44", "pair_id": "pair_a0c36ec0a7996ca0", "statement_latex": "(n) \\\\ne (chi'_l(K_{n,n}))", "inference_order": 2199} {"schema_version": "1.0", "item_id": "rcd2_2a362a899d5e2dc9", "pair_id": "pair_610e19c03533835b", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(P(C_1 n log n <\\operatorname{argmin}_{z\\in D}\\delta_D(z,(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))", "inference_order": 2200} {"schema_version": "1.0", "item_id": "rcd2_0f5adc18d213ab75", "pair_id": "pair_2e66d390b061b2d9", "statement_latex": "(2^{2^{r^{2^{2^{k+9}}}}}) - (W(r,k)) \\ge 0", "inference_order": 2201} {"schema_version": "1.0", "item_id": "rcd2_9587acb5bcacbe28", "pair_id": "pair_6fb05331512d6780", "statement_latex": "\\nexists S \\subseteq D: (((L(-v)) \\in S \\land (-L(v)) \\notin S) \\lor ((-L(v)) \\in S \\land (L(-v)) \\notin S))", "inference_order": 2202} {"schema_version": "1.0", "item_id": "rcd2_125aa2859642e72a", "pair_id": "pair_ed69d8521880c339", "statement_latex": "\\nexists S \\subseteq D: ((((a/p)) \\in S \\land (\\varepsilon(pi_a)) \\notin S) \\lor ((\\varepsilon(pi_a)) \\in S \\land ((a/p)) \\notin S))", "inference_order": 2203} {"schema_version": "1.0", "item_id": "rcd2_c71ad74a0911080b", "pair_id": "pair_4d02a10055eda3e9", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((|A_d|))=F((\\frac{w(d)}{d}X + R_d))", "inference_order": 2204} {"schema_version": "1.0", "item_id": "rcd2_9ba8f667c7f84eef", "pair_id": "pair_1f605285e008e05b", "statement_latex": "\\frac{\\exp(Pr[Intersection(X_i ≤ u_i)])}{\\exp(Pr[Intersection(X_i ≤ u_i)])+\\exp(Pr[Intersection(Y_i ≤ u_i)])}\\le\\frac{1}{2}", "inference_order": 2205} {"schema_version": "1.0", "item_id": "rcd2_2157838478ca50db", "pair_id": "pair_48b02de8a3f2db85", "statement_latex": "\\operatorname{dist}((P(∪_{i=1}^∞ A_i))-(∑_{i=1}^∞ P(A_i)),(-\\infty,0])=0", "inference_order": 2206} {"schema_version": "1.0", "item_id": "rcd2_e341ce7bfbc967e4", "pair_id": "pair_8de08bdcfcb842ee", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(P(S_n in A))+s\\\\ne(binom(n,floor(n/2))/2^n)", "inference_order": 2207} {"schema_version": "1.0", "item_id": "rcd2_5d3ec810f8f3d5e7", "pair_id": "pair_f6e2cea990f695d0", "statement_latex": "(sum_{n \\in S} (1/n)) \\in {(1)}", "inference_order": 2208} {"schema_version": "1.0", "item_id": "rcd2_933d1f7c4a570618", "pair_id": "pair_1d9fb287ba536efb", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((f(x)))=F((⟨Aφ(x), φ(x)⟩))", "inference_order": 2209} {"schema_version": "1.0", "item_id": "rcd2_911dbed887181438", "pair_id": "pair_f250270489ac1d01", "statement_latex": "\\max\\{0,(0)-(d_)\\}\\\\ne0\\;\\land\\;(0)\\ne (d_)", "inference_order": 2210} {"schema_version": "1.0", "item_id": "rcd2_ade69f2fe4b188c6", "pair_id": "pair_043336a5b4a5a121", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\sum _{k=(0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,({\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}))) + \\eta", "inference_order": 2211} {"schema_version": "1.0", "item_id": "rcd2_914ba9e86b167a6d", "pair_id": "pair_d6e0b0b00cca44a6", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(n)\\\\nez\\;\\land\\;z=(p1 + p2 + p3)", "inference_order": 2212} {"schema_version": "1.0", "item_id": "rcd2_041f731cd54aaa3f", "pair_id": "pair_6587a84d7e078c85", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (n max_{|z|=1}|P(z)|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (max_{|z|=1}|P'(z)|)\\}", "inference_order": 2213} {"schema_version": "1.0", "item_id": "rcd2_3f1f578d0fa8eb45", "pair_id": "pair_6d8a1c052d5de90f", "statement_latex": "\\max\\{0,(|d(X,Y) - d(A,B)|)-(ε)\\}=(0) + \\eta", "inference_order": 2214} {"schema_version": "1.0", "item_id": "rcd2_0f0179732403d141", "pair_id": "pair_ef4a40a8f58ebe6e", "statement_latex": "(\\exp(-(sigma^2 / a^2) * h(at / sigma^2))) - (Pr(S_n > t)) \\ge 0", "inference_order": 2215} {"schema_version": "1.0", "item_id": "rcd2_c2ffadc42182df9b", "pair_id": "pair_219ead495b7f1496", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(z=((x_{1}+x_{2}+\\cdots +x_{m})^{n})\\Longleftrightarrow z=(\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}))\\right)", "inference_order": 2216} {"schema_version": "1.0", "item_id": "rcd2_275e02b7ca817fd0", "pair_id": "pair_ca4dac014aa2bef9", "statement_latex": "\\{(([I - A]x),z):z\\in D\\}=\\{((d),z):z\\in D\\}", "inference_order": 2217} {"schema_version": "1.0", "item_id": "rcd2_19a65c1daf3586a1", "pair_id": "pair_7b95bf8aadcdb6a6", "statement_latex": "(\\{(max_matching_size)\\}\\times\\{(1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|))\\})\\cap\\{(z,z):z\\in D\\}=\\{((max_matching_size),(1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)))\\}", "inference_order": 2218} {"schema_version": "1.0", "item_id": "rcd2_34a380b8d5e4a737", "pair_id": "pair_29ffa827a7235267", "statement_latex": "\\delta_D(([x^2 + y^2 + (x+y)^2]^2),(2[x^4 + y^4 + (x+y)^4]))=0", "inference_order": 2219} {"schema_version": "1.0", "item_id": "rcd2_576a83b0949109be", "pair_id": "pair_b13f62c3d632b667", "statement_latex": "(c * d^L) - (p(a,d)) \\\\le 0", "inference_order": 2220} {"schema_version": "1.0", "item_id": "rcd2_2bc9a87801d3aafc", "pair_id": "pair_8534816df4bcf533", "statement_latex": "\\max\\{0,(Pr(max_{1≤k≤n}|S_{k}|≥λ))-((1/λ^2) * Var[S_{n}])\\}=0", "inference_order": 2221} {"schema_version": "1.0", "item_id": "rcd2_2993bb8d22801e8a", "pair_id": "pair_155f0177343a183c", "statement_latex": "\\exists x \\\\notin \\mathbb{R}: (E(R_b)) \\le x \\land x \\le (E(X^2) / E(X))", "inference_order": 2222} {"schema_version": "1.0", "item_id": "rcd2_893dd8586f6d9ac5", "pair_id": "pair_99b4eda2197da60e", "statement_latex": "\\{(a^2 - b^2)\\}\\\\ne\\{(a + b)(a - b)\\}", "inference_order": 2223} {"schema_version": "1.0", "item_id": "rcd2_a52ae08957c305a0", "pair_id": "pair_46c17cca4c7f4c53", "statement_latex": "\\nexists S \\subseteq D: ((([[u, 0], [0, u^-1]]) \\in S \\land (e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)) \\notin S) \\lor ((e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)) \\in S \\land ([[u, 0], [0, u^-1]]) \\notin S))", "inference_order": 2224} {"schema_version": "1.0", "item_id": "rcd2_e2db6828dafcb10c", "pair_id": "pair_2e5faa5aa8b7a037", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (AB * CD + BC * DA)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (AC * BD)\\}", "inference_order": 2225} {"schema_version": "1.0", "item_id": "rcd2_55d8c68928a5f177", "pair_id": "pair_884e51f3bea2d75d", "statement_latex": "\\delta_D((sum_{k=(-a}^{a} (-1)^k * (2a choose k+a)^3),((3a)! / (a!)^3))=0) + \\eta_{1341}", "inference_order": 2226} {"schema_version": "1.0", "item_id": "rcd2_7ce3f6051346c22b", "pair_id": "pair_edf63d998430db87", "statement_latex": "\\{(SO_{\\gamma _{H}}(1_{K_{H}}))\\}\\\\ne\\{(\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}}))\\}", "inference_order": 2227} {"schema_version": "1.0", "item_id": "rcd2_577277997be6db38", "pair_id": "pair_e84bb06036752d31", "statement_latex": "\\max\\{0,(∫ |∇u*|^p dx)-(∫ |∇u|^p dx)\\}=(0) + \\eta", "inference_order": 2228} {"schema_version": "1.0", "item_id": "rcd2_bb52f4c29cad4ac4", "pair_id": "pair_affd2ac491f91cee", "statement_latex": "\\frac{\\exp(Pr(|X-mu| ≥ k*sigma))}{\\exp(Pr(|X-mu| ≥ k*sigma))+\\exp(1/k^2)}\\le\\frac{1}{2}", "inference_order": 2229} {"schema_version": "1.0", "item_id": "rcd2_c9ad6db8a7f22b02", "pair_id": "pair_fb13a60d725a9c83", "statement_latex": "\\frac{\\exp(φ(Φ^{-1}(γ^n(A))))}{\\exp(φ(Φ^{-1}(γ^n(A))))+\\exp(liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)})}\\le\\frac{1}{2} + \\eta", "inference_order": 2230} {"schema_version": "1.0", "item_id": "rcd2_60cf00adafbbc993", "pair_id": "pair_344fe38d728a1c06", "statement_latex": "\\neg\\left(((x^a - y^b),(1))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 2231} {"schema_version": "1.0", "item_id": "rcd2_19b594f55f72dcd5", "pair_id": "pair_6e3df240b9fbcf9a", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(L))\\\\ne\\operatorname{argmin}_{z\\in D}\\delta_D(z,(lambda*W))", "inference_order": 2232} {"schema_version": "1.0", "item_id": "rcd2_2552d3569686b8b8", "pair_id": "pair_9251a483df3146da", "statement_latex": "(N(X+Y)) \\in (N(X) + N(Y)) + [0,\\infty) + \\eta", "inference_order": 2233} {"schema_version": "1.0", "item_id": "rcd2_52f5c8ef79bb0d67", "pair_id": "pair_d933dc94efced2ce", "statement_latex": "\\forall z\\in D,\\;((∫[0,∞] x^{s-1} f(x) dx),z)=((Γ(s) φ(-s)),z)", "inference_order": 2234} {"schema_version": "1.0", "item_id": "rcd2_dead898a1bad17a9", "pair_id": "pair_a9b53de050384a1c", "statement_latex": "\\forall z\\in D,\\;((\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S}),z)=((\\int_{V} (\\nabla \\cdot \\mathbf{F}) dV),z)", "inference_order": 2235} {"schema_version": "1.0", "item_id": "rcd2_1125bb34b24596e3", "pair_id": "pair_fbc21740d6f87c50", "statement_latex": "\\neg\\left((\\pi(x;q,a)) - (2x / (phi(q) log(x/q))) \\le 0\\right)", "inference_order": 2236} {"schema_version": "1.0", "item_id": "rcd2_86355f4b84e0ed6a", "pair_id": "pair_fcfa08cf5b3fd30e", "statement_latex": "(L(E, s) = sum_{n=1}^∞ a_n / n^s) \\land (0 = 0)", "inference_order": 2237} {"schema_version": "1.0", "item_id": "rcd2_79286948df3bcab7", "pair_id": "pair_4470def769e1c7d1", "statement_latex": "((N^{(i-1)}),(\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}))\\in\\{(z,z):z\\in D\\}", "inference_order": 2238} {"schema_version": "1.0", "item_id": "rcd2_3c6e0e6d859f0877", "pair_id": "pair_a7eb5ac5b60c3682", "statement_latex": "(⟨j m | T_q^(k) | j' m'⟩) \\in {(⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩)}", "inference_order": 2239} {"schema_version": "1.0", "item_id": "rcd2_bac0443442a43608", "pair_id": "pair_fdfcaf1f654e1d01", "statement_latex": "\\{t\\in\\mathbb{R}:t\\le (|d(A,X) - d(B,X)|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (d(A,B))\\} + \\eta", "inference_order": 2240} {"schema_version": "1.0", "item_id": "rcd2_0b17490087fc2380", "pair_id": "pair_0bfcd78a3e6c3b4d", "statement_latex": "((a^n + b^n),(c^n))\\in\\{(z,z):z\\in D\\}", "inference_order": 2241} {"schema_version": "1.0", "item_id": "rcd2_43ebfc9c5f2bf842", "pair_id": "pair_b8ce68c65024a74d", "statement_latex": "(\\text{card}(\\mathbb{K}^I)) = (\\dim(E^*))", "inference_order": 2242} {"schema_version": "1.0", "item_id": "rcd2_c7a342fd7d865d7d", "pair_id": "pair_599a8bc2f784ae61", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(a_n)=(z\\;\\land\\;z=(a_{n+gcd(h,k)})) + \\eta", "inference_order": 2243} {"schema_version": "1.0", "item_id": "rcd2_692f5ed38c16157a", "pair_id": "pair_70586b2f3cce3afe", "statement_latex": "((-1)^n) = (F_{n-1}*F_{n+1} - F_n^2)", "inference_order": 2244} {"schema_version": "1.0", "item_id": "rcd2_e4f7da0f4be53d58", "pair_id": "pair_03c6f040b5a258ee", "statement_latex": "\\frac{\\exp(det(M))}{\\exp(det(M))+\\exp(det(A)*det(C))}\\le\\frac{1}{2} + \\eta", "inference_order": 2245} {"schema_version": "1.0", "item_id": "rcd2_f8a294fba4a8dd4b", "pair_id": "pair_36b45228d893c764", "statement_latex": "\\forall z\\in D,\\;(((x_{1}+x_{2}+\\cdots +x_{m})^{n}),z)=((\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}),z)", "inference_order": 2246} {"schema_version": "1.0", "item_id": "rcd2_b1f90d2e3a782a75", "pair_id": "pair_0a813312aa42184e", "statement_latex": "\\delta_D((1/(1-alpha*z)),(product_{j=(1}^{infinity} (1/(1-z^j))^{M(alpha, j)}))=0) + \\eta", "inference_order": 2247} {"schema_version": "1.0", "item_id": "rcd2_30b6e21721c27e48", "pair_id": "pair_985c72ef90f37f3d", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(H))=(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(H_0 + gV))) + \\eta", "inference_order": 2248} {"schema_version": "1.0", "item_id": "rcd2_373f29e202e68b1d", "pair_id": "pair_fcf9351886792448", "statement_latex": "\\neg\\left(((\\dim(E^*)),(\\text{card}(\\mathbb{K}^I)))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 2249} {"schema_version": "1.0", "item_id": "rcd2_8bdfbdf4986bdfc7", "pair_id": "pair_64196069f8a5e0d0", "statement_latex": "\\forall z\\in D,\\;((P(C_1 n log n > inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n)),z)=((1),z)", "inference_order": 2250} {"schema_version": "1.0", "item_id": "rcd2_f11f844daf54447d", "pair_id": "pair_f0a840eb2d31a7dc", "statement_latex": "\\neg(\\neg (Pr(max_{1\\lek\\len} |S_k| \\ge 3alpha) \\le 3 max_{1\\lek\\len} Pr(|S_k| \\ge alpha))) + \\eta", "inference_order": 2251} {"schema_version": "1.0", "item_id": "rcd2_1594bd9a579dfe2a", "pair_id": "pair_6efa1642aed7867d", "statement_latex": "((A),(k[y_1, ..., y_m]))\\in\\{(z,z):z\\in D\\} + \\eta", "inference_order": 2252} {"schema_version": "1.0", "item_id": "rcd2_40048c68d319c79f", "pair_id": "pair_d7c5cd3d3411e6f6", "statement_latex": "\\neg\\left(\\exists x \\\\in \\mathbb{R}: ([b] for a majorizing b) \\le x \\land x \\le ([a])\\right)", "inference_order": 2253} {"schema_version": "1.0", "item_id": "rcd2_91b1fe1dc43a7827", "pair_id": "pair_2e4e1c9254e8cd66", "statement_latex": "(E) \\in {(F(α))}", "inference_order": 2254} {"schema_version": "1.0", "item_id": "rcd2_75bbf59e9f885039", "pair_id": "pair_63589ffca89c8c5f", "statement_latex": "\\{(n)\\}=(\\{(p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k})\\}) + \\eta", "inference_order": 2255} {"schema_version": "1.0", "item_id": "rcd2_3ead0fb236adbcf7", "pair_id": "pair_7cd8419fc21512bd", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(sum_{k=0", "inference_order": 2256} {"schema_version": "1.0", "item_id": "rcd2_6151f6b035c13da9", "pair_id": "pair_53af381ae41c746a", "statement_latex": "\\forall z\\in D,\\;((max_matching_size),z)=((1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)),z)", "inference_order": 2257} {"schema_version": "1.0", "item_id": "rcd2_61fa5ba1681bf179", "pair_id": "pair_686ebaa6f3b4bf9d", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(prod_{i=(1}^n ||v_i||)\\le t\\Rightarrow (|det(N)|)\\le t) + \\eta", "inference_order": 2258} {"schema_version": "1.0", "item_id": "rcd2_203ad24190faf7f7", "pair_id": "pair_4a1c8057eb955840", "statement_latex": "\\neg\\left((\\neg (f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x))) \\Rightarrow (0 = 1)\\right)", "inference_order": 2259} {"schema_version": "1.0", "item_id": "rcd2_d688439786af3bd7", "pair_id": "pair_ed17e5525b651559", "statement_latex": "(|Irr_{p'}(N_{G}(P))|) = (|Irr_{p'}(G)|)", "inference_order": 2260} {"schema_version": "1.0", "item_id": "rcd2_6631bf8f668d1b46", "pair_id": "pair_5be65a306a45bf54", "statement_latex": "(z) = (x + y)", "inference_order": 2261} {"schema_version": "1.0", "item_id": "rcd2_a4804a253e45aa9d", "pair_id": "pair_db102220af295e7b", "statement_latex": "\\delta_D((Av),(rv))=0", "inference_order": 2262} {"schema_version": "1.0", "item_id": "rcd2_86d3ce6ddcb8f4de", "pair_id": "pair_a6b2fd605d8ba10e", "statement_latex": "\\forall z\\in D,\\;(z=(U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C)\\Longleftrightarrow z=(|\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C))", "inference_order": 2263} {"schema_version": "1.0", "item_id": "rcd2_de4d817ce4fa0a62", "pair_id": "pair_b258ca98dfb46b45", "statement_latex": "\\forall z\\in D,\\;(z=(constant_term(prod_{1≥i≠j≤n} (1-t_i/t_j)^{a_i}))\\Longleftrightarrow z=(sum a_i)! / prod(a_i!))", "inference_order": 2264} {"schema_version": "1.0", "item_id": "rcd2_0bb6ed38a9a784a7", "pair_id": "pair_0d2903de582c3d80", "statement_latex": "\\nexists S \\subseteq D: (((f(x)) \\in S \\land (0 (mod p^n)) \\notin S) \\lor ((0 (mod p^n)) \\in S \\land (f(x)) \\notin S))", "inference_order": 2265} {"schema_version": "1.0", "item_id": "rcd2_2584a59ed2dda73e", "pair_id": "pair_6477fb35b6ce3bb5", "statement_latex": "(p - q) \\in {((1/π) Δ arg f(iy))}", "inference_order": 2266} {"schema_version": "1.0", "item_id": "rcd2_fd594362b7e46094", "pair_id": "pair_02b1f398b3242405", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F(([[u, 0], [0, u^-1]]))=F((e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)))", "inference_order": 2267} {"schema_version": "1.0", "item_id": "rcd2_9c44462f20de09b1", "pair_id": "pair_b0af8de05e4bf912", "statement_latex": "\\neg\\left(\\operatorname{dist}((γ)-(ε^h / 4h),(-\\infty,0])=0\\right)", "inference_order": 2268} {"schema_version": "1.0", "item_id": "rcd2_b61cdf2bc3734510", "pair_id": "pair_2a3b1c84ee52f49f", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\ge (K|A|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|A+A|)\\}\\right)", "inference_order": 2269} {"schema_version": "1.0", "item_id": "rcd2_5292e4401ab86b3e", "pair_id": "pair_a269502a048b2be5", "statement_latex": "\\delta_D((lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)]),(sup_{x∈X} (φ(x) - I(x))))=0", "inference_order": 2270} {"schema_version": "1.0", "item_id": "rcd2_7d5d2027cf7f4c4a", "pair_id": "pair_e2424768e723319d", "statement_latex": "(integral f(x) dLambda(x)) = (V(p))", "inference_order": 2271} {"schema_version": "1.0", "item_id": "rcd2_0f690f47d45d6c84", "pair_id": "pair_3cd7edf7ec153348", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(∫[0,∞] x^{s-1} f(x) dx))=(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Γ(s) φ(-s)))) + \\eta", "inference_order": 2272} {"schema_version": "1.0", "item_id": "rcd2_95fadca70790a1d3", "pair_id": "pair_bd2a70232afd34c6", "statement_latex": "\\forall z\\in D,\\;(z=(SO_{\\gamma _{H}}(1_{K_{H}}))\\Longleftrightarrow z=(\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}})))", "inference_order": 2273} {"schema_version": "1.0", "item_id": "rcd2_f8d1c75cade941c7", "pair_id": "pair_771ee414763e4d8d", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(||u - u_h||)+s=(8/9)", "inference_order": 2274} {"schema_version": "1.0", "item_id": "rcd2_9dee64631a4888a9", "pair_id": "pair_c0fdd39f52248f81", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((f'(x_0)))=F((0))", "inference_order": 2275} {"schema_version": "1.0", "item_id": "rcd2_ae1a00b84b01fc24", "pair_id": "pair_e012754db701b114", "statement_latex": "\\nexists S \\subseteq D: (((Ax) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (Ax) \\notin S)) + \\eta", "inference_order": 2276} {"schema_version": "1.0", "item_id": "rcd2_9b1888ec20602ce1", "pair_id": "pair_ff31293c235667d3", "statement_latex": "\\forall z\\in D,\\;(z\\\\ne(mu(t))\\Longleftrightarrow z=(integral_{-infinity}^{t} A(s)p(s,t) ds))", "inference_order": 2277} {"schema_version": "1.0", "item_id": "rcd2_c9becb5894bec52b", "pair_id": "pair_bffaa425232ec001", "statement_latex": "\\operatorname{dist}((V(K)^((n-1)/n) V(L)^(1/n))-(V_1(K, L)),(-\\infty,0])\\\\ne0", "inference_order": 2278} {"schema_version": "1.0", "item_id": "rcd2_865b04ef61df53d3", "pair_id": "pair_97c0427a308c402c", "statement_latex": "\\neg(\\neg (constant_term(prod_{1\\lei\\nej\\len} (1-t_i/t_j)^{a_i}) =-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3))", "inference_order": 2279} {"schema_version": "1.0", "item_id": "rcd2_10c5bb88a878590f", "pair_id": "pair_a7dfa8a4b6d9a4e8", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (delta |X| |Y|)\\Rightarrow t<((e(X,Y))) + \\eta", "inference_order": 2280} {"schema_version": "1.0", "item_id": "rcd2_53a039950c75f367", "pair_id": "pair_f3c111bc249621b9", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(sum_{i=0", "inference_order": 2281} {"schema_version": "1.0", "item_id": "rcd2_847fe3890b71a87c", "pair_id": "pair_fc98d14aebda5dda", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(E(R_b))+s=(E(X^2) / E(X))", "inference_order": 2282} {"schema_version": "1.0", "item_id": "rcd2_eaa0d79e5cb773a4", "pair_id": "pair_3e8d80eb15a69d88", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(H(t)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))", "inference_order": 2283} {"schema_version": "1.0", "item_id": "rcd2_720e1f7be2e10e65", "pair_id": "pair_37cab1da852fb962", "statement_latex": "\\forall z\\in D,\\;(z=(w(S))\\Longleftrightarrow z=(sum_{x in S} w(x)))", "inference_order": 2284} {"schema_version": "1.0", "item_id": "rcd2_7d96bf4def292ca7", "pair_id": "pair_37525da9a6dfbce0", "statement_latex": "(\\{(F(z,t))\\}\\times\\{(e^{g(z)t})\\})\\cap\\{(z,z):z\\in D\\}=\\{((uv),(xy))\\}", "inference_order": 2285} {"schema_version": "1.0", "item_id": "rcd2_64b60b306ea5cd50", "pair_id": "pair_329b05ca30d70c53", "statement_latex": "([a]) - ([b] for a majorizing b) \\ge 0", "inference_order": 2286} {"schema_version": "1.0", "item_id": "rcd2_e0d71b0af25de4fe", "pair_id": "pair_2033f60eb4132d67", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((nullity A) \\in S \\land (nullity H) \\notin S) \\lor ((nullity H) \\in S \\land (nullity A) \\notin S))\\right)", "inference_order": 2287} {"schema_version": "1.0", "item_id": "rcd2_2440b0622d7ef1e8", "pair_id": "pair_b45782af4a3ca333", "statement_latex": "\\delta_D((f'(x_0)),(0))=0", "inference_order": 2288} {"schema_version": "1.0", "item_id": "rcd2_e3b3df24cf201b86", "pair_id": "pair_454cf526b96d8ff2", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(max|B_k|(|a_n| + |a_n - a_1|))\\le t\\Rightarrow (|∑ a_k b_k|)\\le t", "inference_order": 2289} {"schema_version": "1.0", "item_id": "rcd2_e7b343249d9140b4", "pair_id": "pair_9fd35727e6582bdf", "statement_latex": "\\forall z\\in D,\\;((1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p)),z)=((e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))),z)", "inference_order": 2290} {"schema_version": "1.0", "item_id": "rcd2_b0ffed41821d312b", "pair_id": "pair_787a2865884d14fd", "statement_latex": "\\frac{\\exp(|x(S_j)|)}{\\exp(|x(S_j)|)+\\exp(2t - 1)}\\le\\frac{1}{2}", "inference_order": 2291} {"schema_version": "1.0", "item_id": "rcd2_e29c151c223de9c7", "pair_id": "pair_dec9774504817ef1", "statement_latex": "\\{((\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q}),z):z\\in D\\}=\\{((\\infty),z):z\\in D\\}", "inference_order": 2292} {"schema_version": "1.0", "item_id": "rcd2_8c52647fa2a4dd41", "pair_id": "pair_9df08fa171dd3a32", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (Vol(M))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (prod_{i} d_{i})\\}", "inference_order": 2293} {"schema_version": "1.0", "item_id": "rcd2_81c9a8f9342760d0", "pair_id": "pair_39155343ca7c52d1", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((y_{n+1}))=F((\\sup\\{f(x, \\theta) : x \\in C(\\theta)\\}))", "inference_order": 2294} {"schema_version": "1.0", "item_id": "rcd2_925770f9c0d7615f", "pair_id": "pair_f8c235f3a597c692", "statement_latex": "(lim_{x -) - (1^-} G(x) = sum_{k=0}^\\infty a_k) > 0", "inference_order": 2295} {"schema_version": "1.0", "item_id": "rcd2_d702dd1cb8bf9934", "pair_id": "pair_fd4e5a4cf775a9b1", "statement_latex": "\\{((\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S}),z):z\\in D\\}=\\{((\\int_{V} (\\nabla \\cdot \\mathbf{F}) dV),z):z\\in D\\}", "inference_order": 2296} {"schema_version": "1.0", "item_id": "rcd2_2956e002fed20ab3", "pair_id": "pair_650d60dfd9acb811", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (|x(S_j)|) \\le x \\land x \\le (2t - 1)", "inference_order": 2297} {"schema_version": "1.0", "item_id": "rcd2_de7b936da0b2d977", "pair_id": "pair_9b08d40f18bfdedf", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((dim(H_μ)) \\in S \\land (n!) \\notin S) \\lor ((n!) \\in S \\land (dim(H_μ)) \\notin S))\\right)", "inference_order": 2298} {"schema_version": "1.0", "item_id": "rcd2_3ed80810ff18d883", "pair_id": "pair_e5a51c0d7364c6e2", "statement_latex": "\\forall z\\in D,\\;((f(x)),z)=((s),z)", "inference_order": 2299} {"schema_version": "1.0", "item_id": "rcd2_a254932073f897c8", "pair_id": "pair_f2f41320160ca368", "statement_latex": "\\{(H)\\}=\\{(∑_{x,y} ψ^†(x) F(x, y) ψ(y))\\}", "inference_order": 2300} {"schema_version": "1.0", "item_id": "rcd2_a109039f86ec3e52", "pair_id": "pair_9b08d40f18bfdedf", "statement_latex": "\\nexists S \\subseteq D: (((dim(H_μ)) \\in S \\land (n!) \\notin S) \\lor ((n!) \\in S \\land (dim(H_μ)) \\notin S))", "inference_order": 2301} {"schema_version": "1.0", "item_id": "rcd2_db888f2b9c2f6445", "pair_id": "pair_8a628b56f43fff3e", "statement_latex": "\\neg\\left(\\forall F:D\\to\\{0,1\\},\\;F((E))=F((nhv))\\right)", "inference_order": 2302} {"schema_version": "1.0", "item_id": "rcd2_da425914fdcfab0c", "pair_id": "pair_087ab70c2ef123b0", "statement_latex": "(\\{(I^nM ∩ N)\\}\\times\\{(I^{n-k}(I^kM ∩ N))\\})\\cap\\{(z,z):z\\in D\\}=\\{((I^nM ∩ N),(I^{n-k}(I^kM ∩ N)))\\}", "inference_order": 2303} {"schema_version": "1.0", "item_id": "rcd2_95c6eb39434b8c12", "pair_id": "pair_0874896b58313452", "statement_latex": "\\exists x \\\\notin \\mathbb{R}: (ζ(n) / 2^(n-1)) \\le x \\land x \\le (Δ)", "inference_order": 2304} {"schema_version": "1.0", "item_id": "rcd2_9563b3e84bb19824", "pair_id": "pair_019ca791a0885a9d", "statement_latex": "\\frac{\\exp((1/x) * sum_{n≤x} |f(n) - A(x)|^2)}{\\exp((1/x) * sum_{n≤x} |f(n) - A(x)|^2)+\\exp((2 + epsilon(x)) * B(x)^2)}\\le\\frac{1}{2}", "inference_order": 2305} {"schema_version": "1.0", "item_id": "rcd2_9753fa09ed0b8e9a", "pair_id": "pair_d347f923f1ef78a4", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Integral transform+Spectral terms))", "inference_order": 2306} {"schema_version": "1.0", "item_id": "rcd2_f2fcb4f38217cc6a", "pair_id": "pair_bf2f027430e9633e", "statement_latex": "\\neg\\left(\\{(([[u, 0], [0, u^-1]]),z):z\\in D\\}=\\{((e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)),z):z\\in D\\}\\right)", "inference_order": 2307} {"schema_version": "1.0", "item_id": "rcd2_fdde63d7e0c1d462", "pair_id": "pair_bbb955b1f7b435b2", "statement_latex": "\\frac{\\exp(max|X_j|)}{\\exp(max|X_j|)+\\exp(D^{-1}*sqrt(det(AA^T)))^(1/(N-M))}\\le\\frac{1}{2}", "inference_order": 2308} {"schema_version": "1.0", "item_id": "rcd2_ef7ab71ec22ef43e", "pair_id": "pair_0c7387c59209c4e8", "statement_latex": "\\nexists S \\subseteq D: (((f) \\in S \\land (g ∘ T) \\notin S) \\lor ((g ∘ T) \\in S \\land (f) \\notin S))", "inference_order": 2309} {"schema_version": "1.0", "item_id": "rcd2_07e383b362aa8a51", "pair_id": "pair_5b142a2707249663", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(L)=z\\;\\land\\;z=(lambda*W)", "inference_order": 2310} {"schema_version": "1.0", "item_id": "rcd2_fac3875e17459c54", "pair_id": "pair_a3f122350d6acb42", "statement_latex": "\\delta_D(((x_{1}+x_{2}+\\cdots +x_{m})^{n}),(\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}))=0", "inference_order": 2311} {"schema_version": "1.0", "item_id": "rcd2_89cd721d983ade70", "pair_id": "pair_30c813cf58194699", "statement_latex": "\\{((A(x)),z):z\\in D\\}\\\\ne\\{(((1 / (2*pi*i)) * integral_{c-i*infinity}^{c+i*infinity} g(z) * (x^z / z) dz),z):z\\in D\\}", "inference_order": 2312} {"schema_version": "1.0", "item_id": "rcd2_e0f36614bc9c9d98", "pair_id": "pair_7f358154390755c8", "statement_latex": "\\max\\{0,(|g(z)|)-(|f(z)|)\\}\\\\ne0\\;\\land\\;(|g(z)|)\\ne (|f(z)|)", "inference_order": 2313} {"schema_version": "1.0", "item_id": "rcd2_70d2fbb16e7c30fd", "pair_id": "pair_8d207d5237062be0", "statement_latex": "\\frac{\\exp(∑_{n<1}P(|S_{n}|>εn))}{\\exp(∑_{n≥1}P(|S_{n}|>εn))+\\exp(∞)}<\\frac{1}{2}", "inference_order": 2314} {"schema_version": "1.0", "item_id": "rcd2_2cb28ca6f59a2372", "pair_id": "pair_70439c43ccdf3fb5", "statement_latex": "\\nexists S \\subseteq D: (((\\binom{k}{r} \\frac{r}{k}) \\in S \\land (\\binom{k-1}{r-1}) \\notin S) \\lor ((\\binom{k-1}{r-1}) \\in S \\land (\\binom{k}{r} \\frac{r}{k}) \\notin S))", "inference_order": 2315} {"schema_version": "1.0", "item_id": "rcd2_0078c099eead9d38", "pair_id": "pair_8e20151f31fc6528", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C))=F((|\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C))", "inference_order": 2316} {"schema_version": "1.0", "item_id": "rcd2_2ca7dfac045ea5b3", "pair_id": "pair_425ecc3051c1bf98", "statement_latex": "(det(A)) \\in {(det(A_v^u) * det(A_tilde_v^u))}", "inference_order": 2317} {"schema_version": "1.0", "item_id": "rcd2_bf94654b4f9a2707", "pair_id": "pair_e40ae504348a33a1", "statement_latex": "\\{((det(I_m + AB)),z):z\\in D\\}=\\{((\\text{conv}\\{w(X) : w \\in W\\}),z):z\\in D\\}", "inference_order": 2318} {"schema_version": "1.0", "item_id": "rcd2_379368c4ab265f7d", "pair_id": "pair_65e449574dea906b", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(∫[0,∞] x^{s-1} f(x) dx)=(z\\;\\land\\;z=(Γ(s) φ(-s))) + \\eta", "inference_order": 2319} {"schema_version": "1.0", "item_id": "rcd2_e28d890b7d0071d2", "pair_id": "pair_0c2c708db83a7fee", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (p_{n+1}^2)\\Rightarrow t\\\\ge(p_n#)", "inference_order": 2320} {"schema_version": "1.0", "item_id": "rcd2_85b60f83c70c950f", "pair_id": "pair_93fe24a690c51338", "statement_latex": "(\\{(F(X))\\}\\times\\{(inf_S sup_I (|I|/|S|))\\})\\cap\\{(z,z):z\\in D\\}=\\{((F(X)),(inf_S sup_I (|I|/|S|)))\\}", "inference_order": 2321} {"schema_version": "1.0", "item_id": "rcd2_43745c941b0f0976", "pair_id": "pair_f1bde4e824a4bd79", "statement_latex": "(\\dot{q}) \\in {(f(q,u))} + \\eta", "inference_order": 2322} {"schema_version": "1.0", "item_id": "rcd2_3c74fee1471bead6", "pair_id": "pair_aabc76c8b75a4e65", "statement_latex": "(y_{n+1}) \\in {((x-c_{n})y_{n}-d_{n}y_{n-1})}", "inference_order": 2323} {"schema_version": "1.0", "item_id": "rcd2_040e0c69fc864cb1", "pair_id": "pair_18b356fc7abcd227", "statement_latex": "\\nexists S \\subseteq D: (((U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C) \\in S \\land (|\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C) \\notin S) \\lor ((|\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C) \\in S \\land (U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C) \\notin S))", "inference_order": 2324} {"schema_version": "1.0", "item_id": "rcd2_55b78034bad68fbd", "pair_id": "pair_b9c888410517f10e", "statement_latex": "\\sup_{\\lambda\\in[0,\\infty)}\\lambda((floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)))-(1/2))=+\\infty", "inference_order": 2325} {"schema_version": "1.0", "item_id": "rcd2_7585478edbd6d0c2", "pair_id": "pair_d4467766bca8edf1", "statement_latex": "\\exists x\\in D:x\\in \\{prime p\\in D:n < p < 2*n\\}", "inference_order": 2326} {"schema_version": "1.0", "item_id": "rcd2_a60654c119413958", "pair_id": "pair_f1a020820d740852", "statement_latex": "\\neg\\left(\\exists z\\in D\\;\\text{such that}\\;(∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c))=z\\;\\land\\;z=(Integral transform+Spectral terms)\\right)", "inference_order": 2327} {"schema_version": "1.0", "item_id": "rcd2_9bdd936dd6f7807f", "pair_id": "pair_d488e304858413d1", "statement_latex": "\\forall z\\in D,\\;(z=(lim_{x→c} f(x)/g(x))\\Longleftrightarrow z=(lim_{x→c} f'(x)/g'(x)))", "inference_order": 2328} {"schema_version": "1.0", "item_id": "rcd2_a4257263dce9415a", "pair_id": "pair_0492dd723f6d2eca", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(sum_{i=0}^{d} binom(n,i))\\le t\\Rightarrow (|F|)\\le t", "inference_order": 2329} {"schema_version": "1.0", "item_id": "rcd2_f4256470099e4754", "pair_id": "pair_892c715e77ffab0a", "statement_latex": "(s(f(x))) = (f(s(x)))", "inference_order": 2330} {"schema_version": "1.0", "item_id": "rcd2_003b90410ebd7a92", "pair_id": "pair_9f12bcb40fea87e0", "statement_latex": "\\neg(\\neg (\\z\\eta(s) = product_{j=1}^{\\infty} (1/(1-z^j))^{M(alpha, j)}))", "inference_order": 2331} {"schema_version": "1.0", "item_id": "rcd2_8e85f992e49f523e", "pair_id": "pair_433fcacea4695816", "statement_latex": "\\neg(\\neg ((1/x) * sum_{n\\lex} |f(n) - A(x)|^2 \\le (2 + \\varepsilon(x)) * B(x)^2))", "inference_order": 2332} {"schema_version": "1.0", "item_id": "rcd2_8008892ac7412c22", "pair_id": "pair_44a1173943624fe6", "statement_latex": "\\max\\{0,(|∫ Dφ(y)(x) dμ(y)|)-(C(x) ‖φ‖_{∞})\\}=0", "inference_order": 2333} {"schema_version": "1.0", "item_id": "rcd2_72c2fa0be3229da8", "pair_id": "pair_f6a13b8d25eaec18", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(z=(f(x_1, ..., x_n))\\Longleftrightarrow z=(Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))))\\right)", "inference_order": 2334} {"schema_version": "1.0", "item_id": "rcd2_34cc1f8a3a253cf8", "pair_id": "pair_3b5e5e6f6aa930c7", "statement_latex": "\\operatorname{dist}((ζ(n) / 2^(n-1))-(Δ),(-\\infty,0])=0", "inference_order": 2335} {"schema_version": "1.0", "item_id": "rcd2_8792021373e89269", "pair_id": "pair_bb5ff01b221870b2", "statement_latex": "\\max\\{0,(|alpha - beta|)-(|alpha - alpha_i|)\\}=0\\;\\land\\;(p_{n+1}^2)\\ne (p_n#)", "inference_order": 2336} {"schema_version": "1.0", "item_id": "rcd2_2bceec7978731fc6", "pair_id": "pair_f55fe451c92d8686", "statement_latex": "\\frac{\\exp(P(X_N - X_0 ≤ epsilon))}{\\exp(P(X_N - X_0 ≥ epsilon))+\\exp(exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2)))}\\le\\frac{1}{2}", "inference_order": 2337} {"schema_version": "1.0", "item_id": "rcd2_c7ac6cc761d45e5f", "pair_id": "pair_18c902475bdd0794", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(sum_{i\\\\ne0}^n binom(n,i))=z\\;\\land\\;z=(2^n)", "inference_order": 2338} {"schema_version": "1.0", "item_id": "rcd2_9179421b66927cd1", "pair_id": "pair_e237bd794be802d1", "statement_latex": "\\forall z\\in D,\\;(z=((Pr(Σ_t=A) / Pr(Σ_t=-A))\\Longleftrightarrow z=(e^(At)))) + \\eta", "inference_order": 2339} {"schema_version": "1.0", "item_id": "rcd2_70bb68cb1e918a99", "pair_id": "pair_a94bf23ca7aa472c", "statement_latex": "\\{t\\in\\mathbb{R}:t\\le (AC * BD)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (AB * CD + BC * DA)\\}", "inference_order": 2340} {"schema_version": "1.0", "item_id": "rcd2_c9dd53980c8b0774", "pair_id": "pair_da8b5f16a6f1ec06", "statement_latex": "(SMS^T) \\in {(D ⊕ D)} + \\eta", "inference_order": 2341} {"schema_version": "1.0", "item_id": "rcd2_5e7780cb999e6894", "pair_id": "pair_c708422eeac02032", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (1/2)\\Rightarrow t<((floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)))) + \\eta", "inference_order": 2342} {"schema_version": "1.0", "item_id": "rcd2_b2d7a71a952e771c", "pair_id": "pair_73dfc1d4003e5e81", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(f^{-1})'(b)=z\\;\\land\\;z=(1 / f'(a))", "inference_order": 2343} {"schema_version": "1.0", "item_id": "rcd2_385ade6cf64fb932", "pair_id": "pair_987558619cdf15e6", "statement_latex": "\\neg\\left(\\{((\\nu(\\chi(1))),z):z\\in D\\}=\\{((\\nu(|G:D|)),z):z\\in D\\}\\right)", "inference_order": 2344} {"schema_version": "1.0", "item_id": "rcd2_eae07cc59e08625f", "pair_id": "pair_f6e2cea990f695d0", "statement_latex": "(sum_{n \\in S} (1/n)) \\in {(1)} + \\eta", "inference_order": 2345} {"schema_version": "1.0", "item_id": "rcd2_8d2f66ad3b4dbba8", "pair_id": "pair_fcfa08cf5b3fd30e", "statement_latex": "(L(E, s) \\\\ne sum_{n=1}^∞ a_n / n^s) \\land (0 = 0)", "inference_order": 2346} {"schema_version": "1.0", "item_id": "rcd2_ffec2bc0daca0392", "pair_id": "pair_a182580d78d10065", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (max(|A + A|, |A * A|))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (c|A|^{1 + ε})\\}", "inference_order": 2347} {"schema_version": "1.0", "item_id": "rcd2_4cd8d9cae9ab5d29", "pair_id": "pair_cfce204f4169051a", "statement_latex": "\\exists x \\\\notin \\mathbb{R}: (sqrt(x*y)) \\le x \\land x \\le ((x + y)/2)", "inference_order": 2348} {"schema_version": "1.0", "item_id": "rcd2_8ac3fbd373644595", "pair_id": "pair_09d4cdd1f7021c91", "statement_latex": "\\neg\\left((\\{(φ(+∞) − φ(0))\\}\\times\\{(−π(1/2*n0 + nb))\\})\\cap\\{(z,z):z\\in D\\}=\\{((φ(+∞) − φ(0)),(−π(1/2*n0 + nb)))\\}\\right)", "inference_order": 2349} {"schema_version": "1.0", "item_id": "rcd2_ccbf35656f90e8fb", "pair_id": "pair_72926438a6fa0134", "statement_latex": "\\frac{\\exp(\\sum_{n=0}^{\\infty} n|a_n|^2)}{\\exp(\\sum_{n=0}^{\\infty} n|a_n|^2)+\\exp(1)}\\le\\frac{1}{2}", "inference_order": 2350} {"schema_version": "1.0", "item_id": "rcd2_bdfeb27a0cfd2b26", "pair_id": "pair_e509ffee2b4ca75c", "statement_latex": "\\{((E[sum_{x in N} f(x)]),z):z\\in D\\}=\\{((integral f(x) dLambda(x)),z):z\\in D\\}", "inference_order": 2351} {"schema_version": "1.0", "item_id": "rcd2_c7e9f232585dc265", "pair_id": "pair_142e900971ec82b3", "statement_latex": "\\neg\\left(\\frac{\\exp(Pr[A])}{\\exp(Pr[A])+\\exp(x(A) * Product_{B in Gamma(A)} (1 - x(B)))}\\le\\frac{1}{2}\\right)", "inference_order": 2352} {"schema_version": "1.0", "item_id": "rcd2_f60a443a376b7450", "pair_id": "pair_2030cfcd72207041", "statement_latex": "\\neg\\left(\\exists x \\\\in \\mathbb{R}: (det(M)) \\le x \\land x \\le (det(A)*det(C))\\right)", "inference_order": 2353} {"schema_version": "1.0", "item_id": "rcd2_1debddf48a890c05", "pair_id": "pair_ad6be3869f88f81e", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(\\nu(\\chi(1))))+\\delta_D(z,(\\nu(|G:D|))))\\\\ne0", "inference_order": 2354} {"schema_version": "1.0", "item_id": "rcd2_b46056b00fdcea6d", "pair_id": "pair_3fac4342f01fe341", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,({m+n \\choose r}))+\\delta_D(z,(\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k})))=0", "inference_order": 2355} {"schema_version": "1.0", "item_id": "rcd2_23ee629edc8aa69e", "pair_id": "pair_414a08db6e185d76", "statement_latex": "\\forall r\\in\\mathbb{Q}_{>0}\\exists N\\forall n\\ge N:d((X × Y),(X)) 0) + \\eta", "inference_order": 2390} {"schema_version": "1.0", "item_id": "rcd2_43493e93f21a6269", "pair_id": "pair_4797b17b01882b18", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(L(E, s)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{n=1}^∞ a_n / n^s))", "inference_order": 2391} {"schema_version": "1.0", "item_id": "rcd2_7c09cdce6a70d80a", "pair_id": "pair_96c40e4b184a3cee", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (V(K)^((n-1)/n) V(L)^(1/n)) \\le x \\land x \\le (V_1(K, L))) + \\eta_{277}", "inference_order": 2392} {"schema_version": "1.0", "item_id": "rcd2_58a79061f633fe57", "pair_id": "pair_0bb95d3c9989a17b", "statement_latex": "\\frac{\\exp(Pr[P(r1, ..., rn) =( 0])}{\\exp(Pr[P(r1, ..., rn) = 0])+\\exp(d/|S|)}\\le\\frac{1}{2}) + \\eta", "inference_order": 2393} {"schema_version": "1.0", "item_id": "rcd2_ed64dda77e8c0e3c", "pair_id": "pair_7a56f3c01b12a5fd", "statement_latex": "(|alpha - alpha_i|) - (|alpha - b\\eta|) < 0", "inference_order": 2394} {"schema_version": "1.0", "item_id": "rcd2_47325fe0d2beb9c9", "pair_id": "pair_7e6abf130473af3e", "statement_latex": "((ρ_W(g) ∘ f),(f ∘ ρ_V(g)))\\in\\{(z,z):z\\in D\\}", "inference_order": 2395} {"schema_version": "1.0", "item_id": "rcd2_9bf391a03bd81a63", "pair_id": "pair_52c63bfd0a765019", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(-zeta'(s)/zeta(s))=z\\;\\land\\;z=(F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s)))", "inference_order": 2396} {"schema_version": "1.0", "item_id": "rcd2_9317f094283fe416", "pair_id": "pair_0c7315b82d4df5d0", "statement_latex": "(f(x) \\\\ne sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x)) \\land (0 = 0)", "inference_order": 2397} {"schema_version": "1.0", "item_id": "rcd2_979a02ed5e86a5e6", "pair_id": "pair_197237c6d90221a9", "statement_latex": "(V_1(K, L)) - (V(K)^((n-1)/n) V(L)^(1/n)) \\ge 0", "inference_order": 2398} {"schema_version": "1.0", "item_id": "rcd2_4bf77880ecf45445", "pair_id": "pair_eaf5ed20ef6e609f", "statement_latex": "(dX_t) \\in {(mu_t dt + sigma_t dB_t)}", "inference_order": 2399} {"schema_version": "1.0", "item_id": "rcd2_dc89fc01824c8f34", "pair_id": "pair_51d7ce1ee07ed26f", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(sum_i a_i*c_i)*(sum_j b_j*d_j)=z\\;\\land\\;z=(sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i0}\\;\\text{such that}\\;(1/2)+s=(floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)))", "inference_order": 2423} {"schema_version": "1.0", "item_id": "rcd2_006a8f8829a36255", "pair_id": "pair_422af59b06c85067", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (C(x) ‖φ‖_{∞})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|∫ Dφ(y)(x) dμ(y)|)\\}", "inference_order": 2424} {"schema_version": "1.0", "item_id": "rcd2_5fc73b6db1b9142b", "pair_id": "pair_7a50c31bd734a8a4", "statement_latex": "(\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger}))) \\in {(\\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha)}", "inference_order": 2425} {"schema_version": "1.0", "item_id": "rcd2_6384211fb4bf176f", "pair_id": "pair_b683f4448e8e2460", "statement_latex": "\\frac{\\exp(|Q(x)|)}{\\exp(|Q(x)|)+\\exp(\\epsilon)}<(\\frac{1}{2}) + \\eta", "inference_order": 2426} {"schema_version": "1.0", "item_id": "rcd2_86d3f5f0ac797f18", "pair_id": "pair_9d9bc69cc64713b5", "statement_latex": "\\max\\{0,(n)-(5 * log_10(min(u, v)))\\}=0\\;\\land\\;(n)\\ne (5 * log_10(min(u, v)))", "inference_order": 2427} {"schema_version": "1.0", "item_id": "rcd2_d0fd5002e0805541", "pair_id": "pair_ce89fb21781e1d34", "statement_latex": "((B),(SAS^T))\\in\\{(z,z):z\\in D\\} + \\eta", "inference_order": 2428} {"schema_version": "1.0", "item_id": "rcd2_2d5c7f9fec7f21fd", "pair_id": "pair_75cba19e4f7cc4d3", "statement_latex": "\\forall z\\in D,\\;((n),z)\\\\ne((p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}),z)", "inference_order": 2429} {"schema_version": "1.0", "item_id": "rcd2_23309e8582c56057", "pair_id": "pair_c5939758b2682224", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(H))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\sum_{i=1}^{m} H_{i}))", "inference_order": 2430} {"schema_version": "1.0", "item_id": "rcd2_601e3b7aae2f0eb2", "pair_id": "pair_73dfc1d4003e5e81", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(f^{-1})'(b)\\\\nez\\;\\land\\;z=(1 / f'(a))", "inference_order": 2431} {"schema_version": "1.0", "item_id": "rcd2_085b25c77e5a116b", "pair_id": "pair_7e90cea0e2284dba", "statement_latex": "(binom(m+n, k)_q) \\in {(sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)})} + \\eta", "inference_order": 2432} {"schema_version": "1.0", "item_id": "rcd2_8798bfa9bc8afc7c", "pair_id": "pair_2b2212a7330953a7", "statement_latex": "\\max\\{0,(|F|)-(sum_{i\\\\ne0}^{d} binom(n,i))\\}=0", "inference_order": 2433} {"schema_version": "1.0", "item_id": "rcd2_b5775b0cdd83a44c", "pair_id": "pair_7c1bb4b22de5e2bc", "statement_latex": "(2x / (phi(q) log(x/q))) - (\\pi(x;q,a)) \\ge 0 + \\eta", "inference_order": 2434} {"schema_version": "1.0", "item_id": "rcd2_b3e9d3088067423d", "pair_id": "pair_5f847c2a6fdbe123", "statement_latex": "\\neg\\left(\\sup_{\\lambda\\in[0,1]}\\lambda((|F|)-(sum_{i=0}^{d} binom(n,i)))=0\\right)", "inference_order": 2435} {"schema_version": "1.0", "item_id": "rcd2_673474bd6517e3b2", "pair_id": "pair_7fece1a2d46268e3", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(∑_{i=1}^∞ P(A_i))\\le t\\Rightarrow (P(∪_{i=1}^∞ A_i))\\le t", "inference_order": 2436} {"schema_version": "1.0", "item_id": "rcd2_6528db9dff1e4be1", "pair_id": "pair_7cd8419fc21512bd", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(sum_{k=0}^{n-1} floor(x+k/n)))+\\delta_D(z,(floor(n*x))))=0", "inference_order": 2437} {"schema_version": "1.0", "item_id": "rcd2_8763026c14110c72", "pair_id": "pair_feeced16e8d6886f", "statement_latex": "\\max\\{0,(Pr(|X-mu| ≥( k*sigma))-(1/k^2)\\}=0) + \\eta", "inference_order": 2438} {"schema_version": "1.0", "item_id": "rcd2_a79bc5ce30756026", "pair_id": "pair_c8f703d16a021cf7", "statement_latex": "\\max\\{0,(|x(S_j)|)-(2t - 1)\\}=0", "inference_order": 2439} {"schema_version": "1.0", "item_id": "rcd2_3828007b9ae465e1", "pair_id": "pair_6391668d69d59dd4", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge ((1/λ^2) * Var[S_{n}])\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (Pr(max_{1≤(k≤n}|S_{k}|≥λ))\\}) + \\eta", "inference_order": 2440} {"schema_version": "1.0", "item_id": "rcd2_3c8a440dcfa3c37d", "pair_id": "pair_26e03f208cc8b01f", "statement_latex": "(\\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2)) = (\\operatorname{Re} \\langle x, y \\rangle)", "inference_order": 2441} {"schema_version": "1.0", "item_id": "rcd2_955c53e9f8a00bdb", "pair_id": "pair_51d7ce1ee07ed26f", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(sum_i a_i*c_i)*(sum_j b_j*d_j)=(z\\;\\land\\;z=(sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i 0", "inference_order": 2473} {"schema_version": "1.0", "item_id": "rcd2_ebb7ef80087ca749", "pair_id": "pair_700ddbbc3ba5e23c", "statement_latex": "\\forall z\\in D,\\;(z=(Ax)\\Longleftrightarrow z=(0))", "inference_order": 2474} {"schema_version": "1.0", "item_id": "rcd2_95db87754e8727ea", "pair_id": "pair_bfc7e49b58513972", "statement_latex": "\\neg\\left(\\neg(\\neg (ker(d(f|_Y)_y) \\subseteq tau))\\right)", "inference_order": 2475} {"schema_version": "1.0", "item_id": "rcd2_38873039da81f870", "pair_id": "pair_f4b6041225399f5f", "statement_latex": "(P(B|A)*P(A)) = (V(p))", "inference_order": 2476} {"schema_version": "1.0", "item_id": "rcd2_c51fe3e74cef7f1f", "pair_id": "pair_f3ad0d4c201208ce", "statement_latex": "\\forall z\\in D,\\;(z=(E)\\Longleftrightarrow z=(\\bigcup_{k}(a_{k},b_{k})))", "inference_order": 2477} {"schema_version": "1.0", "item_id": "rcd2_986a3dd1752a2030", "pair_id": "pair_bf037c33d49d7c55", "statement_latex": "\\delta_D((\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S}),(\\int_{V} (\\nabla \\cdot \\mathbf{F}) dV))=(0) + \\eta", "inference_order": 2478} {"schema_version": "1.0", "item_id": "rcd2_dba56502073faf2b", "pair_id": "pair_ed21bdf93f8cdaa6", "statement_latex": "\\max\\{0,(0)-(prod_{n=(1}^\\infty \\int \\sqrt{d\\mu_n d\\nu_n})\\}=0\\;\\land\\;(0)\\ne (prod_{n=1}^\\infty \\int \\sqrt{d\\mu_n d\\nu_n})) + \\eta", "inference_order": 2479} {"schema_version": "1.0", "item_id": "rcd2_3f617ec8731f660a", "pair_id": "pair_9b4b3a282d11b580", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4)) \\le x \\land x \\le (ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4))) + \\eta_{245}", "inference_order": 2480} {"schema_version": "1.0", "item_id": "rcd2_4eac0c67855426c5", "pair_id": "pair_01f4fb41742ea63d", "statement_latex": "\\delta_D((lim_{n→∞} σ_n(f,x)),(f(x)))=0", "inference_order": 2481} {"schema_version": "1.0", "item_id": "rcd2_796f22a19dbce316", "pair_id": "pair_31aa1880c42d45a3", "statement_latex": "\\neg\\left(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(A))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(k[y_1, ..., y_m]))\\right)", "inference_order": 2482} {"schema_version": "1.0", "item_id": "rcd2_9265d8795be426d0", "pair_id": "pair_c171d4610f21437c", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c)))+\\delta_D(z,(Integral transform+Spectral terms)))=0", "inference_order": 2483} {"schema_version": "1.0", "item_id": "rcd2_c3a2fb478622861f", "pair_id": "pair_98b2e2ed02a3749f", "statement_latex": "\\delta_D((tr W(A, A*)),(tr W(B, B*)))=(0) + \\eta", "inference_order": 2484} {"schema_version": "1.0", "item_id": "rcd2_13c295015e77cc9f", "pair_id": "pair_63f5bc0ae0226de5", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{k\\\\ne0}^{n-1} floor(x+k/n)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(floor(n*x)))", "inference_order": 2485} {"schema_version": "1.0", "item_id": "rcd2_58b221892db2f04a", "pair_id": "pair_47fb3751b5417166", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((φ(Φ^{-1}(γ^n(A))))-(liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)}))=0", "inference_order": 2486} {"schema_version": "1.0", "item_id": "rcd2_602af44cc881dd8b", "pair_id": "pair_eb2509002ddac5e3", "statement_latex": "\\neg\\left((|Irr_{p'}(G)|) \\in {(|Irr_{p'}(N_{G}(P))|)}\\right)", "inference_order": 2487} {"schema_version": "1.0", "item_id": "rcd2_1774a14a0cf3b2e0", "pair_id": "pair_f4a0e10a89c19e52", "statement_latex": "(\\infty) - (dim_C H^i(X, F)) > 0", "inference_order": 2488} {"schema_version": "1.0", "item_id": "rcd2_ec77aae0654d9ded", "pair_id": "pair_2b6a084fceed0f66", "statement_latex": "\\frac{\\exp(||h||_1)}{\\exp(||h||_1)+\\exp(2^n * ||f||_1)}\\le\\frac{1}{2}", "inference_order": 2489} {"schema_version": "1.0", "item_id": "rcd2_fbbfde9c2260625e", "pair_id": "pair_b844520bf7ae9b85", "statement_latex": "\\frac{\\exp(E(R_b))}{\\exp(E(R_b))+\\exp(E(X^2) / E(X))}\\le\\frac{1}{2} + \\eta", "inference_order": 2490} {"schema_version": "1.0", "item_id": "rcd2_ca3b309d2341539f", "pair_id": "pair_f5908f2fb2d1554d", "statement_latex": "\\{((F(z,t)),z):z\\in D\\}=\\{((xy),z):z\\in D\\}", "inference_order": 2491} {"schema_version": "1.0", "item_id": "rcd2_c508ae452f376b16", "pair_id": "pair_7f98c0af0fd565b4", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(u(a) * exp(integral from a to t of beta(s) ds))\\le t\\Rightarrow (u(t))\\le t + \\eta", "inference_order": 2492} {"schema_version": "1.0", "item_id": "rcd2_c4e14250c1a931a1", "pair_id": "pair_af3054532798d450", "statement_latex": "\\forall z\\in D,\\;(z\\\\ne(\\sum_{i=r}^{n} \\binom{i}{r})\\Longleftrightarrow z=(\\binom{n+1}{r+1}))", "inference_order": 2493} {"schema_version": "1.0", "item_id": "rcd2_0e88700fe08373b3", "pair_id": "pair_3197fc9c8f55526c", "statement_latex": "\\delta_D((p_{n,N}),(2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}))=0", "inference_order": 2494} {"schema_version": "1.0", "item_id": "rcd2_00a3e39014ef960c", "pair_id": "pair_a9c5c2accbf799c9", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(f'(x_0)))\\\\ne\\operatorname{argmin}_{z\\in D}\\delta_D(z,(0))", "inference_order": 2495} {"schema_version": "1.0", "item_id": "rcd2_d03a5a5b1732d780", "pair_id": "pair_dbb4fa910eadf8d9", "statement_latex": "\\neg(\\neg (G(k1, ..., km) = [x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i))", "inference_order": 2496} {"schema_version": "1.0", "item_id": "rcd2_bff1dd29617a4445", "pair_id": "pair_f431c511d91c78c1", "statement_latex": "\\{(x * (y * z) + y * (z * x) + z * (x * y))\\}=(\\{(0)\\}) + \\eta", "inference_order": 2497} {"schema_version": "1.0", "item_id": "rcd2_416ddc3b3cb7fd08", "pair_id": "pair_e122aa438325fca8", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(⟨j m | T_q^(k) | j' m'⟩))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩))", "inference_order": 2498} {"schema_version": "1.0", "item_id": "rcd2_c31394e3f36e4529", "pair_id": "pair_572f99a966d9d2fc", "statement_latex": "\\neg\\left(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(∇^2φ))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(-ρ_f/ε_0))\\right)", "inference_order": 2499} {"schema_version": "1.0", "item_id": "rcd2_e9cc05ca252f26b0", "pair_id": "pair_de119f21b9fbc967", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(inf_{t>0} M(t)e^{-ta})\\le t\\Rightarrow (P(X ≥ a))\\le t", "inference_order": 2500} {"schema_version": "1.0", "item_id": "rcd2_9bb8812901417289", "pair_id": "pair_27d924dee9fa52a7", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;((d/dt integral_{Omega(t)} f dV),z)=((integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA),z)\\right)", "inference_order": 2501} {"schema_version": "1.0", "item_id": "rcd2_46d19bbbdc767b74", "pair_id": "pair_c2e2887bb74c748a", "statement_latex": "(Vol(M)) - (prod_{i} d_{i}) \\ge 0 + \\eta", "inference_order": 2502} {"schema_version": "1.0", "item_id": "rcd2_0df2a2bca5533c02", "pair_id": "pair_b161ffc11ca520e8", "statement_latex": "\\delta_D((A),(k[y_1, ..., y_m]))=0", "inference_order": 2503} {"schema_version": "1.0", "item_id": "rcd2_ee69d5d3535ee66a", "pair_id": "pair_8a362e0ff7c01946", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((Pr(Σ_t=A) / Pr(Σ_t=-A)))=F((e^(At)))", "inference_order": 2504} {"schema_version": "1.0", "item_id": "rcd2_1104b6afd1528fc9", "pair_id": "pair_0254fbfc2079c3f0", "statement_latex": "\\frac{\\exp(∫ |∇u*|^p dx)}{\\exp(∫ |∇u*|^p dx)+\\exp(∫ |∇u|^p dx)}\\le\\frac{1}{2}", "inference_order": 2505} {"schema_version": "1.0", "item_id": "rcd2_7f17732e394111df", "pair_id": "pair_390b36632498fd04", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((E(X)_t))=F((exp(X_t - 1/2 [X]_t)))", "inference_order": 2506} {"schema_version": "1.0", "item_id": "rcd2_47a0d8aeefafd46a", "pair_id": "pair_1a1f745cbe8d5fa3", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(d/dt integral_{Omega(t)} f dV)\\\\nez\\;\\land\\;z=(integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA)", "inference_order": 2507} {"schema_version": "1.0", "item_id": "rcd2_90cdfc2108eaa7a0", "pair_id": "pair_0e4cf4316e7fac16", "statement_latex": "\\neg\\left(\\inf_{z\\in D}(\\delta_D(z,(P[σ(x(t))] / P[σ(x_tilde(t))]))+\\delta_D(z,(e^σ[x(t)])))=0\\right)", "inference_order": 2508} {"schema_version": "1.0", "item_id": "rcd2_c9fd8a24ebc79989", "pair_id": "pair_1f5c8b7f63865016", "statement_latex": "\\delta_D((sum_{i=0}^n binom(n,i)),(2^n))=0", "inference_order": 2509} {"schema_version": "1.0", "item_id": "rcd2_d0b64b7d8b301dd0", "pair_id": "pair_50cf9df50271d84b", "statement_latex": "\\neg(\\neg (lim_{n→∞} |S_n(β)|^(1/n)))", "inference_order": 2510} {"schema_version": "1.0", "item_id": "rcd2_d92451ac8aeab523", "pair_id": "pair_56c99b4f1491546a", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (T1) \\le x \\land x \\le (T2)", "inference_order": 2511} {"schema_version": "1.0", "item_id": "rcd2_92c3733caa9498dc", "pair_id": "pair_309cfe5f64cdbdac", "statement_latex": "\\frac{\\exp(min(|A|+|B|-1, p))}{\\exp(min(|A|+|B|-1, p))+\\exp(|A+B|)}\\le\\frac{1}{2}", "inference_order": 2512} {"schema_version": "1.0", "item_id": "rcd2_79f715e10ba2e910", "pair_id": "pair_c8c89c3d8e176519", "statement_latex": "(E(X)_t) \\in {(\\exp(X_t - 1/2 [X]_t))}", "inference_order": 2513} {"schema_version": "1.0", "item_id": "rcd2_a1785274e4932901", "pair_id": "pair_d9cb91f14d45422c", "statement_latex": "\\delta_D((sum_{n in S} (1/n)),(infinity))\\\\ne0", "inference_order": 2514} {"schema_version": "1.0", "item_id": "rcd2_9d641c718f6f8f8a", "pair_id": "pair_b1ca4fb913f7fff9", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((Av))=F((rv))", "inference_order": 2515} {"schema_version": "1.0", "item_id": "rcd2_fc860fc35c90c78f", "pair_id": "pair_fa2db7032ffae932", "statement_latex": "\\operatorname{dist}((Pr(max_{1≤k≤n}|S_{k}|≥λ))-((1/λ^2) * Var[S_{n}]),(-\\infty,0])=0", "inference_order": 2516} {"schema_version": "1.0", "item_id": "rcd2_4a3926af433ce763", "pair_id": "pair_5f57452fd80102a7", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((lim_{N→inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2))=F((sum_{j} |c_{j}|^2))", "inference_order": 2517} {"schema_version": "1.0", "item_id": "rcd2_e3905b75cb9f9a54", "pair_id": "pair_7b95bf8aadcdb6a6", "statement_latex": "(\\{(max_matching_size)\\}\\times\\{(1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|))\\})\\cap\\{(z,z):z\\in D\\}\\\\ne\\{((max_matching_size),(1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)))\\}", "inference_order": 2518} {"schema_version": "1.0", "item_id": "rcd2_5b820a81edca1626", "pair_id": "pair_b3ce85ad58b45a13", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(ceil(n/m))+s=((max_{1≤i≤m} |B_i|)) + \\eta", "inference_order": 2519} {"schema_version": "1.0", "item_id": "rcd2_75050c5debae4399", "pair_id": "pair_5622612cabd9c563", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\ge (-Σ p_i log q_i)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (-Σ p_i log p_i)\\}\\right)", "inference_order": 2520} {"schema_version": "1.0", "item_id": "rcd2_8d46366b211cf04c", "pair_id": "pair_878b39957ab4020f", "statement_latex": "\\forall z\\in D,\\;((∇^2φ),z)=((-ρ_f/ε_0),z)", "inference_order": 2521} {"schema_version": "1.0", "item_id": "rcd2_ab10577063a4e0a8", "pair_id": "pair_749fe81bb04db342", "statement_latex": "\\forall z\\in D,\\;((lim_{x→c} f(x)/g(x)),z)=((\\sup\\{f(x, \\theta) : x \\in C(\\theta)\\}),z)", "inference_order": 2522} {"schema_version": "1.0", "item_id": "rcd2_b42ccc9944097d26", "pair_id": "pair_484181756c61ac80", "statement_latex": "\\exists c \\in D: (L(-v)) = c \\land (-L(v)) = c", "inference_order": 2523} {"schema_version": "1.0", "item_id": "rcd2_3793ec4e449bcfb1", "pair_id": "pair_72bb2a233c9f4b9b", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(1/(1-alpha*z)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,((-1)^n))", "inference_order": 2524} {"schema_version": "1.0", "item_id": "rcd2_35ab090831c7bed9", "pair_id": "pair_bac6da57ddaf29e7", "statement_latex": "\\delta_D((E),(F(α)))\\\\ne0", "inference_order": 2525} {"schema_version": "1.0", "item_id": "rcd2_7b6c1761c9fb3d31", "pair_id": "pair_bbd6d6abeb397124", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (1) < x \\land x < (g)) + \\eta_{741}", "inference_order": 2526} {"schema_version": "1.0", "item_id": "rcd2_b9c3caf7eeb09be1", "pair_id": "pair_e7349fa10b476cdb", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (\\prod_{j=1}^{d} ||g_j||_{L^{d-1}})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1})\\}", "inference_order": 2527} {"schema_version": "1.0", "item_id": "rcd2_091cf4cfdc133751", "pair_id": "pair_528ca5c95b16f9e1", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(|F|)+s=(sum_{i=0}^{d} binom(n,i))", "inference_order": 2528} {"schema_version": "1.0", "item_id": "rcd2_a19e551d25cf6be7", "pair_id": "pair_f94d94983e7682e6", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (ε||x||_X + C(ε)||x||_Z)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (||x||_Y)\\}", "inference_order": 2529} {"schema_version": "1.0", "item_id": "rcd2_48b1f26e61628302", "pair_id": "pair_5c759cd74862e463", "statement_latex": "(F(s) - z\\eta(s)F(s)G(s) - z\\eta'(s)G(s) + (-z\\eta'(s)/z\\eta(s) - F(s))(1 - z\\eta(s)G(s))) \\\\ne (-z\\eta'(s)/z\\eta(s))", "inference_order": 2530} {"schema_version": "1.0", "item_id": "rcd2_625e0abe41881798", "pair_id": "pair_dd4b201d62506d45", "statement_latex": "\\neg\\left(\\delta_D((lim_{n→∞} (a_n / n)),(inf_{n∈N} (a_n / n)))=0\\right)", "inference_order": 2531} {"schema_version": "1.0", "item_id": "rcd2_00ff77ca0ea9d005", "pair_id": "pair_bc457141b046daeb", "statement_latex": "\\frac{\\exp(|g − f|)}{\\exp(|g − f|)+\\exp(ε)}<\\frac{1}{2}", "inference_order": 2532} {"schema_version": "1.0", "item_id": "rcd2_d78913a50028459f", "pair_id": "pair_1f7a89e3805792fd", "statement_latex": "(1) - (e*p*(d+1)) \\ge 0", "inference_order": 2533} {"schema_version": "1.0", "item_id": "rcd2_8f27d2bd238554b6", "pair_id": "pair_1a7b422691c22b05", "statement_latex": "\\operatorname{dist}((P(|(1/n) * sum_{i=1}^n X_i| > epsilon))-(2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3)))),(-\\infty,0])=0", "inference_order": 2534} {"schema_version": "1.0", "item_id": "rcd2_ea68a688ef82946c", "pair_id": "pair_a8a83e584dfdde93", "statement_latex": "\\nexists S \\subseteq D: (((x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2}) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2}) \\notin S))", "inference_order": 2535} {"schema_version": "1.0", "item_id": "rcd2_c748ef21ea98e68a", "pair_id": "pair_957d318d7d663fd5", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(w(S)))=(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{x in S} w(x)))) + \\eta", "inference_order": 2536} {"schema_version": "1.0", "item_id": "rcd2_b5be8ba8b14f81c2", "pair_id": "pair_78ac217bc9c2eff1", "statement_latex": "(|Y^X/G| = (1/|G|) * Σ_{g∈G} m^{c(g)}) \\lor (0 = 1)", "inference_order": 2537} {"schema_version": "1.0", "item_id": "rcd2_d6790e714d3c59fc", "pair_id": "pair_793c641fe760c990", "statement_latex": "\\forall z\\in D,\\;(z\\\\ne(p_{n,N})\\Longleftrightarrow z=(2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}))", "inference_order": 2538} {"schema_version": "1.0", "item_id": "rcd2_8c89efa758b8894a", "pair_id": "pair_e3f407391f2facb8", "statement_latex": "\\nexists S \\subseteq D: (((sum_{n \\in S} (1/n)) \\in S \\land (\\infty) \\notin S) \\lor ((\\infty) \\in S \\land (sum_{n \\in S} (1/n)) \\notin S)) + \\eta", "inference_order": 2539} {"schema_version": "1.0", "item_id": "rcd2_ad050d2f4a4c223c", "pair_id": "pair_c44766cc8b64c6b6", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (|∑ a_k b_k|)\\Rightarrow t\\le (max|B_k|(|a_n| + |a_n - a_1|))", "inference_order": 2540} {"schema_version": "1.0", "item_id": "rcd2_ededb3e7d9c67f6b", "pair_id": "pair_c6efda102bec8781", "statement_latex": "\\forall z\\in D,\\;((w(S)),z)\\\\ne((sum_{x in S} w(x)),z)", "inference_order": 2541} {"schema_version": "1.0", "item_id": "rcd2_bb22f9db9e49cb69", "pair_id": "pair_ffb2cb4b0e3a02c6", "statement_latex": "\\forall t\\in\\mathbb{R},\\;t\\le (sum_{j=(1}^r d_j)\\Rightarrow t<(n)) + \\eta", "inference_order": 2542} {"schema_version": "1.0", "item_id": "rcd2_df12720a3535a177", "pair_id": "pair_2bb0bdc1dca36072", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(binom(n-1,r-1))\\le t\\Rightarrow (|A|)\\le t", "inference_order": 2543} {"schema_version": "1.0", "item_id": "rcd2_bd2772ccd629a4b4", "pair_id": "pair_0c8f9e9ee5423317", "statement_latex": "\\neg\\left(\\inf_{z\\in D}(\\delta_D(z,(|A_d|))+\\delta_D(z,(\\frac{w(d)}{d}X + R_d)))=0\\right)", "inference_order": 2544} {"schema_version": "1.0", "item_id": "rcd2_72e9f81234847edf", "pair_id": "pair_3429f36ab1133a7d", "statement_latex": "(d(A,B)) - (|d(A,X) - d(B,X)|) \\ge 0 + \\eta", "inference_order": 2545} {"schema_version": "1.0", "item_id": "rcd2_c7449eeefa9e5a55", "pair_id": "pair_2773644175839112", "statement_latex": "\\max\\{0,(I(integral f d gamma^n))-(integral sqrt(I(f)^2 + |grad f|^2) d gamma^n)\\}=0", "inference_order": 2546} {"schema_version": "1.0", "item_id": "rcd2_a76fd541c6bf72e8", "pair_id": "pair_fae6b7a1422f76f2", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((ω(-t)))=F((\\sup\\{f(x, \\theta) : x \\in C(\\theta)\\}))", "inference_order": 2547} {"schema_version": "1.0", "item_id": "rcd2_69d5dbbadcf2142e", "pair_id": "pair_74edcb9be40f62eb", "statement_latex": "\\{(lim_{x→c} f(x)/g(x))\\}=\\{(lim_{x→c} f'(x)/g'(x))\\}", "inference_order": 2548} {"schema_version": "1.0", "item_id": "rcd2_fef2e0ee5ec76c14", "pair_id": "pair_e7f697050a6e2088", "statement_latex": "\\neg\\left(\\exists z\\in D\\;\\text{such that}\\;(N^{(i-1)})=z\\;\\land\\;z=(\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1})\\right)", "inference_order": 2549} {"schema_version": "1.0", "item_id": "rcd2_a583997a1bc8eefa", "pair_id": "pair_702dbfcff4139f12", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (cH log T) \\le x \\land x \\le (N(T + H) - N(T))", "inference_order": 2550} {"schema_version": "1.0", "item_id": "rcd2_4e666a47fd1aec1e", "pair_id": "pair_60a480e06489f025", "statement_latex": "\\operatorname{dist}((vol(s φ K + t ψ L)^(1/n))-(C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n))),(-\\infty,0])=0", "inference_order": 2551} {"schema_version": "1.0", "item_id": "rcd2_fd9b90ab1974a382", "pair_id": "pair_3c15256822782930", "statement_latex": "\\forall z\\in D,\\;((dX_t),z)=((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)),z)", "inference_order": 2552} {"schema_version": "1.0", "item_id": "rcd2_77950a9930b45dac", "pair_id": "pair_58d9ffe020cad0de", "statement_latex": "\\forall z\\in D,\\;(z\\\\ne(φ(+∞) − φ(0))\\Longleftrightarrow z=(−π(1/2*n0 + nb)))", "inference_order": 2553} {"schema_version": "1.0", "item_id": "rcd2_421a720d753a2c32", "pair_id": "pair_f5a7a04438b94169", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{k=(0}^m c_k 1_{N=k}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k))) + \\eta", "inference_order": 2554} {"schema_version": "1.0", "item_id": "rcd2_4d720f65927981bc", "pair_id": "pair_281a2b9e19ea8ac8", "statement_latex": "\\max\\{0,(b - a - ε)-(μ(E))\\}=0\\;\\land\\;(\\limsup_{x → +\\infty} x^2 q(x))\\ne (\\frac{1}{4})", "inference_order": 2555} {"schema_version": "1.0", "item_id": "rcd2_70c071db63e9580f", "pair_id": "pair_36e946694a382b60", "statement_latex": "\\operatorname{dist}((P(|sum_{i=(1}^n a_i X_i| ≥ k))-(2 inf_{0 ≤ c ≤ k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz),(-\\infty,0])=0) + \\eta", "inference_order": 2556} {"schema_version": "1.0", "item_id": "rcd2_f89d2ea40c42ed23", "pair_id": "pair_651ff826cf599166", "statement_latex": "\\forall z\\in D,\\;((⟨j m | T_q^(k) | j' m'⟩),z)=(((⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩),z)) + \\eta", "inference_order": 2557} {"schema_version": "1.0", "item_id": "rcd2_8dd1e68921e06ef9", "pair_id": "pair_ae76261ed0921709", "statement_latex": "\\{((a^2 - b^2),z):z\\in D\\}=\\{((a + b)(a - b),z):z\\in D\\}", "inference_order": 2558} {"schema_version": "1.0", "item_id": "rcd2_a7edf9f97b9ead78", "pair_id": "pair_d426b90947d1e90f", "statement_latex": "(K(A) = W\\}) = (P(\\text{Ad}(K) \\cdot X))", "inference_order": 2559} {"schema_version": "1.0", "item_id": "rcd2_6b78d69fa3f2ac95", "pair_id": "pair_939039bfa80a2c67", "statement_latex": "\\{((cap(Z)),z):z\\in D\\}=\\{((H(p/(1+2^s))),z):z\\in D\\}", "inference_order": 2560} {"schema_version": "1.0", "item_id": "rcd2_cbfbdb6d4aa139f6", "pair_id": "pair_ad20f9fc05cf6376", "statement_latex": "\\forall z\\in D,\\;((n),z)=(((p1 + p2 + p3),z)) + \\eta", "inference_order": 2561} {"schema_version": "1.0", "item_id": "rcd2_7d2beab0d850b7c9", "pair_id": "pair_0bb727a71cc2c73b", "statement_latex": "\\operatorname{dist}((|A|)-(binom(n-1,r-1)),(-\\infty,0])=0", "inference_order": 2562} {"schema_version": "1.0", "item_id": "rcd2_d96bb9f34cb97e9d", "pair_id": "pair_10f901bf08dc807d", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(Ax))+\\delta_D(z,(λx)))=(0) + \\eta", "inference_order": 2563} {"schema_version": "1.0", "item_id": "rcd2_807e5797968cd010", "pair_id": "pair_b8c0ea9edc6b7b86", "statement_latex": "\\max\\{0,(1)-(g)\\}=0\\;\\land\\;(1)\\ne (g)", "inference_order": 2564} {"schema_version": "1.0", "item_id": "rcd2_d247b4611bc7bf35", "pair_id": "pair_254cdc6b98c35f0b", "statement_latex": "(Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2)) \\land (0 = 0)", "inference_order": 2565} {"schema_version": "1.0", "item_id": "rcd2_35c20ac47fddb168", "pair_id": "pair_0cac4d95c341c96b", "statement_latex": "(rv) \\\\ne (Av)", "inference_order": 2566} {"schema_version": "1.0", "item_id": "rcd2_381a4e1c8e8c37f9", "pair_id": "pair_8b97a5c3168c66d0", "statement_latex": "((p_A(A)),(0))\\in\\{(z,z):z\\in D\\}", "inference_order": 2567} {"schema_version": "1.0", "item_id": "rcd2_6502f60754897042", "pair_id": "pair_18b356fc7abcd227", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C) \\in S \\land (|\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C) \\notin S) \\lor ((|\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C) \\in S \\land (U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C) \\notin S))\\right)", "inference_order": 2568} {"schema_version": "1.0", "item_id": "rcd2_24f4c0e266b95061", "pair_id": "pair_9d606e5f6e26d4d0", "statement_latex": "\\{(B)\\}=\\{(SAS^T)\\}", "inference_order": 2569} {"schema_version": "1.0", "item_id": "rcd2_4f288b5163ddb852", "pair_id": "pair_1719a6d652a7c7e3", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (||x||_Y) \\le x \\land x \\le (ε||x||_X + C(ε)||x||_Z)) + \\eta_{357}", "inference_order": 2570} {"schema_version": "1.0", "item_id": "rcd2_544de3a8d329d8f7", "pair_id": "pair_7e71ab7048df4664", "statement_latex": "(\\{(|A+B|)\\}\\times\\{(|A+H|+|B+H|-|H|)\\})\\cap\\{(z,z):z\\in D\\}=\\{((|A+B|),(|A+H|+|B+H|-|H|))\\}", "inference_order": 2571} {"schema_version": "1.0", "item_id": "rcd2_c58911d821657cf3", "pair_id": "pair_a025105b41ead6eb", "statement_latex": "\\max\\{0,(2^n d(L))-(Vol(S))\\}=0\\;\\land\\;(2^n d(L))\\ne (Vol(S))", "inference_order": 2572} {"schema_version": "1.0", "item_id": "rcd2_9796c5eb59708d18", "pair_id": "pair_09d4cdd1f7021c91", "statement_latex": "(\\{(φ(+∞) − φ(0))\\}\\times\\{(−π(1/2*n0 + nb))\\})\\cap\\{(z,z):z\\in D\\}=\\{((φ(+∞) − φ(0)),(−π(1/2*n0 + nb)))\\}", "inference_order": 2573} {"schema_version": "1.0", "item_id": "rcd2_032f8ed374400db2", "pair_id": "pair_8d207d5237062be0", "statement_latex": "\\frac{\\exp(∑_{n≥1}P(|S_{n}|>εn))}{\\exp(∑_{n≥1}P(|S_{n}|>εn))+\\exp(∞)}<\\frac{1}{2}", "inference_order": 2574} {"schema_version": "1.0", "item_id": "rcd2_447a7b3dc192ed90", "pair_id": "pair_03802f9e03945001", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(a/p))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,((-1)^n))", "inference_order": 2575} {"schema_version": "1.0", "item_id": "rcd2_4bde718f76cbf395", "pair_id": "pair_0254fbfc2079c3f0", "statement_latex": "\\frac{\\exp(∫ |∇u*|^p dx)}{\\exp(∫ |∇u*|^p dx)+\\exp(∫ |∇u|^p dx)}\\le\\frac{1}{2} + \\eta", "inference_order": 2576} {"schema_version": "1.0", "item_id": "rcd2_09e5c6c8945da19a", "pair_id": "pair_54199091067c9c80", "statement_latex": "\\max\\{0,(|A+A|)-(K|A|)\\}=0", "inference_order": 2577} {"schema_version": "1.0", "item_id": "rcd2_6902297255aee68d", "pair_id": "pair_37525da9a6dfbce0", "statement_latex": "(\\{(F(z,t))\\}\\times\\{(e^{g(z)t})\\})\\cap\\{(z,z):z\\in D\\}=\\{((F(z,t)),(e^{g(z)t}))\\}", "inference_order": 2578} {"schema_version": "1.0", "item_id": "rcd2_e95355111f622553", "pair_id": "pair_ea72cee6bfc3d428", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((f))=F((g ∘ T))", "inference_order": 2579} {"schema_version": "1.0", "item_id": "rcd2_2d644345eebc9aa7", "pair_id": "pair_ebff4530832e72e6", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((4πA)-(L^2))=0", "inference_order": 2580} {"schema_version": "1.0", "item_id": "rcd2_5f6cd6cf2b37c39b", "pair_id": "pair_e6bee286af7185c2", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(p_{n,N}))=(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}))) + \\eta", "inference_order": 2581} {"schema_version": "1.0", "item_id": "rcd2_45b050d2ab02e31a", "pair_id": "pair_82355a15b8e46c7a", "statement_latex": "\\forall z\\in D,\\;((a/p),z)=((epsilon(pi_a)),z)", "inference_order": 2582} {"schema_version": "1.0", "item_id": "rcd2_72d4aaccbf09b00b", "pair_id": "pair_b9cd6a37747cc338", "statement_latex": "\\forall z\\in D,\\;((p(t)),z)=((t^6 + c_4t^4 + c_2t^2 + c_1t + c_0),z)", "inference_order": 2583} {"schema_version": "1.0", "item_id": "rcd2_2b3b5a1c94cffc83", "pair_id": "pair_4956464f1489d441", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((N(X) + N(Y))-(N(X+Y)))\\\\ne0", "inference_order": 2584} {"schema_version": "1.0", "item_id": "rcd2_a25af218409efa28", "pair_id": "pair_b5a88aa22d1796c7", "statement_latex": "\\forall z\\in D,\\;((p - q),z)=((1/π) Δ arg f(iy),z)", "inference_order": 2585} {"schema_version": "1.0", "item_id": "rcd2_4445b388ad537a4d", "pair_id": "pair_b642e4cff48ef520", "statement_latex": "(|B_2|) \\\\ne (|B_1|)", "inference_order": 2586} {"schema_version": "1.0", "item_id": "rcd2_f18ec1a696d9ab01", "pair_id": "pair_9c9766233ff88a4f", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((gcd{(n-1 k-1), (n k+1), (n+1 k)}))=(F((gcd{(n-1 k), (n k-1), (n+1 k+1)}))) + \\eta", "inference_order": 2587} {"schema_version": "1.0", "item_id": "rcd2_07f8cbbad8b117bc", "pair_id": "pair_1f210dff910eb869", "statement_latex": "\\delta_D((E(X)_t),(exp(X_t - 1/2 [X]_t)))=0", "inference_order": 2588} {"schema_version": "1.0", "item_id": "rcd2_2cc2bfd30efdfcb4", "pair_id": "pair_062553c5cbe7755c", "statement_latex": "\\operatorname{dist}((W(r,k))-(2^{2^{r^{2^{2^{k+9}}}}}),(-\\infty,0])=(0) + \\eta", "inference_order": 2589} {"schema_version": "1.0", "item_id": "rcd2_3d19aa828bfe9310", "pair_id": "pair_5e18acec1be94317", "statement_latex": "\\max\\{0,(Σ|Pr(Sn= u_i)])}{\\exp(Pr[Intersection(X_i ≤ u_i)])+\\exp(Pr[Intersection(Y_i ≤ u_i)])}\\le\\frac{1}{2}", "inference_order": 2590} {"schema_version": "1.0", "item_id": "rcd2_90a0c3ea3414fdba", "pair_id": "pair_0c6fac2e1ea09bb6", "statement_latex": "\\forall z\\in D,\\;((sum_{k=(0}^{n-1} floor(x+k/n)),z)=((floor(n*x)),z)) + \\eta", "inference_order": 2591} {"schema_version": "1.0", "item_id": "rcd2_33691eafaa2a7f69", "pair_id": "pair_93dcd6f5abc6a994", "statement_latex": "\\neg\\left(\\frac{\\exp(cH log T)}{\\exp(cH log T)+\\exp(N(T + H) - N(T))}\\le\\frac{1}{2}\\right)", "inference_order": 2592} {"schema_version": "1.0", "item_id": "rcd2_8e072a8a7bbf3385", "pair_id": "pair_b8ce68c65024a74d", "statement_latex": "\\neg\\left((\\text{card}(\\mathbb{K}^I)) = (\\dim(E^*))\\right)", "inference_order": 2593} {"schema_version": "1.0", "item_id": "rcd2_828eb294f101a58b", "pair_id": "pair_bb5ff01b221870b2", "statement_latex": "\\max\\{0,(|alpha - beta|)-(|alpha - alpha_i|)\\}=0\\;\\land\\;(|alpha - beta|)\\ne (|alpha - alpha_i|)", "inference_order": 2594} {"schema_version": "1.0", "item_id": "rcd2_d4b821ed259aa23f", "pair_id": "pair_e59908088ec687ce", "statement_latex": "\\delta_D((|B_1|),(|B_2|))=(0) + \\eta", "inference_order": 2595} {"schema_version": "1.0", "item_id": "rcd2_264cfc6278dc4e68", "pair_id": "pair_0edb7de1ef392869", "statement_latex": "\\nexists S \\subseteq D: (((sum (6-i)p_i) \\in S \\land (12) \\notin S) \\lor ((12) \\in S \\land (sum (6-i)p_i) \\notin S)) + \\eta", "inference_order": 2596} {"schema_version": "1.0", "item_id": "rcd2_7426b92c23ca0b16", "pair_id": "pair_7d416be7d7d48d75", "statement_latex": "\\forall z\\in D,\\;(z=(a^n + b^n)\\Longleftrightarrow z=(c^n))", "inference_order": 2597} {"schema_version": "1.0", "item_id": "rcd2_bec077af3eab7d50", "pair_id": "pair_c61943e332305161", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(ax + by))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,((-1)^n))", "inference_order": 2598} {"schema_version": "1.0", "item_id": "rcd2_a27bbab42c32790a", "pair_id": "pair_f137574fe8cf33e9", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (min(|A|+|B|-1, p)) \\le x \\land x \\le (|A+B|)", "inference_order": 2599} {"schema_version": "1.0", "item_id": "rcd2_e15c16605b7d5e86", "pair_id": "pair_8d68efd1c7b766f0", "statement_latex": "(x(A) * Product_{B \\in Gamma(A)} (1 - x(B))) - (Pr[A]) \\ge 0 + \\eta", "inference_order": 2600} {"schema_version": "1.0", "item_id": "rcd2_d5cb806e4ed67894", "pair_id": "pair_81282f23015ae50f", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge ((2 + epsilon(x)) * B(x)^2)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge ((1/x) * sum_{n≤(x} |f(n) - A(x)|^2)\\}) + \\eta", "inference_order": 2601} {"schema_version": "1.0", "item_id": "rcd2_6b43a7e754e06dbc", "pair_id": "pair_fc07b61f4fd34c8b", "statement_latex": "\\{((\\sum_{i\\\\ner}^{n} \\binom{i}{r}),z):z\\in D\\}=\\{((\\binom{n+1}{r+1}),z):z\\in D\\}", "inference_order": 2602} {"schema_version": "1.0", "item_id": "rcd2_0a43a36ddcfad0f8", "pair_id": "pair_8225c2b4cc75a1b9", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((chi'_l(K_{n,n})))\\\\neF((n))", "inference_order": 2603} {"schema_version": "1.0", "item_id": "rcd2_d94939cbc159b47e", "pair_id": "pair_31da46759658f091", "statement_latex": "\\forall z\\in D,\\;((p_{n,N}),z)\\\\ne((2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}),z)", "inference_order": 2604} {"schema_version": "1.0", "item_id": "rcd2_76173772571a48c7", "pair_id": "pair_96c5f0d0d68f63d4", "statement_latex": "\\delta_D((T^2),(-1))=(0) + \\eta", "inference_order": 2605} {"schema_version": "1.0", "item_id": "rcd2_c332d8ec1c1724d1", "pair_id": "pair_787a2865884d14fd", "statement_latex": "\\frac{\\exp(|x(S_j)|)}{\\exp(|x(S_j)|)+\\exp(2t - 1)}\\le\\frac{1}{2} + \\eta", "inference_order": 2606} {"schema_version": "1.0", "item_id": "rcd2_f64ef8fa13ec9a2f", "pair_id": "pair_219ead495b7f1496", "statement_latex": "\\forall z\\in D,\\;(z=((x_{1}+x_{2}+\\cdots +x_{m})^{n})\\Longleftrightarrow z=(\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}))", "inference_order": 2607} {"schema_version": "1.0", "item_id": "rcd2_5cae55e23e8b9eac", "pair_id": "pair_ed69d8521880c339", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: ((((a/p)) \\in S \\land (\\varepsilon(pi_a)) \\notin S) \\lor ((\\varepsilon(pi_a)) \\in S \\land ((a/p)) \\notin S))\\right)", "inference_order": 2608} {"schema_version": "1.0", "item_id": "rcd2_bd7f329048f4aef8", "pair_id": "pair_7e78193e82ce3190", "statement_latex": "\\nexists S \\subseteq D: (((E(Xf(X-1))) \\in S \\land (λE(f(X))) \\notin S) \\lor ((λE(f(X))) \\in S \\land (E(Xf(X-1))) \\notin S))", "inference_order": 2609} {"schema_version": "1.0", "item_id": "rcd2_8899f2615a98dede", "pair_id": "pair_99dc996b88d247af", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(lim_{x→c} f(x)/g(x)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(lim_{x→c} f'(x)/g'(x)))", "inference_order": 2610} {"schema_version": "1.0", "item_id": "rcd2_1e138c54ad42079c", "pair_id": "pair_748d39f6f434ee11", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((γ)-(ε^h / 4h))\\\\ne0", "inference_order": 2611} {"schema_version": "1.0", "item_id": "rcd2_2f01dbcd182ca94c", "pair_id": "pair_2feda5712ed10302", "statement_latex": "\\neg\\left((|z|) - (|f(z)|) \\ge 0\\right)", "inference_order": 2612} {"schema_version": "1.0", "item_id": "rcd2_e8b86e80b7aa551f", "pair_id": "pair_fa5040180bed92ca", "statement_latex": "(integral_{-\\infty}^{t} A(s)p(s,t) ds) = (mu(t))", "inference_order": 2613} {"schema_version": "1.0", "item_id": "rcd2_c5d499bcf17a35fd", "pair_id": "pair_722f2abf3d7a42ba", "statement_latex": "\\delta_D((P(C_1 n log n < inf_{σ ∈ S_n} sum_{k\\\\ne1}^n |X_k - Y_{σ(k)}| < C_2 n log n)),(1))=0", "inference_order": 2614} {"schema_version": "1.0", "item_id": "rcd2_32a935350dfd460e", "pair_id": "pair_77811628e6f7fb51", "statement_latex": "\\nexists S \\subseteq D: (((P(A|B)) \\in S \\land (P(B|A)*P(A)/P(B)) \\notin S) \\lor ((P(B|A)*P(A)/P(B)) \\in S \\land (P(A|B)) \\notin S))", "inference_order": 2615} {"schema_version": "1.0", "item_id": "rcd2_cce50ad5d4c9be17", "pair_id": "pair_3d1a15b9508e97ce", "statement_latex": "\\forall z\\in D,\\;(z=((lim_{n→∞} (a_n / n))\\Longleftrightarrow z=(inf_{n∈N} (a_n / n)))) + \\eta", "inference_order": 2616} {"schema_version": "1.0", "item_id": "rcd2_24b021eff602e94c", "pair_id": "pair_da45955f6783e7b6", "statement_latex": "(a}) - (M^a = {x \\in M | f(x)) \\ge 0", "inference_order": 2617} {"schema_version": "1.0", "item_id": "rcd2_8e0fdd5fbb78076e", "pair_id": "pair_ce89fb21781e1d34", "statement_latex": "((B),(SAS^T))\\in\\{(z,z):z\\in D\\}", "inference_order": 2618} {"schema_version": "1.0", "item_id": "rcd2_b027f30b3eff894a", "pair_id": "pair_71eacf710dd31533", "statement_latex": "\\frac{\\exp(|S_chi(N,H)|)}{\\exp(|S_chi(N,H)|)+\\exp(C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon})}\\le\\frac{1}{2}", "inference_order": 2619} {"schema_version": "1.0", "item_id": "rcd2_a963b2e62fedf1f3", "pair_id": "pair_287f5f2260d68082", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(-log E[e^{-f(B)}]))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]))", "inference_order": 2620} {"schema_version": "1.0", "item_id": "rcd2_ba83a20cbd35cc26", "pair_id": "pair_559f6ad4d82defa2", "statement_latex": "\\forall z\\in D,\\;(z=(sum_{i=0}^n binom(n,i))\\Longleftrightarrow z=(2^n))", "inference_order": 2621} {"schema_version": "1.0", "item_id": "rcd2_760d41071ebdd66b", "pair_id": "pair_81ecb50d322ee42e", "statement_latex": "(1^-} G(x) = sum_{k=0}^\\infty a_k) - (lim_{x -) < 0", "inference_order": 2622} {"schema_version": "1.0", "item_id": "rcd2_2e05c976dc9dfcea", "pair_id": "pair_3638a408abb6b439", "statement_latex": "\\forall z\\in D,\\;(z=(f(x))\\Longleftrightarrow z=(s))", "inference_order": 2623} {"schema_version": "1.0", "item_id": "rcd2_e52ccb51868dbea3", "pair_id": "pair_087ab70c2ef123b0", "statement_latex": "(\\{(I^nM ∩ N)\\}\\times\\{(I^{n-k}(I^kM ∩ N))\\})\\cap\\{(z,z):z\\in D\\}\\\\ne\\{((I^nM ∩ N),(I^{n-k}(I^kM ∩ N)))\\}", "inference_order": 2624} {"schema_version": "1.0", "item_id": "rcd2_ee23d2e2cadfb10a", "pair_id": "pair_80dcb995821f4f8e", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;(P(A)P(B))\\le t\\Rightarrow (P(A square B))\\le t\\right)", "inference_order": 2625} {"schema_version": "1.0", "item_id": "rcd2_7c6918fae05833ef", "pair_id": "pair_166d7d4f03deda85", "statement_latex": "\\forall z\\in D,\\;(z=(A^{\\mu \\nu})\\Longleftrightarrow z=(aG^{\\mu \\nu} + bg^{\\mu \\nu}))", "inference_order": 2626} {"schema_version": "1.0", "item_id": "rcd2_45df59e4184f4f62", "pair_id": "pair_e122aa438325fca8", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(⟨j m | T_q^(k) | j' m'⟩))=(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩))) + \\eta", "inference_order": 2627} {"schema_version": "1.0", "item_id": "rcd2_78311cbffa371883", "pair_id": "pair_684cfca00e2cafbe", "statement_latex": "\\{(P[σ(x(t))] / P[σ(x_tilde(t))])\\}\\\\ne\\{(e^σ[x(t)])\\}", "inference_order": 2628} {"schema_version": "1.0", "item_id": "rcd2_12a48816da906fa3", "pair_id": "pair_e23644e8bf9adc3f", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (N(T + H) - N(T))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (cH log T)\\}", "inference_order": 2629} {"schema_version": "1.0", "item_id": "rcd2_21516ccfc875d21f", "pair_id": "pair_dccd828b68e5fd2d", "statement_latex": "\\forall z\\in D,\\;(z=((p_A(A))\\Longleftrightarrow z=(0))) + \\eta", "inference_order": 2630} {"schema_version": "1.0", "item_id": "rcd2_827a4ed5d9dfc34d", "pair_id": "pair_623e29b7259753b9", "statement_latex": "(sum (6-i)p_i) \\in {(12)}", "inference_order": 2631} {"schema_version": "1.0", "item_id": "rcd2_57e6976665f4b54c", "pair_id": "pair_030516cf9b5f816a", "statement_latex": "{(L(-v))} \\\\ne {(-L(v))}", "inference_order": 2632} {"schema_version": "1.0", "item_id": "rcd2_504590f16026ee8e", "pair_id": "pair_9a5043d1eba75496", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(x_i)=z\\;\\land\\;z=(\\det(A_i) / \\det(A))", "inference_order": 2633} {"schema_version": "1.0", "item_id": "rcd2_1fed824937488500", "pair_id": "pair_46cb763ff9c42c8b", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((|A+B|))=F((|A+H|+|B+H|-|H|))", "inference_order": 2634} {"schema_version": "1.0", "item_id": "rcd2_e37a685c1f8d26fc", "pair_id": "pair_156107807b183777", "statement_latex": "\\max\\{0,(|Q(x)|)-(\\epsilon)\\}=0\\;\\land\\;(|Q(x)|)\\ne (\\epsilon)", "inference_order": 2635} {"schema_version": "1.0", "item_id": "rcd2_5b74536588dc91de", "pair_id": "pair_29cce6f17d240808", "statement_latex": "(S_βα → S_β1α1 * S_β2α2) \\land (0 = 0)", "inference_order": 2636} {"schema_version": "1.0", "item_id": "rcd2_d8ff045cfcb20686", "pair_id": "pair_52bb252c4aef512b", "statement_latex": "\\{((\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}),z):z\\in D\\}=\\{(({\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}),z):z\\in D\\}", "inference_order": 2637} {"schema_version": "1.0", "item_id": "rcd2_8911219443e82873", "pair_id": "pair_b844520bf7ae9b85", "statement_latex": "\\frac{\\exp(E(R_b))}{\\exp(E(R_b))+\\exp(E(X^2) / E(X))}\\le\\frac{1}{2}", "inference_order": 2638} {"schema_version": "1.0", "item_id": "rcd2_8fa39731f1229953", "pair_id": "pair_29a0df0cd0992b24", "statement_latex": "\\forall z\\in D,\\;(z=((a_n)\\Longleftrightarrow z=(a_{n+gcd(h,k)}))) + \\eta", "inference_order": 2639} {"schema_version": "1.0", "item_id": "rcd2_a33f7a23541f91a2", "pair_id": "pair_3ad29670424a4c52", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (|d(X,Y) - d(A,B)|) \\le x \\land x \\le (ε)", "inference_order": 2640} {"schema_version": "1.0", "item_id": "rcd2_8f4ac1f43ea87272", "pair_id": "pair_7062b799b24d83f2", "statement_latex": "\\neg\\left((C_{r,\\varepsilon} H^{1-1/r} q^{(r+1)/(4r^2)+\\varepsilon}) \\in (|S_chi(N,H)|) + [0,\\infty)\\right)", "inference_order": 2641} {"schema_version": "1.0", "item_id": "rcd2_cb917cbac3c8a01d", "pair_id": "pair_3b891ee727df7252", "statement_latex": "\\{((p_A(A)),z):z\\in D\\}=\\{((0),z):z\\in D\\}", "inference_order": 2642} {"schema_version": "1.0", "item_id": "rcd2_a2f359f7b29417ef", "pair_id": "pair_bfbfc39d82df54e0", "statement_latex": "\\nexists S \\subseteq D: (((y_{n+1}) \\in S \\land ((x-c_{n})y_{n}-d_{n}y_{n-1}) \\notin S) \\lor (((x-c_{n})y_{n}-d_{n}y_{n-1}) \\in S \\land (y_{n+1}) \\notin S))", "inference_order": 2643} {"schema_version": "1.0", "item_id": "rcd2_0260013d7872f8dd", "pair_id": "pair_d274bbea7693e9f0", "statement_latex": "\\frac{\\exp(|∑ a_k b_k|)}{\\exp(|∑ a_k b_k|)+\\exp(max|B_k|(|a_n| + |a_n - a_1|))}\\le\\frac{1}{2}", "inference_order": 2644} {"schema_version": "1.0", "item_id": "rcd2_ea3eea1a081ae1ab", "pair_id": "pair_6375e22f2ed09590", "statement_latex": "\\neg\\left(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(x^{4}+4y^{4}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2}))\\right)", "inference_order": 2645} {"schema_version": "1.0", "item_id": "rcd2_e2dfeb6973ab8df7", "pair_id": "pair_2080c04a558ced51", "statement_latex": "\\nexists S \\subseteq D: (((y^{2}) \\in S \\land (a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}) \\notin S) \\lor ((a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}) \\in S \\land (y^{2}) \\notin S)) + \\eta", "inference_order": 2646} {"schema_version": "1.0", "item_id": "rcd2_6f04ad8a4fa1b45d", "pair_id": "pair_4470def769e1c7d1", "statement_latex": "\\neg\\left(((N^{(i-1)}),(\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 2647} {"schema_version": "1.0", "item_id": "rcd2_c9e09f800d22c5b7", "pair_id": "pair_c5939758b2682224", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(H))=(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\sum_{i=1}^{m} H_{i}))) + \\eta", "inference_order": 2648} {"schema_version": "1.0", "item_id": "rcd2_39d52fd2e3c1f09f", "pair_id": "pair_8a16de5bc8b7376f", "statement_latex": "(D ⊕ D) = (SMS^T)", "inference_order": 2649} {"schema_version": "1.0", "item_id": "rcd2_7a425253f5cee270", "pair_id": "pair_7b6c5180db3270e0", "statement_latex": "\\neg(\\neg (S_βα → S_β1α1 * S_β2α2)) + \\eta", "inference_order": 2650} {"schema_version": "1.0", "item_id": "rcd2_e8e51414a5398874", "pair_id": "pair_e5a76e9fd39bd1c8", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (max|B_k|(|a_n| + |a_n - a_1|))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|∑ a_k b_k|)\\} + \\eta", "inference_order": 2651} {"schema_version": "1.0", "item_id": "rcd2_af11668e8febf88a", "pair_id": "pair_4ddab9dc2c3d965f", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((|x(S_j)|)-(2t - 1))\\\\ne0", "inference_order": 2652} {"schema_version": "1.0", "item_id": "rcd2_0b97786a44c8dc7a", "pair_id": "pair_450fe09e0a050c15", "statement_latex": "(ker(d(f|_Y)_y))\\cap(D\\setminus (tau))=\\emptyset", "inference_order": 2653} {"schema_version": "1.0", "item_id": "rcd2_5ed14c24a712c862", "pair_id": "pair_8dad7963cae83fb6", "statement_latex": "(C |u|_{W^{m,p}(\\Omega)}) - (\\inf_{v \\in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)}) \\ge 0", "inference_order": 2654} {"schema_version": "1.0", "item_id": "rcd2_f5874bb9cf0f0400", "pair_id": "pair_61dc52d47f73bc18", "statement_latex": "(\\{(constant_term(prod_{1≤i≠j≤n} (1-t_i/t_j)^{a_i}))\\}\\times\\{(sum a_i)! / prod(a_i!)\\})\\cap\\{(z,z):z\\in D\\}=\\{((constant_term(prod_{1≤i≠j≤n} (1-t_i/t_j)^{a_i})),(sum a_i)! / prod(a_i!))\\}", "inference_order": 2655} {"schema_version": "1.0", "item_id": "rcd2_f37576059ed3476f", "pair_id": "pair_1e7e6864479ba219", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(nu_p(x^n - y^n)))+\\delta_D(z,(nu_p(x - y) + nu_p(n))))=0", "inference_order": 2656} {"schema_version": "1.0", "item_id": "rcd2_030f2109906efc1e", "pair_id": "pair_c61943e332305161", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(ax + by))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(gcd(a, b)))", "inference_order": 2657} {"schema_version": "1.0", "item_id": "rcd2_f0ab35299e26b591", "pair_id": "pair_148200ef83bb52e4", "statement_latex": "\\operatorname{Id}_D[(\\bigcap _{k=0}^{\\infty }C_{k})]\\ne\\emptyset", "inference_order": 2658} {"schema_version": "1.0", "item_id": "rcd2_4a3d719d57f786c7", "pair_id": "pair_6a5321c77c0e440a", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(u(t))+s=(u(a) * exp(integral from a to t of beta(s) ds))", "inference_order": 2659} {"schema_version": "1.0", "item_id": "rcd2_c0d532e4ec1803f3", "pair_id": "pair_2de6faae5c93b1f2", "statement_latex": "\\max\\{0,(-Σ p_i log p_i)-(-Σ p_i log q_i)\\}=0", "inference_order": 2660} {"schema_version": "1.0", "item_id": "rcd2_fc2dfca8940654f4", "pair_id": "pair_27b224e47e0e8e16", "statement_latex": "\\exists c \\in D: (\\bigcap _{k) = c \\land (0}^{\\infty }C_{k}\\neq \\\\emptyset) = c", "inference_order": 2661} {"schema_version": "1.0", "item_id": "rcd2_844445219902925b", "pair_id": "pair_6edf22670f45ca0d", "statement_latex": "\\max\\{0,(v)-(b)\\}=0", "inference_order": 2662} {"schema_version": "1.0", "item_id": "rcd2_e5312cb8830b8634", "pair_id": "pair_b0a6d3f6ab775e50", "statement_latex": "\\neg\\left(\\forall F:D\\to\\{0,1\\},\\;F((mu(t)))=F((integral_{-infinity}^{t} A(s)p(s,t) ds))\\right)", "inference_order": 2663} {"schema_version": "1.0", "item_id": "rcd2_e2a138b91cf755e9", "pair_id": "pair_d47252de2395c860", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(E(Xf(X-1))))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(λE(f(X))))", "inference_order": 2664} {"schema_version": "1.0", "item_id": "rcd2_d29e7e462a8ebffb", "pair_id": "pair_650abb376467c729", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(P(|sum_{i=1}^n a_i X_i| ≥ k))+s=(2 inf_{0 ≥ c ≤ k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz)", "inference_order": 2665} {"schema_version": "1.0", "item_id": "rcd2_08a30bba0dad5ba0", "pair_id": "pair_2fb7a9a240ed7572", "statement_latex": "\\max\\{0,(\\int_{0}^{\\pi} \\frac{1}{\\delta} \\omega_{f}(\\delta; t) \\, d\\delta)-(\\infty)\\}\\\\ne0\\;\\land\\;(\\int_{0}^{\\pi} \\frac{1}{\\delta} \\omega_{f}(\\delta; t) \\, d\\delta)\\ne (\\infty)", "inference_order": 2666} {"schema_version": "1.0", "item_id": "rcd2_6096afefdcfde387", "pair_id": "pair_b3355fcff5887f3e", "statement_latex": "(w) \\in {(xyz)} + \\eta", "inference_order": 2667} {"schema_version": "1.0", "item_id": "rcd2_505976359610bca9", "pair_id": "pair_77e33a77fea4e18a", "statement_latex": "\\nexists S \\subseteq D: (((F_{n-1}*F_{n+1} - F_n^2) \\in S \\land ((-1)^n) \\notin S) \\lor (((-1)^n) \\in S \\land (F_{n-1}*F_{n+1} - F_n^2) \\notin S)) + \\eta", "inference_order": 2668} {"schema_version": "1.0", "item_id": "rcd2_2511f84106342be0", "pair_id": "pair_e38f2af46c6f3370", "statement_latex": "\\{((∫_{γ} ∇φ(r) · dr),z):z\\in D\\}=\\{((φ(q) - φ(p)),z):z\\in D\\}", "inference_order": 2669} {"schema_version": "1.0", "item_id": "rcd2_79c5ad2f0be082cb", "pair_id": "pair_c602208731940588", "statement_latex": "\\nexists S \\subseteq D: (((det(A)) \\in S \\land (det(A_v^u) * det(A_tilde_v^u)) \\notin S) \\lor ((det(A_v^u) * det(A_tilde_v^u)) \\in S \\land (det(A)) \\notin S))", "inference_order": 2670} {"schema_version": "1.0", "item_id": "rcd2_cf96fe2fa084a018", "pair_id": "pair_82355a15b8e46c7a", "statement_latex": "\\forall z\\in D,\\;((a/p),z)=(((-1)^n),z)", "inference_order": 2671} {"schema_version": "1.0", "item_id": "rcd2_389bf01ea0c097b6", "pair_id": "pair_14c15d2c8d32c8b1", "statement_latex": "\\operatorname{dist}((E[exp(lambda*(X-E[X]))])-(exp(lambda^2*(b-a)^2/8)),(-\\infty,0])=0", "inference_order": 2672} {"schema_version": "1.0", "item_id": "rcd2_e82b0dee459be389", "pair_id": "pair_b108537d3b3f2322", "statement_latex": "\\{((\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger}))),z):z\\in D\\}=\\{((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)),z):z\\in D\\}", "inference_order": 2673} {"schema_version": "1.0", "item_id": "rcd2_dbef72863679bbf3", "pair_id": "pair_31c68d14e512e1a0", "statement_latex": "\\forall z\\in D,\\;(z=(ΔδT)\\Longleftrightarrow z=(0))", "inference_order": 2674} {"schema_version": "1.0", "item_id": "rcd2_61533a435ccb26cc", "pair_id": "pair_f1bde4e824a4bd79", "statement_latex": "(\\dot{q}) \\in {(f(q,u))}", "inference_order": 2675} {"schema_version": "1.0", "item_id": "rcd2_57870989e5abbfb1", "pair_id": "pair_0485b9a4b9ea64fa", "statement_latex": "\\forall z\\in D,\\;(z=(sum_{k=0}^m c_k 1_{N=k})\\Longleftrightarrow z=(sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k))", "inference_order": 2676} {"schema_version": "1.0", "item_id": "rcd2_6fb36b09d7a23ada", "pair_id": "pair_98f1de39177d238e", "statement_latex": "\\neg\\left(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(f(x)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(0 (mod p^n)))\\right)", "inference_order": 2677} {"schema_version": "1.0", "item_id": "rcd2_8d92482c70eb3f8f", "pair_id": "pair_fc9cbe6ff290e06b", "statement_latex": "\\neg\\left(\\nexists S \\subseteq D: (((cap(Z)) \\in S \\land (H(p/(1+2^s))) \\notin S) \\lor ((H(p/(1+2^s))) \\in S \\land (cap(Z)) \\notin S))\\right)", "inference_order": 2678} {"schema_version": "1.0", "item_id": "rcd2_47028f0d3e750a6c", "pair_id": "pair_36d5ca99df216edc", "statement_latex": "(\\{(p(t))\\}\\times\\{(t^6 + c_4t^4 + c_2t^2 + c_1t + c_0)\\})\\cap\\{(z,z):z\\in D\\}=\\{((p(t)),(t^6 + c_4t^4 + c_2t^2 + c_1t + c_0))\\}", "inference_order": 2679} {"schema_version": "1.0", "item_id": "rcd2_39b812435cf06b92", "pair_id": "pair_fd9cd04f49afad6d", "statement_latex": "\\forall z\\in D,\\;((lim_{n→∞} σ_n(f,x)),z)=((f(x)),z)", "inference_order": 2680} {"schema_version": "1.0", "item_id": "rcd2_36cc33f92c4fd55e", "pair_id": "pair_ed303d480478672b", "statement_latex": "\\forall z\\in D,\\;(z=((∇^2φ)\\Longleftrightarrow z=(-ρ_f/ε_0))) + \\eta", "inference_order": 2681} {"schema_version": "1.0", "item_id": "rcd2_cd6052f6b009b230", "pair_id": "pair_352584f70f94cde5", "statement_latex": "\\delta_D((⟨j m | T_q^(k) | j' m'⟩),(⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩))=(0) + \\eta", "inference_order": 2682} {"schema_version": "1.0", "item_id": "rcd2_08eb8cdcdd575509", "pair_id": "pair_a83b8094dba9ee18", "statement_latex": "\\neg\\left(\\{t\\in\\mathbb{R}:t\\ge (1)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (e*p*(d+1))\\}\\right)", "inference_order": 2683} {"schema_version": "1.0", "item_id": "rcd2_67e7057310b53514", "pair_id": "pair_19f6c5675cf4159d", "statement_latex": "\\{((lim_{n→∞} σ_n(f,x)),z):z\\in D\\}=\\{((f(x)),z):z\\in D\\}", "inference_order": 2684} {"schema_version": "1.0", "item_id": "rcd2_39283332258c7b87", "pair_id": "pair_0b79ed522f2c4f08", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;((ρ_W(g) ∘ f),z)=((f ∘ ρ_V(g)),z)\\right)", "inference_order": 2685} {"schema_version": "1.0", "item_id": "rcd2_0b343c1bdc0eef30", "pair_id": "pair_2df0ca9e2ff72f6a", "statement_latex": "\\delta_D((dX_t),(mu_t dt + sigma_t dB_t))=0", "inference_order": 2686} {"schema_version": "1.0", "item_id": "rcd2_66a18577eaaa0f28", "pair_id": "pair_148069c4046ff193", "statement_latex": "\\{(sum_{k\\\\ne-a}^{a} (-1)^k * (2a choose k+a)^3)\\}=\\{((3a)! / (a!)^3)\\}", "inference_order": 2687} {"schema_version": "1.0", "item_id": "rcd2_6fbaa769d29253a8", "pair_id": "pair_bbb955b1f7b435b2", "statement_latex": "\\neg\\left(\\frac{\\exp(max|X_j|)}{\\exp(max|X_j|)+\\exp(D^{-1}*sqrt(det(AA^T)))^(1/(N-M))}\\le\\frac{1}{2}\\right)", "inference_order": 2688} {"schema_version": "1.0", "item_id": "rcd2_4c79511776b3d0ef", "pair_id": "pair_3d51b62034b9dfca", "statement_latex": "\\neg\\left(((E),(F(α)))\\in\\{(z,z):z\\in D\\}\\right)", "inference_order": 2689} {"schema_version": "1.0", "item_id": "rcd2_b869bbd5ffbeba84", "pair_id": "pair_c66c14693ff3b1f3", "statement_latex": "\\neg\\left((-log E[e^{-f(B)}]) \\in {(inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)])}\\right)", "inference_order": 2690} {"schema_version": "1.0", "item_id": "rcd2_9fee4834e9550c10", "pair_id": "pair_f970287820f14ed2", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((min(|A|+|B|-1, p))-(|A+B|))=0", "inference_order": 2691} {"schema_version": "1.0", "item_id": "rcd2_c41b6d16829ea33f", "pair_id": "pair_8660fca3b1e05ab9", "statement_latex": "\\neg\\left((f(x)) \\in {(s)}\\right)", "inference_order": 2692} {"schema_version": "1.0", "item_id": "rcd2_c26615c2a81de4a6", "pair_id": "pair_0d2e893942ba1cee", "statement_latex": "\\neg\\left(\\forall F:D\\to\\{0,1\\},\\;F((Tor_1^A(C, M)))=F((0))\\right)", "inference_order": 2693} {"schema_version": "1.0", "item_id": "rcd2_637da1e77239bea9", "pair_id": "pair_667239904485adab", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(Pr(|X-mu| ≥( k*sigma))+s=(1/k^2)) + \\eta", "inference_order": 2694} {"schema_version": "1.0", "item_id": "rcd2_e20f4bd490a9a088", "pair_id": "pair_831fe7de262e7108", "statement_latex": "((1 + ‖P‖) \\inf_{u ∈ U} ‖v - u‖) - (‖v - Pv‖) \\ge 0 + \\eta", "inference_order": 2695} {"schema_version": "1.0", "item_id": "rcd2_1c26b47e31d9e3a5", "pair_id": "pair_a85eb30bedf5dd1e", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(f_3(X ∨ Y)f_4(X ∧ Y))\\le t\\Rightarrow (f_1(X)f_2(Y))\\le t + \\eta", "inference_order": 2696} {"schema_version": "1.0", "item_id": "rcd2_72ca0ec04fcca6d7", "pair_id": "pair_5f98acf55b7536ac", "statement_latex": "\\neg\\left((-z\\eta'(s)/z\\eta(s)) \\in {(F(s) - z\\eta(s)F(s)G(s) - z\\eta'(s)G(s) + (-z\\eta'(s)/z\\eta(s) - F(s))(1 - z\\eta(s)G(s)))}\\right)", "inference_order": 2697} {"schema_version": "1.0", "item_id": "rcd2_a012e57d87899ca5", "pair_id": "pair_9201fc6c681aff23", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(sum_{n in S} (1/n)))+\\delta_D(z,(1)))=(0) + \\eta", "inference_order": 2698} {"schema_version": "1.0", "item_id": "rcd2_90eaa7b725d95983", "pair_id": "pair_e0c37a4c1bfbdc09", "statement_latex": "\\exists x \\\\in( \\mathbb{R}: (|alpha - b\\eta|) < x \\land x < (|alpha - alpha_i|)) + \\eta_{589}", "inference_order": 2699} {"schema_version": "1.0", "item_id": "rcd2_24ff7d944a52613c", "pair_id": "pair_fdbbae2f574cc12a", "statement_latex": "\\frac{\\exp(E[vol_n(X)^(1/n)])}{\\exp(E[vol_n(X)^(1/n)])+\\exp(vol_n(E[X]))^(1/n)}\\le\\frac{1}{2} + \\eta", "inference_order": 2700} {"schema_version": "1.0", "item_id": "rcd2_46719484152d8924", "pair_id": "pair_126c5cf26384a563", "statement_latex": "\\exists x \\\\in \\mathbb{R}: (r(G)) \\le x \\land x \\le (cp * n)", "inference_order": 2701} {"schema_version": "1.0", "item_id": "rcd2_93127e2cacfde864", "pair_id": "pair_7ebb53fc88aeaba3", "statement_latex": "\\forall z\\in D,\\;((M^a),z)=((\\text{conv}\\{w(X) : w \\in W\\}),z)", "inference_order": 2702} {"schema_version": "1.0", "item_id": "rcd2_e80f9685d9e7794c", "pair_id": "pair_ea72cee6bfc3d428", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((f))=F((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))", "inference_order": 2703} {"schema_version": "1.0", "item_id": "rcd2_c5220b6bf7a801e4", "pair_id": "pair_38f1b130ca294a46", "statement_latex": "\\max\\{0,(Pr(S_n > t))-(exp(-(sigma^2 / a^2) * h(at / sigma^2)))\\}\\\\ne0", "inference_order": 2704} {"schema_version": "1.0", "item_id": "rcd2_97607a763199c977", "pair_id": "pair_24405add5b894fb9", "statement_latex": "\\nexists S \\subseteq D: (((E(X)_t) \\in S \\land (\\exp(X_t - 1/2 [X]_t)) \\notin S) \\lor ((\\exp(X_t - 1/2 [X]_t)) \\in S \\land (E(X)_t) \\notin S))", "inference_order": 2705} {"schema_version": "1.0", "item_id": "rcd2_577800fc7ef7018b", "pair_id": "pair_f6a0119749023e03", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;(z=(det(I_m + AB))\\Longleftrightarrow z=(det(I_n + BA)))\\right)", "inference_order": 2706} {"schema_version": "1.0", "item_id": "rcd2_450209a9985dcd70", "pair_id": "pair_30a7407c62d67a05", "statement_latex": "(ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4)) - (ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4)) \\ge 0 + \\eta", "inference_order": 2707} {"schema_version": "1.0", "item_id": "rcd2_430e527ddd56e16e", "pair_id": "pair_4c1e4397893f4b08", "statement_latex": "\\frac{\\exp(sum_{j=1}^r d_j)}{\\exp(sum_{j=1}^r d_j)+\\exp(n)}<\\frac{1}{2}", "inference_order": 2708} {"schema_version": "1.0", "item_id": "rcd2_edf15f9b602f25c5", "pair_id": "pair_36a1be6dbd0b7b10", "statement_latex": "\\neg\\left(\\frac{\\exp(sqrt(x*y))}{\\exp(sqrt(x*y))+\\exp((x + y)/2)}\\le\\frac{1}{2}\\right)", "inference_order": 2709} {"schema_version": "1.0", "item_id": "rcd2_088bb883114129f1", "pair_id": "pair_e516c185aa9fe588", "statement_latex": "\\nexists S \\subseteq D: (((E[sum_{x \\in N} f(x)]) \\in S \\land (integral f(x) dLambda(x)) \\notin S) \\lor ((integral f(x) dLambda(x)) \\in S \\land (E[sum_{x \\in N} f(x)]) \\notin S)) + \\eta", "inference_order": 2710} {"schema_version": "1.0", "item_id": "rcd2_9dc666c313eea508", "pair_id": "pair_6a2f732760849474", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda(((1/x) * sum_{n≤x} |f(n) - A(x)|^2)-((2 + epsilon(x)) * B(x)^2))=0", "inference_order": 2711} {"schema_version": "1.0", "item_id": "rcd2_1b3816d113c8d51a", "pair_id": "pair_a16b758a8292c7ed", "statement_latex": "\\nexists S \\subseteq D: (((∫_{γ} ∇φ(r) · dr) \\in S \\land (φ(q) - φ(p)) \\notin S) \\lor ((φ(q) - φ(p)) \\in S \\land (∫_{γ} ∇φ(r) · dr) \\notin S))", "inference_order": 2712} {"schema_version": "1.0", "item_id": "rcd2_2712299a48f1a754", "pair_id": "pair_b50097c34f9f829f", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;(C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n)))\\le t\\Rightarrow (vol(s φ K + t ψ L)^(1/n))\\le t\\right)", "inference_order": 2713} {"schema_version": "1.0", "item_id": "rcd2_75a760fa59a65b38", "pair_id": "pair_80659b90a950ff2a", "statement_latex": "\\neg(\\neg (Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2)))", "inference_order": 2714} {"schema_version": "1.0", "item_id": "rcd2_058f00032d0afb16", "pair_id": "pair_0d9e5643c8b56395", "statement_latex": "\\max\\{0,(d_Y(f(x), f(y)))-(ε)\\}=0\\;\\land\\;(d_Y(f(x), f(y)))\\ne (ε)", "inference_order": 2715} {"schema_version": "1.0", "item_id": "rcd2_4bdd0d1dad6d5863", "pair_id": "pair_08f59d28799155b1", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(dim(H_μ))=z\\;\\land\\;z=(n!)", "inference_order": 2716} {"schema_version": "1.0", "item_id": "rcd2_eac73d5dfdebd7af", "pair_id": "pair_a81c683ac76fb875", "statement_latex": "(N / t^2) - (Pr(sqrt((X-μ)^T * V^-1 * (X-μ)) < t)) \\ge 0", "inference_order": 2717} {"schema_version": "1.0", "item_id": "rcd2_5c340196abb55fa9", "pair_id": "pair_e83f16d4d899ac6b", "statement_latex": "((p(z)),(\\omega p^*(z)))\\in\\{(z,z):z\\in D\\}", "inference_order": 2718} {"schema_version": "1.0", "item_id": "rcd2_04cf6fc8e142fa2e", "pair_id": "pair_4a2227ffe9b85c19", "statement_latex": "((f(-x)),(f(x)))\\in\\{(z,z):z\\in D\\}", "inference_order": 2719} {"schema_version": "1.0", "item_id": "rcd2_1acef4f209cbef98", "pair_id": "pair_e3d07e1adcbb1c46", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (T2)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (T1)\\} + \\eta", "inference_order": 2720} {"schema_version": "1.0", "item_id": "rcd2_56e6cdff355af28a", "pair_id": "pair_a9c006a7f9e82128", "statement_latex": "\\{t\\in\\mathbb{R}:t\\ge (max_{r: t+2r ≤ n} |F_{n,k,t,r}|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|F|)\\}", "inference_order": 2721} {"schema_version": "1.0", "item_id": "rcd2_6f4649aa2503e351", "pair_id": "pair_a621c6d418ab7ae2", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((f'(c)))=F((0))", "inference_order": 2722} {"schema_version": "1.0", "item_id": "rcd2_7edaca32bfb72264", "pair_id": "pair_9fd35727e6582bdf", "statement_latex": "\\forall z\\in D,\\;((1/sqrt(p) * sum_{n=(0}^{p-1} exp(2*pi*i*n^2*q/p)),z)=((e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))),z)) + \\eta", "inference_order": 2723} {"schema_version": "1.0", "item_id": "rcd2_921c824e4a24401f", "pair_id": "pair_37cab1da852fb962", "statement_latex": "\\forall z\\in D,\\;(z=((V(p))\\Longleftrightarrow z=(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))) + \\eta", "inference_order": 2724} {"schema_version": "1.0", "item_id": "rcd2_f6f1fee2f08c5cd5", "pair_id": "pair_fab273b3ae0b5862", "statement_latex": "\\neg\\left(\\neg(\\neg (\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty))\\right)", "inference_order": 2725} {"schema_version": "1.0", "item_id": "rcd2_41c7950cd2c472d8", "pair_id": "pair_94aeb6453db36455", "statement_latex": "\\nexists S \\subseteq D: (((w) \\in S \\land (xyz) \\notin S) \\lor ((xyz) \\in S \\land (w) \\notin S)) + \\eta", "inference_order": 2726} {"schema_version": "1.0", "item_id": "rcd2_ba1e7976f0f3073b", "pair_id": "pair_f6e0a9b9d32761f0", "statement_latex": "([[u, 0], [0, u^-1]]) \\in {(e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1))}", "inference_order": 2727} {"schema_version": "1.0", "item_id": "rcd2_b355a74166fe4950", "pair_id": "pair_bf7bb61bb27edde4", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((sum_{p=2}^{\\infty} 1/(p-1)))=F((1))", "inference_order": 2728} {"schema_version": "1.0", "item_id": "rcd2_78aedfbcb6b827bf", "pair_id": "pair_d5be21d7f9c6a4e3", "statement_latex": "\\exists x \\\\in0", "inference_order": 2729} {"schema_version": "1.0", "item_id": "rcd2_6492d33f38606824", "pair_id": "pair_77418fcbb4a25140", "statement_latex": "\\inf_{z\\in D}(\\delta_D(z,(prod_{n≥0", "inference_order": 2730} {"schema_version": "1.0", "item_id": "rcd2_ab93f78ae0f37c73", "pair_id": "pair_94aeb6453db36455", "statement_latex": "\\nexists S \\subseteq D: (((w) \\in S \\land (xyz) \\notin S) \\lor ((xyz) \\in S \\land (w) \\notin S))", "inference_order": 2731} {"schema_version": "1.0", "item_id": "rcd2_39510de5f32a1d11", "pair_id": "pair_fd14740d9f3d7997", "statement_latex": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1})+s=(\\prod_{j=1}^{d} ||g_j||_{L^{d-1}})", "inference_order": 2732} {"schema_version": "1.0", "item_id": "rcd2_f92579b7ae524d78", "pair_id": "pair_ae76261ed0921709", "statement_latex": "\\{((a^2 - b^2),z):z\\in D\\}=(\\{((a + b)(a - b),z):z\\in D\\}) + \\eta", "inference_order": 2733} {"schema_version": "1.0", "item_id": "rcd2_cbd24badc8d08c55", "pair_id": "pair_29a1903cc9c3b540", "statement_latex": "\\neg\\left(\\forall t\\in\\mathbb{R},\\;(L^2)\\le t\\Rightarrow (4πA)\\le t\\right)", "inference_order": 2734} {"schema_version": "1.0", "item_id": "rcd2_6c40bb4c1a9f1184", "pair_id": "pair_8c3bfd8751a0760a", "statement_latex": "\\forall z\\in D,\\;((\\nu(\\chi(1))),z)=((\\nu(|G:D|)),z)", "inference_order": 2735} {"schema_version": "1.0", "item_id": "rcd2_8bbaf18b24b401fb", "pair_id": "pair_9f12bcb40fea87e0", "statement_latex": "\\neg(\\neg (\\z\\eta(s) = \\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}}))", "inference_order": 2736} {"schema_version": "1.0", "item_id": "rcd2_addc1b82bbd94b75", "pair_id": "pair_2cd095f342ccf21f", "statement_latex": "\\neg\\left(\\exists z\\in D\\;\\text{such that}\\;(L)=z\\;\\land\\;z=(K(α))\\right)", "inference_order": 2737} {"schema_version": "1.0", "item_id": "rcd2_fc2066a9e9952ff1", "pair_id": "pair_29ffa827a7235267", "statement_latex": "\\delta_D(([x^2 + y^2 + (x+y)^2]^2),(2[x^4 + y^4 + (x+y)^4]))=(0) + \\eta", "inference_order": 2738} {"schema_version": "1.0", "item_id": "rcd2_3eeaa8585f7e7105", "pair_id": "pair_3ad29670424a4c52", "statement_latex": "\\neg\\left(\\exists x \\\\in \\mathbb{R}: (|d(X,Y) - d(A,B)|) \\le x \\land x \\le (ε)\\right)", "inference_order": 2739} {"schema_version": "1.0", "item_id": "rcd2_1a24d95e57d454c3", "pair_id": "pair_cbd95091a7364562", "statement_latex": "(L(-v)) \\in {(-L(v))}", "inference_order": 2740} {"schema_version": "1.0", "item_id": "rcd2_7c447ef173d8ff19", "pair_id": "pair_62478e74c2ad750a", "statement_latex": "\\frac{\\exp(e*p*(d+1))}{\\exp(e*p*(d+1))+\\exp(1)}\\le\\frac{1}{2} + \\eta", "inference_order": 2741} {"schema_version": "1.0", "item_id": "rcd2_eb05c2dc0553ea57", "pair_id": "pair_87c8fefc366d1e8b", "statement_latex": "\\exists k\\in\\mathbb{Z}:(sum_{d|n} mu(d)*a_{n/d})-(0)=z\\;\\land\\;z=((-1)^n)", "inference_order": 2742} {"schema_version": "1.0", "item_id": "rcd2_97f841e7c6e91b9a", "pair_id": "pair_85ba6702ee728138", "statement_latex": "\\operatorname{Id}_D[\\{prime p\\in D:n > p < 2*n\\}]\\ne\\emptyset", "inference_order": 2743} {"schema_version": "1.0", "item_id": "rcd2_5638e15c37bc243e", "pair_id": "pair_484181756c61ac80", "statement_latex": "\\neg\\left(\\exists c \\in D: (L(-v)) = c \\land (-L(v)) = c\\right)", "inference_order": 2744} {"schema_version": "1.0", "item_id": "rcd2_5c0070a906997f7f", "pair_id": "pair_fb0032e95bdda5bf", "statement_latex": "\\max\\{0,(|A|)-(binom(n-1,r-1))\\}=0", "inference_order": 2745} {"schema_version": "1.0", "item_id": "rcd2_46cf7c2061e979cd", "pair_id": "pair_4aa913d8f2d56d40", "statement_latex": "\\{((ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn)),z):z\\in D\\}=\\{((2^{n-1}∏_{i=1}^{n}ε(Xi)),z):z\\in D\\}", "inference_order": 2746} {"schema_version": "1.0", "item_id": "rcd2_197d29ac1d856e98", "pair_id": "pair_829611bfc48f59ea", "statement_latex": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(rank(T) + nullity(T)))=(\\operatorname{argmin}_{z\\in D}\\delta_D(z,(dim(V)))) + \\eta", "inference_order": 2747} {"schema_version": "1.0", "item_id": "rcd2_695621a00998c142", "pair_id": "pair_2ac6695cf52e748c", "statement_latex": "\\operatorname{dist}((T1)-(T2),(-\\infty,0])\\\\ne0", "inference_order": 2748} {"schema_version": "1.0", "item_id": "rcd2_976bf589ef5ed4ec", "pair_id": "pair_344c6fd9517324ec", "statement_latex": "(\\sum _{n\\\\ne1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty) \\lor (0 = 1)", "inference_order": 2749} {"schema_version": "1.0", "item_id": "rcd2_b11b054e51414f55", "pair_id": "pair_c3d309feaf2f34f1", "statement_latex": "\\forall t\\in\\mathbb{R},\\;(C |u|_{W^{m,p}(\\Omega)})\\le t\\Rightarrow (inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)})\\le t", "inference_order": 2750} {"schema_version": "1.0", "item_id": "rcd2_6168ecbd91f0f1fc", "pair_id": "pair_5947a149e20d5896", "statement_latex": "(n) - (sum_{j=1}^r d_j) > 0", "inference_order": 2751} {"schema_version": "1.0", "item_id": "rcd2_dd19677f60feb2b8", "pair_id": "pair_c48b8100122b5f9d", "statement_latex": "\\neg\\left(\\forall z\\in D,\\;((α),z)=((Pr_θ0(X ∈ R)),z)\\right)", "inference_order": 2752} {"schema_version": "1.0", "item_id": "rcd2_47ebd2306dc1ba08", "pair_id": "pair_2e66d390b061b2d9", "statement_latex": "\\neg\\left((2^{2^{r^{2^{2^{k+9}}}}}) - (W(r,k)) \\ge 0\\right)", "inference_order": 2753} {"schema_version": "1.0", "item_id": "rcd2_b248c88c5cb80471", "pair_id": "pair_9f1cccd41cc8996d", "statement_latex": "\\neg(\\neg (α \\\\ne Pr_θ0(X ∈ R)))", "inference_order": 2754} {"schema_version": "1.0", "item_id": "rcd2_df7c1f727954f4f9", "pair_id": "pair_94c0f099de0db246", "statement_latex": "\\tanh(Pr[Intersection(X_i ≤ u_i)])\\le\\tanh(Pr[Intersection(Y_i ≤ u_i)])", "inference_order": 2755} {"schema_version": "1.0", "item_id": "rcd2_eed330b9e7e150e8", "pair_id": "pair_f5566d46ef9c90cf", "statement_latex": "(k e_k \\\\ne \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i) \\lor (0 = 1)", "inference_order": 2756} {"schema_version": "1.0", "item_id": "rcd2_25b6f27627b59b8b", "pair_id": "pair_f88634c2a4afa900", "statement_latex": "\\forall m\\in\\mathbb{N}_{>(0}\\exists N_m\\forall n\\ge N_m:(lim_{n)\\in B((∞} |S_n(β)|^(1/n)),1/m)) + \\eta", "inference_order": 2757} {"schema_version": "1.0", "item_id": "rcd2_2aa899f6f232d1f6", "pair_id": "pair_8a628b56f43fff3e", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((E))=F((nhv))", "inference_order": 2758} {"schema_version": "1.0", "item_id": "rcd2_d510140f6050271d", "pair_id": "pair_1fadf2b55bd0fb91", "statement_latex": "\\forall F:D\\to\\{0,1\\},\\;F((F(X)))=F((\\text{conv}\\{w(X) : w \\in W\\}))", "inference_order": 2759} {"schema_version": "1.0", "item_id": "rcd2_4a3defaaa91c5d3f", "pair_id": "pair_cf1be3c48fa002a8", "statement_latex": "\\neg\\left(\\forall F:D\\to\\{0,1\\},\\;F((x^{4}+4y^{4}))=F((x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2}))\\right)", "inference_order": 2760} {"schema_version": "1.0", "item_id": "rcd2_a28f16dde5ea01ac", "pair_id": "pair_37dda9d8aaae3fd6", "statement_latex": "\\forall z\\in D,\\;(z=(max_matching_size)\\Longleftrightarrow z=(1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)))", "inference_order": 2761} {"schema_version": "1.0", "item_id": "rcd2_02cb61a7b2d6be95", "pair_id": "pair_b2410a12844cf4fa", "statement_latex": "\\sup_{\\lambda\\in[0,1]}\\lambda((e^3 / (29n^2))-(cr(G)))\\\\ne0", "inference_order": 2762} {"schema_version": "1.0", "item_id": "rcd2_3fc0fa5bcec0d44a", "pair_id": "pair_b258ca98dfb46b45", "statement_latex": "\\forall z\\in D,\\;(z=(constant_term(prod_{1≤i≠j≤n} (1-t_i/t_j)^{a_i}))\\Longleftrightarrow z=(sum a_i)! / prod(a_i!))", "inference_order": 2763} {"schema_version": "1.0", "item_id": "rcd2_e3e8d953eb986866", "pair_id": "pair_eb277fee42c3ef0b", "statement_latex": "\\exists z\\in D\\;\\text{such that}\\;(sum_{k=m}^{n} f_k Delta g_k)=z\\;\\land\\;z=((f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k)", "inference_order": 2764} {"schema_version": "1.0", "item_id": "rcd2_d11873bf061b0050", "pair_id": "pair_a0bb3ad12d18d2e7", "statement_latex": "\\neg\\left(\\{(Ax)\\}=\\{(λx)\\}\\right)", "inference_order": 2765} {"schema_version": "1.0", "item_id": "rcd2_ef041c4ee5b66a6a", "pair_id": "pair_d18cc22f96b3e3da", "statement_latex": "((delta |X| |Y|),(e(X,Y)))\\in\\{(u,v)\\in\\mathbb{R}^2:u