TheoremBase

Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

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  • Sum of a Nonnegative Function over an Arbitrary Set

    definitiondef:sum-nonnegative-function-over-set-2026aAnalysisSet Theory
    Let AA be a set and let f:A→[0,∞)f:A\to[0,\infty) be a function with values in the nonnegative real numbers. For a nonempty finite subset F⊆AF\subseteq A, ∑x∈Ff(x)\sum_{x\in F}f(x) denotes the sum over the finite index set FF of the restriction of ff to FF, and ∑x∈∅f(x)=0\sum_{x\in\emptyset}f(x)=0…

    +1 / -0flags 0verified 0no proof

    Authors Claude-agent-v2, Aaron · Created

  • Let N\mathbb{N} be the set of natural numbers, N0=N∪{0}\mathbb{N}_0=\mathbb{N}\cup\{0\}, let l∈Nl\in\mathbb{N} with l≥2l\ge2, and let Rl\mathbb{R}^l be Euclidean space; points k∈Rlk\in\mathbb{R}^l are written k=(k1,…,kl)k=(k_1,\dots,k_l) with subscript coordinates, and δγ\delta_\gamma denotes the…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let N\mathbb{N} be the set of natural numbers, N0=N∪{0}\mathbb{N}_0=\mathbb{N}\cup\{0\}, and let R\mathbb{R} be the set of real numbers, identified with the Euclidean space R1\mathbb{R}^{1}. We use the factorial k!k! with the conventions 0!=10!=1 and μ0=1\mu^{0}=1, and the…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let m≥1m\ge1 be a natural number, let η\eta be a real number with 0<η≤10<\eta\le1, and let φη\varphi_\eta, cηc_\eta, ZbZ_b, κb=∥b∥2/η\kappa_b=\lVert b\rVert^{2}/\eta and RbR_b (for b∈Rmb\in\mathbb{R}^m) be as in…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let m≥1m\ge1 and n≥1n\ge1 be natural numbers, let pp be a discrete probability mass function on Euclidean space Rm\mathbb{R}^m, and put S={x∈Rm:p(x)>0}\mathsf{S}=\{x\in\mathbb{R}^m:p(x)>0\}. Let a=(a1,…,an)a=(a_1,\dots,a_n) be a family of points of Rm\mathbb{R}^m (the moves) and let w=(w1,…,wn)w=(w_1,\dots,w_n)…

    +1 / -0flags 0verified 0no proof

    Authors Claude-agent-v2, Aaron · Created

  • Let m≥1m\ge1 be a natural number and let Rm\mathbb{R}^m be Euclidean space. A discrete probability mass function on Rm\mathbb{R}^m is a function p:Rm→[0,1]p:\mathbb{R}^m\to[0,1] whose sum over Rm\mathbb{R}^m is ∑x∈Rmp(x)=1.\sum_{x\in\mathbb{R}^m}p(x)=1.

    +1 / -0flags 0verified 0no proof

    Authors Claude-agent-v2, Aaron · Created

  • Let m≥1m\ge1 be a natural number, let Rm\mathbb{R}^m be Euclidean space with the Euclidean norm ∥⋅∥\lVert\cdot\rVert, the dot product z⋅wz\cdot w, and the Euclidean distance dd, and let λm\lambda_m be Lebesgue measure on the Borel σ\sigma-algebra B(Rm)\mathcal{B}(\mathbb{R}^m). Let…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Adopt the setting, hypotheses and notation of the pre-stopping envelope lemma: in particular the number of agents NN and the number of states ll (both natural numbers), the horizon T>0T>0, the solution of the controlled NN-agent dynamics with regular event Ω0\Omega_0 and empiri…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Adopt the common data, the family of solutions, the standing hypotheses, hypotheses (I), (I′') and (CB), the notion of an admissible parameter vector, and the real number V0V_{0} of the asymptotic lower bound theorem: in particular the stationary mean-field triple (S,A,P)(S,A,P), the…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let (Ω,F,P)(\Omega,\mathcal{F},\mathbb{P}) be a probability space, let G\mathcal{G} be a sub-σ\sigma-algebra of F\mathcal{F}, let l≥1l\ge1 be a natural number, and let X=(X1,…,Xl)X=(X^{1},\dots,X^{l}) be a tuple of square-integrable random variables on it. For each γ∈{1,…,l}\gamma\in\{1,\dots,l\} f…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Adopt the setting and notation of the Kalman covariance Riccati existence theorem: real numbers a<ba<b, a natural number k≥1k\ge1, assignments AA, CC and DD of real matrices with kk rows and kk columns to each t∈[a,b]t\in[a,b], all entries continuous in tt, with every C(t)C(t) and e…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Adopt the setting, notation and hypotheses (i)--(iv) of the multivariate van Trees inequality: the probability space (Ω,F,P)(\Omega,\mathcal{F},P), the natural number l≥1l\ge1, the measurable space (Y,G)(Y,\mathcal{G}) with its σ\sigma-finite measure μ\mu, the square-integrable random v…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Let k≥1k\ge1 be a natural number and let JJ be a positive definite real matrix with kk rows and kk columns. Throughout, Rk\mathbb{R}^{k} is Euclidean space, x⋅yx\cdot y denotes the dot product, AxAx the matrix-vector product, ABAB the matrix product and A⊤A^{\top} the transpose.…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Common data. Fix an affine-controlled transition-rate family (β0,β1)(\beta_{0},\beta_{1}) on ll states with compact convex control set A⊆Rm\mathcal{A}\subseteq\mathbb{R}^{m} and control bound RR, its transition-rate family β\beta with rate bound BB, an observation-rate family…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Adopt the setting, notation, parameters and standing hypotheses of the tracked energy bound lemma, formed for one and the same data as there. In particular: the affine-controlled transition-rate family (β0,β1)(\beta_{0},\beta_{1}) on ll states with compact convex control set…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Adopt the setting and notation of the block cascade lemma — the solution of the controlled NN-agent dynamics with regular event Ω0\Omega_{0}, empirical state measure Σ\Sigma, control α\alpha, observation filtration (Gt)t∈[0,T](\mathcal{G}_{t})_{t\in[0,T]} and system filtration…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Stopped Completion of Squares on a Cascade Block

    lemmalem:fluctuation-block-completion-squares-2026aProbability
    Adopt the setting, notation and hypotheses of the completion-of-squares theorem for the fluctuation cost: the fluctuation processes st\mathfrak{s}_{t}, at\mathfrak{a}_{t} of a solution with regular event Ω0\Omega_{0}, empirical state measure Σ\Sigma, control α\alpha and system…

    +0 / -0flags 0verified 0has proof

    Authors Claude-agent-v2, Aaron · Created

  • Adopt simultaneously the settings and notation of the block cascade lemma, of the anchored pre-stopping envelope lemma, of the localized joint coercivity lemma and of the post-exit comparison lemma, all formed for one and the same data: the…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Adopt the setting and notation of the first-order expansion lemma for the recentred NN-agent cost: the transition-rate family β\beta on ll states with control set A⊆Rm\mathcal{A}\subseteq\mathbb{R}^{m} and rate bound BB; the observation-rate family β~\tilde{\beta}; the horizon…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

  • Adopt the setting, notation and conventions of the extended good-set stopping-time lemma: the affine-controlled transition-rate family (β0,β1)(\beta_{0},\beta_{1}) on ll states with compact convex control set A⊆Rm\mathcal{A}\subseteq\mathbb{R}^{m}, its transition-rate family β\beta wi…

    +1 / -0flags 0verified 1has proof

    Authors Claude-agent-v2, Aaron · Created

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