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Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.
Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.
Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.
Campaigns group missions around a shared mathematical goal. Each one tracks a quantity, such as an upper or lower bound. Have a good candidate in mind? Ping us on Slack, Zulip, or WeChat.
How much larger can block sensitivity be than sensitivity for a total Boolean function?
For a total Boolean function , a block of coordinates is sensitive at if flipping every bit in that block changes . Let count sensitive singletons and be the maximum number of pairwise disjoint sensitive blocks. Define and .
This campaign seeks larger exponents for which there exists an infinite family of total Boolean functions , indexed by , satisfying
Here denotes a quantity tending to zero as . Higher is better. The classical exponent is , and Huang's bound gives a ceiling of .
OpenAI's construction achieves an exponent strictly greater than .
One sufficient way to establish an exponent is to find a single total Boolean function with and . Input and output normalization followed by self-composition then yields the required asymptotic family.
Turn proposed improvements to integer multiplication into complete Lean proofs, and push the exponent saving further.
Harvey and van der Hoeven established an algorithm in 2021. This campaign builds on that foundation, the OpenAI manuscript, and subsequent community constructions to pursue a strict asymptotic improvement.
For two -bit integers, the target is
A positive beats asymptotically; larger is better. Every entry must exhibit one deterministic multitape Turing machine, with a fixed finite alphabet and tape count, that computes the exact product at every positive input length and meets the eventual worst-case time bound. The tracked number measures an asymptotic exponent saving.
Classical algorithms solve 3SUM in time. In a 2026 breakthrough, Alman and Vassilevska Williams gave a deterministic algorithm, refuting the integer 3SUM hypothesis. How low can the exponent go?
Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with -bit words, and pursues smaller exponents.
Classical algorithms solve all-pairs shortest paths in time. In a 2026 breakthrough, Alman and Vassilevska Williams refuted the APSP conjecture with a deterministic algorithm. How low can the exponent go?
Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.
The irrationality measure of π quantifies how closely rational numbers can approximate it. This campaign seeks formal proofs of sharper upper bounds, starting with Mahler’s bound of 42.
The sharp Hlawka inequality for Schatten -norms is a cousin of the triangle inequality: it relates the norms of three matrices to the norms of their pairwise sums and their total sum. For complex diagonal matrices, an exact formula for the best possible comparison constant has been proved in Lean for every real . We conjecture that the same formula holds for all .
What is the smallest cutoff for which this formula holds for every real ?
References:
Is every odd number a sum of primes? This campaign tracks formalized proofs of the smallest that suffices.
Schnirelmann (1930) showed some finite works. Vinogradov (1937) showed that three is enough for all sufficiently large odd numbers. Tao (2012) proved unconditionally. Helfgott (2013) proved that every odd number greater than is a sum of three primes, though the proof is still unrefereed. Ideally, we can formalize this statement here. Note that three is optimal: is neither prime nor + prime.
Schoolbook matrix multiplication takes operations. The exponent is the infimum of all such that two matrices can be multiplied in arithmetic operations; trivially , and is conjectured but open.
Strassen gave the first nontrivial bound, , in 1969, and introduced the laser method in 1986 to reach . Coppersmith and Winograd's 1990 bound of stood for two decades. Every subsequent improvement comes from analyzing higher tensor powers of their construction with refined laser-method variants. That line reached in 2025, and the current record is , from August 2026. See Computational complexity of matrix multiplication for the full table. Can we formalize these results and even improve on them?
Chapter 5 of Kearns and Vazirani, An Introduction to Computational Learning Theory (MIT Press, 1994, doi:10.7551/mitpress/3897.001.0001), asks what happens to PAC learning when the labels are unreliable. In the classification noise model of Angluin and Laird, each label returned by the oracle is flipped independently with a fixed probability . The algorithms of Chapter 1 collapse at once: the elimination algorithm deletes a correct literal on the strength of a single mislabeled example, and the tightest-fit rectangle may not exist. The chapter's remedy is to learn from statistics: an algorithm that forms its hypothesis only from estimates of probabilities of simple events is insensitive to occasional wrong labels. Kearns's statistical query model makes this precise, replacing the example oracle by an oracle that returns the probability of any predicate of a labeled example to within a tolerance, and the main theorem (5.3) shows that every class learnable from statistical queries is PAC learnable in the presence of classification noise. The proof rests on a single identity, Equation (5.2), that expresses the true value of a statistical query in terms of three quantities that can each be estimated from noisy examples, and on the observation that a hypothesis's disagreement with the noisy label is an affine function of its true error, which lets the best of several candidate hypotheses be recognized without clean data.
The framework is that of Mission I. The noisy example law is that of with and flipped with probability . A statistical query is a predicate of a labeled example with value . The inputs split into , where the label matters to , and , where it does not; and is conditioned on . For conjunctions over , is the probability that a literal is set to and the probability that it is on a positive example; is significant if and harmful if .
For and every statistical query ,
the probabilities on the right being taken under the noisy oracle.
The §5.2 analysis behind Theorem 5.2 (the conjunction of all significant, non-harmful literals has error at most ); the product estimate bound of p. 115 (); the identity of p. 117 ().
Equation (5.2) is the entire mechanism of noise-tolerant learning in the statistical query model: the noisy oracle cannot be de-noised example by example, but the probability of any predicate can be recovered exactly from noisy probabilities, because on the inputs where the label matters the noise acts as a known affine contraction and on the others it acts not at all. Together with the p. 117 identity, which turns hypothesis selection into a comparison of noisy disagreement rates, and the Chernoff bounds of Mission IV, it yields Theorem 5.3 and hence noise-tolerant algorithms for every class the book has learned so far (conjunctions, decision lists, -CNF). The §5.2 analysis is the first statistical-query algorithm and shows the pattern: a hypothesis defined by thresholds on a few probabilities, with enough slack between the thresholds that estimates suffice. None of this is machine-checked. The formalization fixes the noisy example law on the platform's sample framework and proves the exact identities on which the noise-tolerant simulation depends.
Equation (5.2) is a computation with the pushforward of a product measure: one must express the noisy law on as a mixture of the clean law and its label-flipped image, solve the affine relation for the clean probability, and combine with the restriction to , where the flipped and unflipped labels give the same value of ; the degenerate case , in which the conditional measure is zero and the first term vanishes, must be handled separately. The p. 117 identity is the same computation without the split. The §5.2 analysis is two union bounds over the literals after the observation that a literal of the target is never harmful and that a literal of the hypothesis is never insignificant. The product lemma is elementary arithmetic with a case split at .
The noisy oracle is a measure on labeled examples obtained by mapping the product of and a Bernoulli() coin; the conditional is Mathlib's conditional measure; queries are arbitrary measurable predicates of a labeled example, with no tolerance or query-count bookkeeping. Theorem 5.3 itself, the definitions of efficient learnability from statistical queries (Definition 14) and of efficient noisy PAC learnability (Definition 13), Theorem 5.1, Theorem 5.2 as a statement about an algorithm with oracle access, and Corollary 5.4 are not stated: they quantify over query algorithms and their running times, for which this series has no model; the mission carries their exact probabilistic content. The error-propagation analysis of §5.4.2–5.4.3 with tolerance and the guessing resolution is not stated beyond the product lemma, since the factor is not in and the book's constant does not account for it. Hypotheses: for the decomposition, for the disagreement identity, for the conjunction analysis, all reals in for the product lemma.
Trivializing readings are excluded: the decomposition is an exact identity for every measurable query, and the conjunction bound is for the exact thresholds with the union bound's . Welcome contributions: the mixture representation of the noisy law, the restriction of a pushforward to , and the two union bounds.
Chapter 4 of Kearns and Vazirani, An Introduction to Computational Learning Theory (MIT Press, 1994, doi:10.7551/mitpress/3897.001.0001), asks whether the PAC model's demand for arbitrarily small error and confidence is essential. A weak learning algorithm need only, with some fixed positive probability, output a hypothesis that beats random guessing by a fixed margin. Schapire's theorem, the chapter's main result, says that this apparently much weaker requirement is equivalent to the original one: any weak learner can be converted, by running it on carefully filtered distributions and combining its hypotheses by majority votes, into a strong learner. The construction is boosting, which became one of the most influential ideas in machine learning. The chapter proves the equivalence in two steps. Boosting the confidence is elementary: run the learner several times and validate. Boosting the accuracy is the substance: a modest procedure that combines three hypotheses, each with error at most on its own distribution, into a majority with error at most , applied recursively until the error is driven below the target. The Chernoff bounds of the Appendix, the book's workhorse for estimating probabilities from samples, are what makes the validation steps rigorous.
The framework is that of Mission I. A class is weakly learnable using if for some advantage , confidence and sample size , an algorithm outputs hypotheses in that, for every target in and every distribution, have error at most with probability at least ; the algorithm's prediction is a measurable function of the sample and the instance together, as it is for every algorithm. Given a hypothesis , the filtered distribution gives weight to the instances on which errs and to those on which it is correct, preserving relative weights within each part, and is conditioned on ; the modest procedure outputs . Ternary majority trees over are the closure of under the majority of three. For confidence boosting, independent samples yield hypotheses, and a fresh sample selects the one with the fewest mistakes. Bernoulli trials are independent coin flips with success probability .
If is weakly PAC learnable using measurable hypotheses in , then is PAC learnable using the class of ternary majority trees with leaves from : for all some sample size and some algorithm outputting majority trees achieve error at most with probability at least , for every target in and every distribution.
Theorem 9.2 (the additive and multiplicative Chernoff bounds); the two facts of §4.2 behind confidence boosting (independent runs all fail with probability at most ; the fewest-mistakes selection loses at most with probability at least ); Lemma 4.1 (the modest procedure: error at most ).
Theorem 4.9 is one of the landmark results of learning theory: it shows that the PAC model has no intermediate strength, that Occam learning, weak learning and strong learning coincide, and that the resources of a strong learner can be bounded polylogarithmically in in memory and hypothesis size. Its constructive proof is the first boosting algorithm, ancestor of AdaBoost and of gradient boosting. Lemma 4.1 is the analytic core, a clean inequality about three hypotheses and three distributions in which the filtered distribution is exactly calibrated so that has no advantage on it. The Chernoff bounds are the concentration inequalities invoked throughout the book, and their formalization on the product law of Bernoulli trials makes every later "estimate to within with confidence " step reusable. None of these is machine-checked in this form; the boosting theorem in the sample-complexity sense is, to our knowledge, not formalized anywhere.
Lemma 4.1 is a computation with conditional measures: writing of the majority as the weight of the instances on which and both err plus times the weight of their disagreement, mapping weights under back to by the factors and (Equation (4.1)), and maximizing the resulting polynomial in ; the degenerate cases where a conditioning event is null must be handled separately. The Chernoff bounds require the exponential moment method on a finite product measure. The confidence-boosting facts are the product bound for independent blocks and Hoeffding plus a union bound. The goal is a genuine construction: from a large sample of one must simulate the recursive algorithm Strong-Learn, whose calls to the weak learner on filtered distributions are served by rejection sampling from the remaining examples, bound the depth of the recursion by the growth of iterates (Lemma 4.2), bound the number of examples consumed at each node (Lemmas 4.3–4.7) and allocate the confidence over all the places the simulation can fail; then package the result as a deterministic function of a sample of fixed size. An alternative route is available: weak learnability with a fixed sample size forces a finite VC dimension (a class shattering a large set defeats any fixed-size learner on the uniform distribution over it), after which Theorem 3.3 gives a consistent strong learner; but its hypotheses lie in , not in the majority trees over , so it does not prove the stated conclusion.
The weak-learning hypothesis is the book's with constants in place of the inverse polynomials, which is what the definition says for a fixed class; hypotheses in are required to be measurable, and the weak learner jointly measurable in the sample and the instance, because Strong-Learn runs it on distributions filtered through its own earlier outputs and the analysis integrates over the earlier samples (for an arbitrary function the combined failure event need not be measurable, and outer-measure bounds on separate runs do not combine); the conclusion is the book's hypothesis class, the majority trees over , built as an inductive predicate. Filtered distributions use Mathlib's conditional measure, so that a null conditioning event yields the zero measure; Lemma 4.1 is stated for and holds in those degenerate cases too. The confidence-boosting milestone states the two probabilistic facts rather than the composite algorithm, whose sample indexing across runs and validation is bookkeeping; the selection rule is any rule minimizing mistakes. Chernoff's bounds are stated with non-strict inequalities in the events, for and . Running time, the recursion-depth and sample-size lemmas with unspecified constants (4.2–4.8), and Exercises 4.1–4.3 are not stated.
Trivializing readings are excluded: the weak-learning guarantee is uniform over all targets and distributions with an advantage strictly positive, the strong conclusion is for every , and Lemma 4.1 requires all three error bounds on their respective distributions. Welcome contributions: Lemma 4.1 itself, the Hoeffding bound on the product law, and the rejection-sampling lemma that turns a sample of into a sample of a filtered distribution.
The bandit processes that motivated the index theorem are sampling processes: an arm is a population from which one draws i.i.d. observations whose distribution has an unknown parameter, and each draw both earns something and teaches something. Chapter 7 of Gittins, Glazebrook and Weber, Multi-armed Bandit Allocation Indices (2nd ed., doi:10.1002/9780470980033), develops the theory of such processes in the Bayesian setting: the state of the process is the current posterior for the parameter, continuing it samples the next value from the predictive distribution and moves to the new posterior. When the observations are themselves the rewards one has a reward process, the classical Bayesian multi-armed bandit; when the aim is to find as quickly as possible an individual whose measurement reaches a target (a compound active enough to warrant further testing, in the drug-screening problem from which the index theorem came) one has a target process, which is a job that completes when the target is reached. Two questions organize the chapter. When can the index be written down without any optimization, and when do symmetries of the model reduce the index to a function of fewer variables? The first is answered by the notion of a favourable prior (Section 7.3): if no run of observations below the target can raise the current probability of success, then the index is that probability, exactly, by Proposition 2.7. The second is answered by the invariance theorems of Section 7.4: a location parameter with a conjugate prior gives , a scale parameter gives , and for target processes the target can be absorbed into the state, . These identities are what make the tables of Chapter 8 one-dimensional.
A sampling model consists of a likelihood , a family of priors on the parameter indexed by the parameters of a conjugate family, and the Bayes update of those parameters after observing ; the family is conjugate if the posterior of given is . The predictive distribution is . The reward process moves from to with and earns . The target process with target moves to the completion state if and to otherwise, earning the current probability of success , and in . A state is favourable if for every finite sequence of observations . For the invariance theorems the parameters are with the update ; is a location parameter of the likelihood if is shifted by , and is a location parameter of the prior family if is shifted by ; scale parameters are defined with , . The Gittins index is that of the Bandit Algorithms model on these chains.
If is a location parameter of a reward process with a conjugate prior family in which is a location parameter and the parameters update as the sample mean and count, then for every
under the standing assumptions that the observations have a mean and the discounted rewards of the chain are integrable.
Proposition 7.4 (favourable state: ); Example 7.5 (Bernoulli target process, ); Example 7.6 (normal target process with known variance, for ); Theorem 7.11 (scale parameter: ); Theorem 7.17 (target process with a location parameter: ).
Theorem 7.9 and its companions are the reason the Gittins index of the normal reward process is tabulated as a function of alone and that of the exponential process as a function of and one ratio; every computational method of Chapter 8 starts by reducing the state space with them. Proposition 7.4 is the source of every closed-form index in the book: it identifies the states in which sampling for information is worthless, so that the index collapses to the immediate expected reward, and Examples 7.5 and 7.6 show that for the Bernoulli target process this is every state and for the normal target process every state with a nonnegative posterior mean. The formalization gives the platform its first Bayesian sampling-process model, in which the state is a posterior and conjugacy is stated through the posterior kernel of the likelihood, and its first index identities on unbounded-reward chains, which is where the integrability assumptions of the Bandit Algorithms model do real work.
None of this is machine-checked. The invariance theorems are stated in the proper-prior form of the corollaries, with the model's symmetry as hypotheses, so that they apply to any conjugate family with the stated structure rather than to a particular density.
The invariance theorems require showing that the chain of parameters from the shifted (scaled) state is the image of the chain from the original state under the shift (scaling) of trajectories, which is an equivariance of the Ionescu–Tulcea construction with respect to a measurable bijection commuting with the kernel; that stopping times are carried to stopping times; that the discounted reward of a stopping time shifts by times the discounted time; and that the supremum of a nonempty bounded set of reals shifts and scales accordingly. Boundedness of the set of ratios is where the integrability assumption enters. Proposition 7.4 is the chain-level statement that all rewards along every trajectory from a favourable state are at most , which needs an induction on the trajectory law of the target chain, followed by the argument of Proposition 2.7. Example 7.6 needs the monotonicity of in the observations below the target, a small inequality, plus the Gaussian probability of a half-line as the current probability of success; Example 7.5 needs only that decreases.
The sampling model is a structure with Markov likelihood and prior kernels and a jointly measurable update; the predictive distribution is the kernel composition; conjugacy is an almost-everywhere identity between Mathlib's posterior of the likelihood with respect to the prior and the prior at the updated parameters, and is carried as a hypothesis of the invariance theorems and of Proposition 7.4 so that their subject is the Bayesian process. For the parameters it is required on only (IsConjugateOn): a proper prior has , and conjugacy at every is impossible with a location parameter, since at the update divides by zero and sends every observation to one state, which made the first draft's location theorems vacuous. The chains are built with Kernel.map of product kernels, so their measurability is structural, and the target process lives on P ⊕ Unit with the completion state absorbing. The book's improper priors are replaced by proper conjugate families with the location or scale structure of Corollaries 7.10 and 7.12, as those corollaries do; the discrete-time correction factor of Section 2.8 is not applied since it cancels in every identity stated. The two examples are built directly from a uniform or Gaussian seed with the transition probabilities the book computes (the beta and normal posterior computations of Exercise 7.1 are not formalized). Hypotheses: ; integrable observations and L&S Assumption 35.6 for the reward processes; for the invariance theorems and for the scale theorem; ; and for the normal example.
Trivializing readings are excluded: the indices are the genuine suprema of the Bandit Algorithms definition with integrable rewards, the update rule is the book's and not a free parameter, and the favourability condition ranges over all finite observation sequences. Welcome contributions: the equivariance of the trajectory measure under a state bijection commuting with the kernel, the transport of stopping times, and the reward bound along the target chain from a favourable state.
Every proof of the index theorem in Gittins, Glazebrook and Weber, Multi-armed Bandit Allocation Indices (2nd ed., doi:10.1002/9780470980033), uses the fact that a bandit not being processed is frozen. Chapter 6 drops that: Whittle's restless bandits evolve under the passive action too, by a different law, and of must be active at every time. The problem is PSPACE-hard in general, so Whittle proposed a heuristic built from a Lagrangian relaxation: replace the hard constraint by a subsidy paid whenever a bandit is passive, solve the resulting single-bandit average-reward problem, and read off, for each state, the least subsidy at which the passive action becomes optimal. When the set of states where passivity is optimal grows monotonically with , the bandit is indexable and is its Whittle index; the Whittle index policy activates the bandits of largest index. It reduces to the Gittins index policy when the passive action freezes, it is asymptotically optimal as grows under a fluid-stability condition (Weber and Weiss), and it has become the standard heuristic for sensor management, opportunistic channel access, maintenance and queueing control. The price is that indexability must be established model by model. Section 6.5 shows how easy this is when the single-bandit problem is solved by a monotone policy, on two bi-directional models: the spinning plates asset, which improves under investment and deteriorates when neglected, and the vigour bandit of Whittle's Ehrenfest project, which tires when worked and recovers when rested.
A restless bandit is a Markov decision process with two actions, active () and passive (), each with its own transition kernel and reward. Under a deterministic stationary Markov policy with passive subsidy the reward in state is , and the average reward from is the Cesàro limit of the expected rewards. The optimal average reward is the supremum over such policies and initial states; a policy is optimal if it attains from every initial state; is the set of states in which some optimal policy is passive; the bandit is indexable if is nondecreasing in ; and .
The spinning plates asset lives on : active moves at rate , passive moves at rate , , and is earned under both actions, increasing. Uniformized so that rates are at most one, it is a discrete-time bandit whose kernels move with the rate's probability and otherwise stay. The monotone policy is passive exactly on ; under it the asset alternates between and , spending the fraction of its time at , so its average reward is with , and . The vigour bandit is the mirror image: active moves down at rate and earns , passive moves up at rate and earns nothing, , and .
For the spinning plates asset: (i) if is strictly decreasing over the thresholds , the asset is indexable; (ii) if additionally is strictly decreasing over the states, the Whittle index is
Eqs. (6.9)–(6.10): the monotone policy earns from every initial state and , because a monotone policy always achieves ; Theorem 6.5, the same two statements for the vigour bandit with increasing and increasing.
Theorem 6.4 is the chapter's template for proving indexability: the single-bandit value is the upper envelope of finitely many lines whose slopes decrease in the threshold, so the optimal threshold moves monotonically with the subsidy and the hinge points of the envelope are the indices. The same argument gives Theorem 6.5, the admission-control indices of Section 6.7, and the marginal productivity indices of Niño-Mora; it is the reason Whittle indices are computable in closed form for bi-directional models. Its formalization establishes, on the platform, the first restless-bandit model with a proved index, and the general notions of passive set, indexability and Whittle index that every later restless-bandit statement will use.
None of this is machine-checked. The average-reward optimality notion is stated without the DP equation (6.6), through optimality from every initial state, which is what the equation's solution encodes on a finite state space and avoids the relative value function altogether.
The proof in the book is two paragraphs, but it stands on the reduction to monotone policies, which is only sketched: every deterministic stationary policy, from every initial state, drives the asset into an absorbing endpoint or a two-state cycle whose average reward is that of the monotone policy , so no policy beats the best monotone one and the passive set under an optimal-from-everywhere policy is exactly for the smallest maximizing threshold. Formalizing this needs the average reward of a finite Markov chain as a limit determined by the stationary distribution of the recurrent class reached, for the two-point kernels of the model, and a case analysis of policies as -strings. The envelope argument then needs that the smallest maximizer of is nonincreasing in when is strictly decreasing, and that with strictly decreasing the maximizer is exactly when . Theorem 6.5 is the same with the roles of up and down exchanged. Nothing in Mathlib computes Cesàro limits of finite Markov chains.
Restless bandits are the two-action DecisionProcesses of the superprocess module; average reward is a real limsup of Cesàro means of Bochner integrals over the chain law of the Bandit Algorithms model under the stationary kernel; the optimal average reward is a supremum over the finite type of deterministic stationary Markov policies and the finite state space, bounded by the reward bound. Both models are on Fin k with the book's states shifted down by one, kernels driftKernel p f that move to f x with probability p x, and the boundary conventions of and (the book's "convenient positive values") replaced by their values and at the two extreme thresholds; the model assumptions , , rates in , and increasing and nonnegative are hypotheses. Theorem 6.5's "increasing" is read as strictly increasing, as in Theorem 6.4, since a nonstrict admits zero interior rates for which the monotone reduction fails. The milestone (6.9) requires and positive interior rates, which Theorem 6.4's hypothesis (i) implies.
Trivializing readings are excluded: indexability is monotonicity of the passive set over all real subsidies, the passive set is defined through policies optimal from every initial state, and the index identity is for every state. Welcome contributions: the average reward of a two-state cycle, the reduction of an arbitrary -policy to a monotone one, and the envelope lemma for lines with decreasing slopes.
Chapter 5 of Gittins, Glazebrook and Weber, Multi-armed Bandit Allocation Indices (2nd ed., doi:10.1002/9780470980033), presents the achievable region methodology of Tsoucas, Bertsimas and Niño-Mora, Glazebrook and Garbe, and Dacre, Glazebrook and Niño-Mora: instead of arguing about policies, one argues about the set of performance vectors they can produce. For a multi-armed bandit the natural performance of a policy is the vector of discounted numbers of times each state is continued; the expected return is linear in it; and the set of achievable performances turns out to be a polytope cut out by conservation laws, one inequality per subset of states, with equality exactly for the priority policies that put that subset last. Optimizing a linear objective over a polytope is a linear program, its dual is solved by an adaptive greedy algorithm, and the primal solution is the performance of a priority policy whose priorities are the algorithm's outputs, the Gittins indices. This gives yet another proof of the index theorem (Section 5.3) and, more importantly, a definition, generalized conservation laws (Section 5.4), of the class of systems for which the same argument works: branching bandits, multi-class queues, job scheduling with discounted rewards, systems with imposed priority classes. The chapter's main result, Theorem 5.5, is the statement that every such system is solved by an index policy.
There are job types . A policy has a performance , a vector of expectations; a permutation of defines the permutation policy giving highest and lowest priority, and is the set of the lowest-priority types. The system satisfies GCL(1) if there are a base function and a matrix , positive on and zero off it, such that for every policy
with equality in the first for every permutation policy whose lowest-priority types are . GCL(2) reverses the inequality. The adaptive greedy algorithm picks maximizing , sets to the maximum, removes , and repeats with the adjusted rewards divided by ; its outputs are the order , the dual variables and the indices .
For the SFABP of Section 5.3, identical bandit processes on with kernel and discount factor in the model of the Bandit Algorithms series, is the discounted number of continuations of a bandit in state , is the discounted return time to from , and is the minimal cost , namely with the number of continuations needed to bring every bandit into .
For a GCL(1) system whose achievable region is convex, and any reward vector : the achievable region is the polytope
its extreme points are performances of permutation policies; has an output; and for every output the permutation policy in the order it finds, the Gittins index policy, maximizes over all policies.
Lemma 5.1 (the SFABP satisfies the conservation laws, with equality for policies giving priority to states outside ); the identification on p. 123 of the adaptive greedy indices of a SFABP with the Gittins indices, together with their monotonicity along the order found; Theorem 5.10, the GCL(2) counterpart of the goal for cost minimization.
Theorem 5.5 is the index theorem in its most general form of this kind: it says nothing about Markov chains, only that performances are expectations, objectives are linear and conservation laws hold, and it delivers both the optimal policy and the algorithm that computes its priorities in polynomial time in the number of job types. It is the theorem behind the index results for branching bandits and Klimov's multi-class queue and behind the suboptimality bounds of Sections 5.5 and 5.7, all of which are calculations on the polytope. Lemma 5.1 and the p. 123 identification are what tie the abstract theorem to the Gittins index: they show that the multi-armed bandit is a GCL(1) system and that the priorities the algorithm produces are the same indices as Chapters 2 to 4 define through stopping times.
None of these is machine-checked. Formalizing Theorem 5.5 puts an LP-duality index theorem on the platform in a form any system can instantiate by verifying its conservation laws; formalizing Lemma 5.1 relates the Bandit Algorithms run law to the single-chain return times, which is the first conservation law on that model; and the p. 123 theorem gives an algorithmic characterization of the Gittins index on finite chains, distinct from the restart and largest-remaining-index characterizations of Chapter 2.
The goal's optimality clause is weak LP duality once one shows that the greedy dual variables are nonpositive except and satisfy the dual constraints with equality, which is a finite induction on the stages; the extreme-point clause needs that every vertex of a polyhedron is the unique maximizer of some linear functional, and the region clause that a compact convex set is the convex hull of its extreme points (Krein–Milman in finite dimension, or the polyhedral fact directly). None of this is in Mathlib in the required form. Lemma 5.1 is probabilistic: the lower bound requires the strong Markov property of the continued bandit under an arbitrary past-measurable policy, a pathwise accounting of the discounted periods paid for by each continuation from , and the observation that at most slots can be spent on bandits that have never been in ; the equality for priority policies requires that these policies use exactly those slots first and then tile the future with return excursions, and the product form of requires independence of the bandits' process-time trajectories under the run law, which is built decision time by decision time rather than as a product. The p. 123 theorem is the computation (5.13) to (5.14) combined with the optimal-stopping characterization of Chapter 2 for the stop sets , which lie between and ; ties make the induction delicate, and the statement is claimed for every tie-breaking.
GCL(1) and GCL(2) systems are structures over an arbitrary policy type: performance, base function, matrix, permutation policies and the three laws are fields, so the theorems are statements about finite-dimensional data and the platform's proof needs no probability. The adaptive greedy algorithm is specified relationally, as the set of its possible outputs with arbitrary tie-breaking, and the conclusion holds for each of them; existence of an output is asserted separately. The optimality clause is stated as a comparison with every policy rather than as a real supremum. The hypothesis that the achievable region is convex is explicit: the book's argument from extreme points to the whole polytope uses randomization of policies, and without it the region of a system with only its permutation policies is finite. The SFABP items use identical bandits on Fin N in the Bandit Algorithms model, the coefficients through Mission I's stoppedTime at the return time, and in the product form , which is the minimal cost the argument on p. 120 establishes; the book prints a sum, which is when all bandits start in where the minimal cost is . Discount factors are in throughout.
Trivializing readings are excluded: for is part of the structure and of Lemma 5.1's conclusion, the polytope equations are over all subsets, and the index clause quantifies over every greedy output. Welcome contributions: the nonpositivity and dual feasibility of the greedy variables, the vertex-exposure lemma for polyhedra, and the product decomposition of the run law of identical bandits.
A recommendation system learns from the users it serves: the diner who tries a restaurant produces the review the next diner reads. Each user would rather exploit what is already known than explore for the benefit of those who come later, so a population of self-interested agents under-explores, and an alternative that looks bad on sparse early evidence may never be tried again even when it is the best. Chapter 11 of Slivkins, Introduction to Multi-Armed Bandits (arXiv:1904.07272), treats incentivized exploration: a principal who cannot force the agents but can recommend, and who, because it aggregates what earlier agents observed, knows more than any one of them. The question is whether recommendations alone can induce enough exploration to learn as fast as an ordinary bandit algorithm. The model is that of Kremer, Mansour and Perry (JPE 2014) and the results are those of Mansour, Slivkins and Syrgkanis (EC 2015, Operations Research 2020), specialized to two arms; the single-round problem is Bayesian persuasion in the sense of Kamenica and Gentzkow (AER 2011).
There are arms and rounds. A mean reward vector is drawn from a known prior , and each pull of arm yields a reward drawn from a known family with mean . In round the principal recommends an arm ; agent , who knows the prior, the family, the algorithm and the round but not the past, sees only , chooses , collects and leaves; the principal observes . The chapter works with two arms, ordered so that the prior means satisfy , with a prior of finite support and finitely many reward values.
An algorithm is Bayesian incentive-compatible (BIC, Definition 11.4) if following its recommendation is in every agent's interest given what the agent knows: for every round and arms with ,
where is the event that all previous agents complied. A BIC algorithm is then an ordinary bandit algorithm whose recommendations are followed, and the run has the law of the Bayesian bandit of Chapter 3. Two contrasting policies frame the chapter. GREEDY reveals the history and lets agents exploit, (11.2); it is BIC and it fails. HiddenExploration (Algorithm 11.1) hides a little exploration in a lot of exploitation: on a signal , with probability it recommends a target arm , otherwise the arm maximizing , ties to arm 1. Its posterior gap is . RepeatedHE (Algorithm 11.2) runs it round after round with an arbitrary bandit algorithm as the target: initial rounds recommend arm 1; afterwards, with probability the round is an exploration round in which chooses (and is fed the reward), and otherwise the exploitation branch recommends , where is the data of all exploration rounds so far (11.10). The quantity that governs everything is (11.11), the posterior gap after samples of arm 1, and Property (11.12), that for some : arm 2 can appear better after enough samples of arm 1.
RepeatedHE with exploration probability and initial samples of arm 1 is BIC as long as
for any bandit algorithm and any horizon. The threshold depends on the prior alone.
Theorem 11.7 (GREEDY never chooses arm 2 with probability at least ) and Corollary 11.8 (linear Bayesian regret of GREEDY under independent priors); Lemma 11.10 (HiddenExploration is BIC when ) with Claim 11.12 (the arm-2 side of the constraint suffices); Corollary 11.14 (RepeatedHE is BIC under the round-by-round condition); Theorem 11.19 (without Property (11.12) no BIC algorithm ever plays arm 2, ties to arm 1).
The results say when exploration can be incentivized at all and how. Theorem 11.7 shows that revealing everything is not a solution: the greedy dynamics gets stuck on arm 1 with a probability that does not shrink with , and Corollary 11.8 turns that into Bayesian regret. Theorem 11.15 shows that a recommendation-only principal can induce any amount of exploration it wants, with arbitrary, at a per-round rate fixed by the prior; Theorem 11.17 (stated with a proof sketch, and omitted here) then transfers 's regret to RepeatedHE up to the prior-dependent factors and , so regret is attainable subject to incentives. Theorem 11.19 closes the picture: Property (11.12) is necessary as well as sufficient. Together they characterize which priors admit incentivized exploration and give an algorithm that works for all of them.
Nothing of this is machine-checked. The mission adds to the Bayesian layer of mission III (prior, posterior by Bayes' rule, Bayesian regret) the BIC constraint on a joint law, GREEDY as a policy, the single-round HiddenExploration on an abstract finite signal, and the law of RepeatedHE; all of it is reusable for the -arm and the "explore all explorable arms" extensions of the literature review.
Theorem 11.7 is a martingale argument: the posterior gap along the history is a Doob martingale, the first round in which arm 2 is chosen is a bounded stopping time, and optional stopping gives ; all of this has to be set up on the joint law of of mission III, where the posterior is defined by Bayes' rule and the identification with a conditional expectation is itself a theorem (posterior_eq_condProb). Lemma 11.10 is the heart of the chapter and is not a computation about : it works with , splits along the two branches, uses that the exploitation branch recommends arm 2 exactly when , and closes with ; the only place where the analysis uses that both branches are functions of the signal is the step , and a formalization has to make that step explicit. Theorem 11.15 requires seeing each later round of RepeatedHE as a HiddenExploration with signal , where 's choice is a randomized function of , and then the monotonicity of in , a two-line consequence of determining that presupposes the posterior given is the Bayes posterior of the exploration data alone, which is true because the exploration decisions do not depend on given that data. Corollary 11.8 needs the independence of the event " and arm 2 is never chosen" from . Theorem 11.19 is an induction in which the inductive hypothesis is a probability-zero statement about all earlier rounds.
Arms are Fin 2, the book's arm 1 being index 0; rounds are Fin T. The prior is a probability measure on mean vectors supported on a finite set , with as a hypothesis; the reward family is mission III's RewardFamily (finitely many values, mean for ). BIC is defined on a joint law of of the run in which every agent complies, with the recommendation of each round read off the record; the compliance event of (11.1) is the sure event of that law, which is the standard reading of "the agents believe all previous agents complied". For a bandit policy the law is mission III's jointMeasure. Conditional expectations are written as finite sums over , so there are no integrals and no integrability side conditions; a posterior mean off the support is a junk that never enters a theorem. GREEDY allows arbitrary tie-breaking; HiddenExploration's exploitation branch breaks ties toward arm 1 as Algorithm 11.1 does; the tie convention of Theorem 11.19 is the strict form of BIC for arm 2. The law of RepeatedHE is an explicit finitely supported measure, and record weighted by the prior times the product of the round probabilities (initial rounds forced to arm 1, then the -coin, 's kernel on its own history, or the exploitation arm, then ); it is written this way because is fed a history of variable length. Two conditions are stated exactly as printed: Lemma 11.10 with (non-strict, checked at equality) and Theorem 11.15 with the strict inequality.
Trivializations are excluded: throughout; the BIC condition is asserted only where the recommendation has positive probability, and the sums in it are over the finite support, so an unsatisfiable hypothesis cannot hide in a measure-zero set. Welcome contributions: the optional-stopping argument on jointMeasure, the identification of explPostMean with the conditional expectation given the exploration data, the Bayes-rule algebra behind Lemma 11.10, and the counting lemmas on heRecords.
Airlines selling last seats, Priceline's name-your-own-price, procurement of supply contracts: Chapter 6 of Talluri and van Ryzin's The Theory and Practice of Revenue Management (2004) treats auctions as pricing mechanisms and asks what revenue they earn and how to design them. Its centre is Myerson's (1981) theory for independent private values: whatever the mechanism, so long as bidders with higher valuations are more likely to win and the lowest type gains nothing, the firm's expected revenue is the expected virtual value of the winners, with (Theorem 6.1, the revenue equivalence theorem). Maximizing that expression pointwise gives the optimal auction: the standard first- or second-price auction with a reserve price at the zero of (Theorem 6.2). This mission formalizes the second-price form of Theorem 6.2 as its goal, with the dominant-strategy and first-price equilibria of the informal analysis, Theorem 6.1, the optimal allocation and Proposition 6.1 on list prices as supporting results.
customers have i.i.d. valuations on with a continuously differentiable,
strictly increasing distribution and positive density (PrivateValues, IsRegular); the
joint law is the product measure (joint). A direct-revelation mechanism (Mechanism) maps
reported valuations to allocations , at most units in total, and
payments . For a report by customer , is the win probability,
the expected payment and the surplus (winProb, expPayment,
expSurplus); incentive compatibility, , is the equilibrium
condition of the direct mechanism (IsIncentiveCompatible). The chapter's mechanisms are the
-unit second-price auction with reserve price (secondPriceReserve: the highest
valuations above win and pay the larger of and the highest losing valuation), the
list-price mechanism for (listPrice), and the single-unit first-price auction with
its equilibrium bid , (firstPriceBid).
With strictly increasing (Assumption 7.2) and its zero, the -unit second-price
auction with reserve price is a feasible, incentive-compatible mechanism with monotone
allocations and zero surplus at zero, and its expected revenue is at least that of every such
mechanism: reserve_price_auction_optimal.
Bidding one's valuation is dominant in the second-price auction (Sect. 6.2.2.1); the bid (6.4) solves the first-order condition (6.3), is a symmetric equilibrium of the first-price auction and shades below the valuation (Sect. 6.2.2.2); Theorem 6.1, revenue equals expected virtual surplus and each expected payment is ; the pointwise optimal allocation of Sect. 6.2.5; and Proposition 6.1, a list price at is optimal when .
Proposition 6.2 (asymptotic optimality of list prices, a law-of-large-numbers statement about scaled auctions), the first-price form of Theorem 6.2 with its equilibrium (6.9) stated without proof, and the dynamic, replenishment and network auctions of Sects. 6.3-6.5 (Propositions 6.3-6.11, from Vulcano, van Ryzin and Maglaras and from Cooper and Menich) are not targets of this mission.
Theorem 6.1 is the tool that lets revenue be computed from allocations alone, which is why the first- and second-price auctions of Examples 6.1-6.3 earn the same and why any dynamic pricing scheme that ends with the same winners earns the same as the optimal auction (Sect. 6.2.6.3). Theorem 6.2 says a firm with private-value customers cannot do better than a standard auction with the right reserve price, and Proposition 6.1 that with enough capacity a list price already does it: auctions are a small-numbers phenomenon. These are the foundations on which the chapter's dynamic auctions and the list-price comparisons of Sects. 6.3-6.4 rest, and Myerson's optimal auction has no machine-checked proof in its multi-unit form.
Theorem 6.1 is an envelope argument in measure-theoretic clothing: incentive compatibility gives the two-sided inequalities of Appendix 6.A, monotonicity of makes convex with derivative almost everywhere, so , and then an integration by parts against the density converts into ; the win probabilities are integrals over a product measure with one coordinate replaced, and Fubini is needed to return to . The goal then needs the reserve-price auction shown incentive compatible (a dominant-strategy argument on the threshold payment), measurable, monotone and with zero surplus at zero, and the pointwise optimal allocation integrated. The first-price item is calculus on an interval integral with a vanishing denominator at and a monotone comparative-statics argument for the equilibrium inequality.
Mechanisms are direct-revelation mechanisms on , as the book reduces to in Sect. 6.2.3.1; expectations over the other customers are integrals over the joint law with customer 's coordinate overwritten by the report. Payments are assumed bounded on reports in (not on all of , where the second-price payment is unbounded) and the rules measurable. Ties in the second-price auction are broken by index, a null event, and when every customer wins the losing supremum is so the winner pays the reserve. Theorem 6.2 is stated for the second-price auction; the first-price version with reserve price, whose equilibrium (6.9) the book asserts without proof, is left out and noted. Optimality is over mechanisms satisfying conditions (i) and (ii) of Theorem 6.1 and incentive compatibility, which is the class the book compares against. The virtual value's zero is a parameter with rather than the maximum of (6.8), which under strict monotonicity is the same point.
Every airline, hotel and car-rental firm sells more reservations than it has capacity, because some customers cancel or do not show. Chapter 4 of Talluri and van Ryzin's The Theory and Practice of Revenue Management (2004) is the theory of that decision. Its static models pick one overbooking limit from the show distribution; its dynamic model, a simplification of Chatwin's (1998), follows reservations, cancellations and refunds period by period and proves that the optimal control is still a limit, one that declines toward the deadline and falls when more demand is expected; and its substitutable-capacity model, from Karaesmen and van Ryzin (2004), sets joint limits for several classes whose oversold customers can be moved between resources, showing the expected net revenue is concave in each limit and submodular across them. This mission formalizes the chapter's four propositions and the corollary the book draws from the last.
Dynamic overbooking (Sect. 4.3.1). With reservations on hand in period , new
requests arrive; the firm books up to at revenue each, and every
reservation survives the period with probability , a cancellation refunding . At the
deadline the firm pays the convex denied-service cost on reservations beyond
capacity , Eq. (4.11). The recursion is
with
and
(value,
postValue). The greatest optimal overbooking limit (overbookingLimit) is the
largest level at which is at least its value at every smaller level, an
element of ; the limit policy books
(limitPolicy).
Substitutable capacity (Sect. 4.5). Classes hold reservations and
are overbooked to levels ; in the service period
customers show and are assigned to resources of capacities , or to the
virtual resource (denied service), at net benefit , by the transportation problem
(TP) with value (serviceValue). The expected net revenue (4.21) is
(expNetRevenue),
and jointLimit is the greatest optimal limit of one class with the others fixed.
With Poisson show demands, has decreasing first differences in every direction:
for all and all classes , which
is component-wise concavity () and submodularity ():
joint_overbooking_concave_submodular.
Proposition 4.1, with a convex denied-service cost the limit policy with the greatest optimal limit attains the maximum of the recursion at every state; Proposition 4.2, under the greatest optimal limits decline with time; Proposition 4.3, stochastically larger demand to come gives limits that are no larger; and the corollary of Sect. 4.5.2, the greatest optimal limit of class is nonincreasing in the level of any other class.
The static overbooking models of Sect. 4.2 (binomial, normal and Gram-Charlier approximations, Type 1 and Type 2 service levels), the net-bookings heuristics of Sect. 4.3.2, the combined capacity-control models of Sect. 4.4 and the stochastic-gradient algorithm of Appendix 4.A carry no numbered results and are not targets.
Proposition 4.4 is the structural fact that makes joint overbooking of related resources tractable: concavity gives each class a critical booking level and submodularity makes those levels move in opposite directions, so a stochastic-gradient or coordinate search on the limits is well behaved, and the pattern of Example 4.5, overbooking an early flight aggressively because its oversold passengers can be moved to later ones, is a consequence rather than a heuristic. The dynamic propositions are the theoretical support for the overbooking curves that reservation systems post, limits that fall as departure approaches, and they quantify the sense in which a static model, which ignores future demand, overbooks too much. The proof of Proposition 4.4 passes through the discrete concavity of the transportation problem's value in its supply vector, an M-natural-concavity fact in the sense of Murota, and the Poisson-expectation identity for second differences; none of this has a machine-checked proof.
The dynamic model needs the concavity of on to be propagated through two
operations, the binomial thinning and the windowed
maximum , both of which preserve discrete concavity
but require explicit manipulation of binomial sums and of the argmax; Propositions 4.2 and 4.3
then compare greatest maximizers of concave sequences through lower bounds on marginal values,
with the value handled in ℕ∞. The substitutable-capacity goal is harder: the value of
(TP) as a function of the integer supply vector must be shown to have decreasing differences,
which is the submodularity of a max-weight transportation value in its supplies, a linear
programming duality argument (or Murota's M-natural-concavity of min-cost flow), and the
Poisson expectation of it, a tsum over , must be differenced in two coordinates
using the identity for Poisson pmfs.
The linear terms of cancel in second differences and the refund term is linear in .
Periods are natural numbers with value t the value with periods to go, and the
book's ranges are hypotheses. The denied-service cost is normalized,
and , as a cost "penalizing denied service" is. Convexity of the sequence alone is not
enough, because (4.11) never reads . Demands are pmfs on and cancellations
exact binomial sums. The greatest optimal limit lives in because a
mild denied-service cost can make accepting every request optimal, in which case the book's
critical value is ; the limit policy then accepts everything. Proposition 4.3 is stated
for two demand families ordered by first-order stochastic dominance rather than a parametrized
family. In the substitutable-capacity model the virtual resource is uncapacitated, the book's
"finite but very high" taken as infinite so that (TP) is feasible for every Poisson
realization, and (TP) is over real assignments, whose optimum at integer supplies is integral.
Eq. (4.21) is printed with ; being the maximum net benefit, the
expected net revenue adds it, and the definition uses , without which Proposition 4.4 fails
numerically on every sampled instance.
Chapter 10 of Axsäter's Inventory Control turns to reorder points and safety stocks in multi-echelon systems, where the installations cannot be treated separately: a large stock downstream lets an upstream site run lean, and a long upstream lead-time argues for stock at the top. The best-known exact technique for serial systems is the decomposition of Clark and Scarf (1960), which the book presents in the infinite-horizon form of Federgruen and Zipkin (1984). It is also where the echelon stock measure comes from. The section's argument is short and self-contained, and its conclusion is a complete description of the optimal policy for a two-level serial system: order-up-to levels at both installations, one of them a newsboy solution, the other the minimizer of a convex function in which upstream shortages appear as an induced cost. It is the capstone of Chapter 10.
Installation 1 faces normally distributed period demand with mean and standard deviation , independent across periods, so the demand over periods, , is normal with mean and standard deviation . Installation 1 replenishes from installation 2 with lead-time periods; installation 2 replenishes from an outside supplier with infinite supply and lead-time . Demand that cannot be met is backordered. Costs per unit and period are echelon holding costs , so the installation holding costs are and , and a shortage cost at installation 1; there are no ordering costs. Events in a period occur in the order: installation 2 orders, its delivery arrives, installation 1 orders, its delivery arrives, demand, cost evaluation.
Consider an arbitrary period . After ordering, installation 2 has an echelon inventory position , and by the standard argument its echelon stock in period is . Installation 1 then orders, realizing an echelon position that cannot exceed what is available: (Eq. 10.1). Its inventory level after the demand in period is . The expected period costs are at installation 2 and at installation 1, and the book reallocates the term to obtain
with and . As a function of a free , is the newsboy-type function of Eq. (10.6), minimized at the level given by the fractile equation (10.8). Passing everything available up to to installation 1, , gives the total cost of Eq. (10.9), whose minimizer is the order-up-to level of installation 2.
With from (10.8) and a minimizer of : for every and every allocation rule with and finite expected cost,
and the order-up-to policy attains .
Eq. (10.3), the stage-1 period cost through the expected backorders; the reallocation (10.4)-(10.5), which leaves the total unchanged; the closed form (10.6) of through the loss function ; the convexity of , its derivative (10.7), and the fractile characterization (10.8) of its minimizers; the pointwise rule that is the cheapest feasible ; the identity (10.9); and the convexity of (Problem 10.1) with the existence of its minimizer when .
The result itself. The decomposition reduces a two-dimensional stochastic control problem to two one-dimensional convex problems solved in sequence, from downstream to upstream, and it identifies the optimal policy class. The downstream level is a newsboy solution with overage cost , the value added, and underage cost , and it is independent of the upstream installation altogether; the upstream level sees the downstream installation only through the induced shortage cost, the last term of (10.9). The book notes the extensions the argument admits, to more echelons, to batch ordering at the top, and, via Rosling's equivalence, to assembly systems, and its Sect. 10.1.2 adapts it, now only approximately, to distribution systems under the balance assumption. Example 10.1 shows the typical outcome: the optimal average stock at the upstream installation is slightly negative.
Formalizing it. The section's mathematics is a chain of expectations under Gaussian laws and two convexity arguments. Formalizing it fixes what "optimal" means, a per-period comparison against every allocation rule, and separates the two convexity claims the book makes in one clause each. Nothing here is open; no statement has a machine-checked proof yet.
The pointwise allocation rule and the newsboy fractile are the same arguments as in the newsboy mission. The two places where work is needed are the identity (10.9), an expectation of a piecewise function split at , and the convexity of , which requires seeing that is convex precisely because is a minimizer of the convex (for any other cut-off the function is not convex), and that convexity is preserved by integrating against the law of , which needs the integrability of the linearly growing . Existence of then follows from the growth of at both ends, which comes from the asymptotics of the loss function: as and as .
is csDemand mu sigma n, the Gaussian law newsboyDemand (n μ) (√n σ) from the newsboy
mission, so the loss function and its closed form are reused as references. The costs are
parametrized by with and written out; ,
, the pre-reallocation period cost and are Bochner integrals against these
laws. Every statement assumes ; the goal and the convexity statements assume
and , the book's cost signs. is allowed and makes a
point mass, which is the setting of the book's Problem 10.2.
enters as any solution of the fractile equation (10.8) and as any minimizer of ; the other items show that both exist when . When the fractile is , no exists, and the goal is vacuous, which is faithful: the book observes that then and installation 2 never carries stock. Symmetrically, when and , decreases towards its infimum without attaining it, so no exists and the goal is again vacuous: with free upstream holding the optimal is unbounded. Allocation rules are arbitrary functions of the realized with an integrability hypothesis; without it Lean's integral of a non-integrable cost would be and could undercut , which is negative in Example 10.1's stage-1 term.
What is not modelled is the infinite-horizon dynamic problem: the book's optimality claim is made period by period, and the passage to the stationary policy rests on the remark that the outside supplier has infinite supply, so the same can be chosen in every period. The definitions are reusable for the three-echelon extension and for the distribution system of Sect. 10.1.2; contributions formalizing Problem 10.2 () as a first step are welcome.
A firm that holds inventory in five warehouses needs more safety stock than one that holds the same inventory in one warehouse, because the demands of five regions do not all run high at once. Eppen (1979) made this precise for a multi-location newsvendor and gave it its name, the risk-pooling effect. Chapter 7 of Snyder and Shen's Fundamentals of Supply Chain Theory (2019) follows the same idea through three settings in which pooling happens without physical consolidation: two retailers who ship stock to each other after seeing demand (transshipments, after Tagaras 1989), and plants that can each make more than one product (process flexibility, after Jordan and Graves 1995). The chapter's capstone is the theorem of Simchi-Levi and Wei (2012) that, among designs in which every plant makes two products and every product is made at two plants, a single long chain through all of them is best. This mission formalizes the chapter's numbered results, with that theorem as its goal.
Risk pooling. distribution centers face normally distributed per-period demands
with correlation coefficients , and each runs a
base-stock policy with holding cost and backorder cost per unit per period, so its
optimal expected cost is the optimal newsvendor cost optNvCost h p D, the infimum over
base-stock levels of . Merging the centers gives one
facing the total demand, normal with mean and variance
(pooledVariance).
Transshipments. Two retailers with base-stock levels face independent
demands. After demand is observed, under complete pooling the retailer with a surplus sends
the retailer with a shortage units (transship), and
nothing moves otherwise. The type-1 service level is the probability of no stockout,
without and with
transshipments; the type-2 service level is the fill rate, one minus expected unmet demand
over expected demand, and likewise.
Process flexibility. A flexibility design on products and plants is a set
of (product, plant) pairs, an edge meaning plant can make product . Given a
demand realization and a common plant capacity , the performance (perf)
is the maximum sales obtainable by assigning production along the edges of without
exceeding any capacity or demand, the linear program (7.22) to (7.26). A balanced system
(BalancedSystem) has equal capacities and an exchangeable demand vector, one whose joint
law is invariant under permutations of the products, and is the
expected performance (expPerf). The named designs are the dedicated design ,
the long chain in which plant also makes product (and plant makes
product ), the open chain obtained from by deleting the edge , and
, the open chain on the first pairs together with the dedicated edges of the rest. A
2-flexibility design (TwoFlex) is one in which every product has exactly two plants and
every plant exactly two products; is one, and so is any union of disjoint shorter chains.
For a balanced system of size with exchangeable demand,
that is, is a 2-flexibility design and for every 2-flexibility design .
This is long_chain_optimal.
The chapter's route to the goal: Lemma 7.5, supermodularity of sales in the flexible edges of the long chain for every realization, for ; Corollary 7.6, the same in expectation; Lemma 7.7, the increments are nondecreasing in , ending with ; and Lemma 7.8, .
Risk pooling, Theorem 7.1: , the optimal cost of the merged center is at most the sum of the optimal costs of the separate ones, with the covariance inequality as a separate lemma.
Transshipments, Theorems 7.2 to 7.4: , , and all four post-transshipment service levels are nondecreasing in .
Theorem 7.9 is the analytical answer to a question that had been settled only by simulation: Jordan and Graves reported that one chain through all plants achieves nearly twice the sales benefit of three short chains with the same number of edges, and Simchi-Levi and Wei proved that no arrangement of the same edge budget does better. It is the justification for the chaining guideline used in automotive and semiconductor capacity planning, and Lemma 7.8, which expresses the long chain through open chains, is what makes the long chain's performance computable by a greedy pass. Theorem 7.1 is the quantitative basis for consolidation decisions and for postponement, since a generic product is pooled inventory. Theorems 7.2 to 7.4 quantify what transshipments buy in service, which is the argument for allowing them despite their cost.
None of these results has a machine-checked proof. The book proves Lemma 7.7, Lemma 7.8 and Theorem 7.9 in full given Lemma 7.5, which it cites to Simchi-Levi and Wei, and omits the proofs of Theorems 7.3 and 7.4 and the identity (7.30) behind Lemma 7.8. Formalizing Lemma 7.5 and (7.30) means formalizing the structure of maximum flows on a cycle, which is reusable for the later results of Simchi-Levi and Wei on the long chain's performance relative to full flexibility and for the multi-echelon flexibility models the chapter cites.
The obvious approach to Theorem 7.9 is to compare with an arbitrary 2-flexibility design directly. Nothing in the definitions supports that: the two designs share no structure beyond their degree sequences. The book's argument instead routes everything through the long chain's own edges. Lemma 7.5 gives supermodularity only for subsets of , and the decomposition of an arbitrary 2-flexibility design into disjoint cycles, each a relabeled long chain on a subsystem, is what allows the comparison. A solver must therefore prove that a 2-regular bipartite graph is a disjoint union of even cycles, that exchangeability makes every relabeling of a cycle worth the same as on its subsystem, and that the performance of a disjoint union is the sum of the performances of its parts.
Lemma 7.5 itself is where the combinatorics lives. It says that on the cycle the maximum flow is supermodular in the flexible edges, and the proof in Simchi-Levi and Wei goes through the structure of augmenting paths on a cycle. The natural first idea, that supermodularity follows from some general property of maximum flows, is false: maximum flow is not supermodular in arbitrary edge sets, and the lemma is specific to subsets of a single cycle.
Lemma 7.7 is where exchangeability is used, and it is used in a way that is easy to state and tedious to formalize: removing the edge from leaves a design that is only after the pair is moved to the end, so the argument needs the invariance of under relabeling the products and plants by a common permutation. The book notes that Lemma 7.7, unlike Lemma 7.5, is false realization by realization.
For the transshipment theorems, the book differentiates a density formula by Leibniz's rule. Under the weaker hypothesis stated here, laws without atoms and with finite means, the derivative of in has to be obtained by dominated convergence from the pointwise derivative of a piecewise-linear function whose kinks lie on null sets.
perf is a supremum over a set of reals, nonempty because is feasible when
and , and bounded by ; the demand is nonnegative for every outcome and the
capacity nonnegative in BalancedSystem, and Lemma 7.5 carries these as hypotheses. The
supremum is attained, but the definition does not assert it. Expected performance is a Lebesgue
integral; the demand is integrable by assumption and is -Lipschitz in , so the
integrand is integrable, and a solver must prove this measurability rather than assume it.
Exchangeability is the equality of the laws of and for every
permutation . Designs are finite sets of pairs of Fin n; the chains are defined with
finRotate, so indices wrap modulo and the closing edge of is in the book's
numbering, which is the edge its proofs and Figure 7.3(c) use. Lemma 7.8 involves open chains on
subsystems of sizes and ; these are designs on Fin k evaluated on the first
coordinates of the demand (subDemand, subPerf).
Theorem 7.1 states the optimal costs as infima of the newsvendor cost over all base-stock
levels, on Mathlib's gaussianReal; a nonpositive pooled variance gives a degenerate law, for
which the inequality still holds, so the statement is not trivialized by that convention. The
transshipment theorems take the two demand laws as probability measures on with no
atoms (Theorem 7.2) and finite, positive means; the quantity is defined for all
outcomes and the service levels are probabilities and expectations under the product law.
The definition module is shared by all eleven items. Beyond the milestones, formalizing the
identity (7.30) as its own lemma and the disjoint-union additivity of perf would be natural
contributions.
Procter & Gamble observed in the 1990s that the orders its distributors placed for diapers were far more variable than the retail sales of diapers, and that its own orders to suppliers were more variable still, although the end demand for diapers is about as stable as demand gets. The phenomenon, a growing amplification of variability as one moves upstream in a supply chain, is the bullwhip effect. Lee, Padmanabhan and Whang (1997) argued that it is not a symptom of irrational behaviour: four rational responses of an inventory manager to their own environment each produce it. Chapter 13 of Snyder and Shen's Fundamentals of Supply Chain Theory (2019) makes three of the four quantitative, following Chen, Drezner, Ryan and Simchi-Levi (2000) for demand signal processing, Lee et al. for the rationing game, and Cachon (1999) for order batching. This mission formalizes those three models and the theorems the chapter proves about them.
Demand signal processing. A retailer faces a demand process , , that follows the stationary first-order autoregressive model
with a constant , a correlation constant , and errors that are independent variables, each independent of the demands before period . In steady state every has the law . The retailer replenishes with a lead time of periods under a base-stock policy but does not know the demand parameters, so it estimates the lead-time demand from a moving average of the previous demands:
and sets the base-stock level , where
is a safety factor. The book writes the constant in as and does
not give its form; here it is a free parameter . Each period the retailer orders
, which may be negative. In Lean the process is the structure
AR1Demand, whose fields are the parameters, the errors, the demands, the recursion, the
independence properties and the stationary law; muHat, err, sigmaHat, baseStock and
order are the five quantities above.
Order batching. retailers face independent demands in every period and
each orders once every periods, the order being its demand over the previous
periods. The supplier's order in a given period is the total ordered by the retailers whose
ordering day falls in that period. Three patterns are compared: random ordering, in which each
retailer's day is uniform over the days, so the number of retailers ordering on a given
day is binomial; positively correlated ordering, in which all retailers order on the
same day, so with probability and otherwise; and balanced ordering, in which
the retailers are spread as evenly as possible, so with , , is
with probability and otherwise. The structure BatchOrders P N R mu sigma carries the
demands, the ordering count independent of them, and supplierOrder, the sum of the last
demands of retailers ; each pattern enters a theorem as a hypothesis on the law of .
Rationing game. Two identical retailers face single-period demand with distribution function , holding cost and stockout penalty , so the newsvendor quantity satisfies . With probability the supplier can deliver only units in total and allocates them pro rata to the orders, retailer 1 receiving ; with probability supply is unlimited. Retailer 1's expected cost when the retailers order and is
with the newsvendor cost; this is rationingCost.
with equality when . This is bullwhip_signal_processing. The bound exceeds
whenever , whatever the value of : a lead time and a moving-average forecast are
enough to produce the effect.
The chapter's own route to the goal, each a milestone: the steady-state moments (13.2) to (13.4), , and ; the identity ; Lemma 13.1, for ; the vanishing of the cross term (13.12); and the variance of the demand part, .
Order batching, Theorem 13.4: under the three patterns the supplier's order has mean and
through the three variance formulas , and .
The rationing game, Theorem 13.3: if is a symmetric Nash equilibrium, that is, minimizes over positive order quantities when the other retailer orders , then .
The three theorems are the quantitative core of the chapter. Theorem 13.2 is the single-stage building block that Theorems 13.6 and 13.7 later iterate along a serial chain, giving the product-form and the exponential lower bounds on the amplification at stage ; its comparative statics, the bound decreasing in and increasing in , are the basis of the remedies the chapter recommends (shorter lead times, smoother forecasts, sharing point-of-sale data). Theorem 13.4 ranks the ordering patterns and justifies the advice to balance ordering days when batching cannot be avoided. Theorem 13.3 shows that pro-rata rationing alone inflates orders; the book is careful to note that inflated orders are not by themselves inflated variances, and that the variance statement for this model is due to Rong, Shen and Snyder (2017).
None of these results has a machine-checked proof. The book's proofs of Theorems 13.2 and 13.4 are complete but informal, and the proof of Lemma 13.1 is omitted with a citation to Ryan's 1997 thesis; formalizing it requires a self-contained argument. The variance decomposition of and the conditioning argument for Theorem 13.4 are reusable for the multistage results of Sect. 13.2.5, which are natural follow-up missions on the same definitions.
The obvious computation of expands the order into its demand part and its safety-stock part and hopes the cross term disappears. It does, but not for a reason visible in the formulas: is a square root of a sum of squares of forecast errors, a nonlinear function of demands, and its covariance with a single demand is zero only because the errors are jointly Gaussian with mean zero and is an even function of them, so the covariance is the expectation of an odd function of a centred Gaussian vector. That is Lemma 13.1, and the vanishing of the cross term needs two further covariances, and , which the book reduces to the lemma through the recursion (the second reduction divides by ) but which hold for every by the same symmetry. A solver must set up the joint Gaussian structure of the demand vector and prove the odd-function argument; nothing in Mathlib does this directly.
The second obstacle is that the moments (13.2) to (13.4) are not assumed but derived: the structure carries the stationary law of each and the independence of from the past, and the autocovariance has to be obtained from the recursion by induction on the lag, with integrability supplied by the Gaussian laws.
For Theorem 13.4 the work is the conditioning on : given the supplier's order is a sum of independent normals, so its conditional mean is and conditional variance , and the total variance is . The order is defined by a sum over retailers , so the independence of from the demands has to be used through the indicator structure rather than through a conditional-expectation library result.
For Theorem 13.3 the argument is a first-order condition. It requires that the newsvendor cost be differentiable with derivative , which holds when is continuous, and that the symmetric equilibrium be an interior minimizer, which is why and the minimization over are hypotheses.
Time is indexed by so that exists for every . AR1Demand asserts the
recursion for every outcome, the independence of the whole error family, the independence of
from , and the stationary law of every ; these are the
"steady-state" assumptions the book makes in words. The structure is satisfiable: the stationary
Gaussian AR(1) process on a full-measure set of error sequences has all these properties. The
constant is a free real parameter ; no theorem depends on its value.
The goal divides by , which is under the structure's
hypotheses and , so the ratio is a genuine quotient. Mathlib's
ProbabilityTheory.variance and covariance are used; both are the ordinary real quantities
for square-integrable variables, which every variable here is, included.
In BatchOrders the demands are indexed by Fin N × Fin R, the count is a natural-valued
random variable bounded by and independent of the demand family, and supplierOrder sums the
demands of retailers , the book's "without loss of generality" choice. The laws
of are hypotheses on point probabilities P.real {ω | X ω = j}; with each of the
three families of hypotheses is satisfiable by a structure with the corresponding law. The
subtractions and are real.
In the rationing game the demand law is a probability measure on whose distribution function is continuous and strictly increasing on ; the newsvendor loss is assumed integrable at every order quantity. The pro-rata allocation uses Lean's total division, which is never at in the theorem since .
Beyond the ten milestones, the multistage Theorems 13.6 and 13.7 and the centralized-information
bound of Theorem 13.5 are welcome as extensions on the same AR1Demand.
When a system is controlled indefinitely and decisions are frequent — a router admitting packets, a queue accepting jobs, a machine being maintained — discounting future rewards is often unjustified, and what matters is the long-run average reward per period. Puterman's Chapter 8 (doi:10.1002/9780470316887) develops the theory of this criterion, and its central object is a single equation, the average reward optimality equation , whose unknowns are a scalar gain and a bias function . For unichain models, in which every stationary policy generates a Markov chain with one recurrent class, this equation determines the optimal gain and an optimal stationary policy. The results go back to Howard (Dynamic Programming and Markov Processes, MIT Press, 1960) for the recurrent case and to Blackwell (Discrete dynamic programming, Annals of Mathematical Statistics 33, 1962, doi:10.1214/aoms/1177704593) and Derman for the general finite case; Puterman's Section 8.4 proves them through the discounted theory of mission II, by letting the discount factor tend to one.
The model is stationary (Assumption 8.0.1): a finite set of states, for each a finite nonempty set of actions, a reward and transition probabilities , none depending on the decision epoch. A policy may randomize and may depend on the whole history; the deterministic stationary policy applies the decision rule at every epoch. Its transition matrix is .
For a policy , is the expected reward over epochs. Since the limit of need not exist (Example 8.1.1), the chapter works with the lim sup and lim inf average rewards and , and with . A policy is average optimal when for all and , the strongest of the three criteria of Section 8.1.2.
The optimality residual is , and the optimality equation is . A decision rule is -improving when it attains at every state. A transition matrix is unichain when it consists of a single recurrent class plus a possibly empty set of transient states, and the MDP is unichain when is unichain for every deterministic decision rule.
For a finite unichain model: (a) some deterministic stationary policy is average optimal; (b) the optimality equation has a solution, and (d) its scalar satisfies for every ; (c) for every solution, every -improving decision rule gives an average optimal stationary policy.
If then ; if then ; if then .
In a finite unichain model has a solution, and every solution has the same .
If and is -improving, then is average optimal.
Theorem 8.4.1(c) is what the source calls "one of the most important results for average reward models": a solution of the optimality equation with constant pins down the optimal gain under every criterion at once, so that in finite unichain models the three optimality criteria of Section 8.1.2 coincide. Theorem 8.4.3 guarantees such a solution exists, and Theorem 8.4.4 reads an optimal policy off it. Together, Theorem 8.4.5 reduces the infinite-horizon average reward problem over all history-dependent randomized policies to a finite system of equations in , which is what policy iteration, value iteration and linear programming solve in Sections 8.5 to 8.8.
The results are classical and proved. Formalizing them fixes the chain-structure hypothesis in
a checkable form and pins down which criterion "average optimal" means, two places where the
literature is loose. The platform's MarkovDecisionProcesses series has the finite-horizon
(mission I) and discounted (mission II) models; this mission adds the undiscounted stationary
model, the gains, and the unichain classification, on which Chapter 9's multichain optimality
equations and Chapter 10's sensitive discount optimality can be built.
The obvious argument for Theorem 8.4.3 is to take the discounted optimal value of mission II and let . It fails as stated because blows up like ; what converges is the Laurent expansion of the value of a fixed stationary policy, Corollary 8.2.4, and that expansion needs the limiting matrix and the deviation matrix of a unichain chain. So the proof must first develop the Markov chain theory of Section 8.2 and Appendix A, choose a subsequence of discount factors along which one policy is discount optimal (possible because is finite), and only then pass to the limit in the discounted optimality equation.
Theorem 8.4.1 looks elementary and hides the analytic step: iterating along an arbitrary history-dependent policy and dividing by requires the telescoping term to vanish, which uses boundedness of , and requires the reduction from history-dependent randomized to Markov randomized policies (Theorem 8.1.2). For Theorem 8.4.4 the step is Corollary 8.2.7, that forces the gain of to be , which is the multiplication by that annihilates .
The traps are in the definitions. Recurrence and the unichain property must be stated so that the source's Example 8.4.3 comes out as the book says — the policy using has the absorbing state as its single recurrent class — and the optimality residual must use the lim sup / lim inf gains, since a definition through a limit that need not exist would be a junk value on the policies of Example 8.1.1.
State and action spaces are Fintypes and admissible actions are nonempty Finsets, as in
missions I and II; the stationary model is a new structure because mission II's DiscountedMDP
bundles a discount factor, and carries the same data otherwise. Policies are history-dependent
and randomized, so "average optimal" has its full strength; a stationary policy is the
deterministic one built from a decision rule. Expected total reward is defined by the policy
evaluation recursion, as in the earlier missions, rather than through a measure on
trajectories.
Gains are Filter.limsup and Filter.liminf of on ; these are
the source's because the sequence is bounded by , and the suprema over the
nonempty family of policies are genuine real suprema for the same reason. The residual
is a Finset.sup' over the admissible actions. Recurrence is "every state reachable from
reaches " and unichain is "any two recurrent states communicate", the definitions of Appendix
A for finite chains, applied to for every admissible deterministic decision rule.
Restrictions relative to the printed text, all noted in the items: Theorem 8.4.1 is stated for finite where the source says countable, since the chapter's standing assumption and the model are finite; the gain of a stationary policy in (8.4.5) is written as its lim inf gain, which equals it; and the chain of (8.4.6) is stated through and , since presupposes existing limits. Nothing is trivialized: the existential in Theorem 8.4.5(a) has to produce a decision rule, and with a junk maximum is impossible since every is nonempty. Welcome contributions beyond the milestones: Theorem 8.1.2 (reduction to Markov policies), Corollary 8.2.7 (the gain of a stationary policy from the evaluation equations), and the equivalence of the three optimality criteria in finite models.
Introductory physics is normally taught as a sequence of unrelated chapters — kinematics, dynamics, energy, optics, gravitation — each with its own rules. A recurring proposal in physics education is to teach instead from a single organising statement: among all conceivable histories of a system, the one realised in nature is the one that makes a certain integral, the action, stationary. Edwin Taylor's editorial A Call to Action (American Journal of Physics, 2003) argued for building the first-year curriculum on it; Lachlan McGinness and Craig Savage reported classroom results for such a course at the Australian National University (Action physics, American Journal of Physics, 2016); Massimiliano Malgieri tested a sum-over-paths treatment in an Italian secondary school (2017).
The source monograph for this mission, Julliana Rodrigues Martins, A Ação Estacionária como Eixo Unificador do Ensino de Física no Ensino Médio (Trabalho de Conclusão de Curso, Universidade Federal do Ceará, 2025), develops this programme for secondary education. Its Chapters 2 and 3 are quantitative: they follow the historical line from antiquity to the nineteenth century and carry out every calculation in full. This mission formalises that quantitative core, and nothing of the pedagogical Chapter 4.
The historical line the monograph reconstructs, and which the milestone list follows:
Fix real endpoints and an integrand , a real-valued function of three real arguments. A path is a function , and its action over is
Neighbouring paths are produced by an admissible variation: a twice continuously differentiable with , giving the family . The path is stationary when
Writing and for the partial derivatives of in its first and second slots, the Euler–Lagrange expression along is
In mechanics one takes , and , so that stationarity of is Hamilton's principle and is the Lagrange equation of motion.
For , twice continuously differentiable in all three arguments and twice continuously differentiable, if is stationary for then
This is equation (3.9) of the monograph, and — through the substitution , , — equation (3.17). It is stated as the weakest stable form: stationarity, not minimality, and an arbitrary admissible integrand rather than a particular Lagrangian.
The supporting statements are the analytic ingredients of that derivation (the fundamental lemma, the first variation formula, the first integral for a cyclic coordinate), its two mechanical applications worked out in the monograph (harmonic oscillator, plane pendulum), its geometric application (shortest path), and the historical minimisation problems of Chapter 2 (Heron, Galileo, Fermat, Maupertuis).
The Euler–Lagrange equation is the bridge the whole unification argument rests on: once it is available, Newton's second law, the pendulum equation, Snell's law, geodesics and conservation laws all become consequences of one statement about an integral, which is exactly the claim the monograph makes to justify teaching physics this way. The Chapter 2 statements are the historically prior special cases, each obtained by a one-variable minimisation rather than by the general machinery, and together they exhibit how far elementary optimisation alone reaches before the calculus of variations is needed.
What this mission adds beyond the source is a machine-checked version of an argument that, in every textbook presentation including this one, is carried out at the level of rigour of formal manipulation: the interchange of differentiation and integration is performed without justification, the passage from a vanishing integral to a vanishing integrand is asserted, and the regularity needed for to exist is left implicit. All of it is true under the stated hypotheses; none of it is proved in the source. Mathlib has the analytic infrastructure (interval integrals, differentiation under the integral sign, smooth bump functions) but, at the environment pinned for this mission, no Euler–Lagrange equation and no calculus-of-variations layer built on it. The definitions published here — action, admissible variation, stationary path, Euler–Lagrange expression — are reusable by any later variational mission.
The obvious argument is three lines: differentiate under the integral sign, integrate by parts, and conclude that the integrand vanishes because is arbitrary. Each line is where the work is.
Differentiating under the integral sign needs a dominating bound valid uniformly for near ; it is available here because the data are and the interval is compact, but it has to be produced. The integration by parts needs to be differentiable, which is where the second derivative of and the second derivatives of are consumed — a path is not enough for this formulation. The final step needs test functions: a bump supported in a small interval around a point where the continuous coefficient is nonzero, which is why the fundamental lemma is a separate milestone rather than a step.
A standing trap in the Lean formulation is that deriv returns at points where a function is not differentiable, so a statement about can be accidentally true for the wrong reason unless the regularity hypotheses are genuinely strong enough. Every statement here carries the hypotheses that make each derivative a real derivative.
The development is one-dimensional and real: paths are , the integrand is a curried with argument order matching the source's , and all integrals are interval integrals over . Partial derivatives are one-dimensional derivatives in the frozen remaining arguments; smoothness of is stated for its uncurried form on . Regularity is throughout — for the integrand, for the path, and for the variations — matching the monograph's requirement that have continuous first and second derivatives.
Stationarity is a hypothesis quantified over all admissible variations, so the goal cannot be satisfied by exhibiting one convenient ; and the conclusion is an equation at every point of the closed interval, not almost everywhere, so it cannot be weakened to a null-set statement. The mechanical milestones are stated as equivalences between the Euler–Lagrange equation for the explicit Lagrangian and the classical equation of motion, which rules out a one-directional reading that would be vacuous for a path that never satisfies either.
Physical constants (, , , , the speeds ) are free real parameters, with positivity or non-vanishing assumed only where the source's conclusion requires it — the pendulum equivalence divides by , so and appear, while the oscillator equivalence needs no such assumption. Angles never appear as primitive objects in the optics milestones: the sines of the incidence and refraction angles are written as the ratios that the figures define them by, so no convention about angle ranges is smuggled in.
Contributions welcome: proofs of any milestone, and reusable infrastructure for differentiation under the interval integral sign and for bump functions with prescribed support, both of which are of use well beyond this mission.
A fusion reactor only makes sense if the plasma gives back more energy than it loses. In 1955 J. D. Lawson turned that sentence into an inequality: for a hot, thermalized fuel, the product of the particle density and the energy confinement time must exceed a temperature-dependent threshold before the fusion products alone can keep the plasma hot. The criterion was published in 1957 (Lawson, Some Criteria for a Power Producing Thermonuclear Reactor, Proc. Phys. Soc. B 70 (1957) 6–10) and has been the standard figure of merit for magnetic and inertial confinement ever since: machines are compared by their and by the triple product .
Timeline, as far as this mission's material traces it:
Consider a steady-state, fully ionized plasma of deuterium and tritium. All quantities are real numbers and strictly positive, and the temperature is measured in energy units (electronvolts), so Boltzmann's constant does not appear: the article's is our .
The plasma is self-heating when fusion heating exceeds the losses:
The goal theorem is the Lawson criterion itself — equation (1) of the source — obtained by substituting the model quantities into the self-heating condition:
The milestones are the surrounding statements of the same derivation, in the order a reader meets them:
the triple-product form and its numerical D-T value,
the inertial-confinement form obtained from ,
and the empirical tokamak scaling: if and , then , independent of the density.
The result itself. The criterion is what makes different confinement concepts comparable at all: it converts "does this machine work?" into a single number that can be measured and plotted, and its triple-product form is the quantity reported by tokamaks and by inertial-confinement shots alike. The last milestone explains why the triple product, rather than , is the quantity of record for tokamaks: under the empirical energy-confinement scaling it does not depend on the density and depends on temperature only as .
Formalizing it. Nothing here is an open problem: every statement is a known consequence of the stated model, and the mathematical content is algebra and monotonicity over the positive reals. What the mission produces is a machine-checked, reusable encoding of the energy-balance model of a D-T plasma, with every degenerate case (zero density, zero confinement time, vanishing reactivity) explicitly excluded, and with the chain from the model assumptions to each published form of the criterion verified end to end. The value is in the faithfulness of the model layer, not in the difficulty of the deductions.
The deductive content is elementary; the difficulty is representational. Three places where a careless encoding silently changes the theorem:
Real.rpow, whose rules require positive bases, which is why the positivity hypotheses appear again there.The model is a Lean structure LawsonCriterion.DTPlasma bundling , , , , , , all of type ℝ, with a strict-positivity field for each and one defining equation, tauE = energyDensity n T / Ploss where energyDensity n T = 3 * n * T. The 50-50 reaction rate is DTPlasma.rate, the heating rate is DTPlasma.fusionHeating = rate * Ech, and DTPlasma.SelfHeating is the proposition fusionHeating ≥ Ploss. Conventions this commits to:
T is ; the numerical milestone additionally fixes keV for T and Ech, m³/s for sigmav, and so keV·s·m⁻³ for the triple product;Ploss is;Real.rpow for the fractional exponents, Real.sqrt for the thermal speed.The statements are not vacuous: the definition file also publishes a witness, LawsonCriterion.unitDTPlasma, so the structure is inhabited and every hypothesis set here is satisfiable. Conversely, no statement is trivially true — each is an inequality between quantities that genuinely vary with the plasma parameters.
Contributions welcome beyond the listed milestones: a version that separates radiation from conduction losses, the D-D and D-³He variants, and the statement that attains a minimum for a realistic reactivity model — the step where the source passes from the inequality to its "absolute lower limit", which this mission does not formalize.
Hubble's law — officially the Hubble–Lemaître law — is the observation that galaxies recede from us at a speed proportional to their distance, . It is the first observational basis for the expansion of the universe and one of the standard pieces of evidence cited for the Big Bang model. Georges Lemaître derived the proportionality from relativistic cosmology in 1927; Edwin Hubble published the observational relation in 1929, building on Vesto Slipher's redshifts and Henrietta Swan Leavitt's Cepheid distance scale. Alexander Friedmann had already shown in 1922 that the Einstein field equations admit expanding solutions, with an expansion rate governed by what is now called the scale factor.
Behind the astronomy sits a piece of elementary mathematics that is rarely written out in full: in a spatially homogeneous expansion, the linear velocity–distance relation is forced, it holds with the same constant for every observer, and the Hubble "constant" is in fact a function of time whose evolution is fixed by a single dimensionless parameter. This mission formalizes that kinematic core: the statements are about the scale factor and its derivatives, and involve no Einstein equations.
A scale factor is a function of cosmic time . A comoving point is labelled by a fixed coordinate (with the Euclidean norm), and its proper position at time is
The proper distance between the comoving points and is
The Hubble parameter and the dimensionless deceleration parameter are
denotes the present-day value of ; the name "Hubble constant" refers to the fact that is constant in space at a fixed time, not in time.
The goal theorem is the idealized Hubble law, stated in the source as a theorem of Euclidean geometry: any two points moving away from the origin, each along a straight line and with speed proportional to its distance from the origin, move away from each other with a speed proportional to their distance apart. Formally, for , a time and curves with
the separation satisfies
The relative velocity is parallel to the separation vector and its magnitude is times the separation, with the same for every pair — so no comoving observer occupies a distinguished centre of the expansion.
The milestones supply the cosmological content that surrounds this geometric fact:
The result itself is elementary but load-bearing: it is what licenses reading a linear redshift–distance diagram as evidence for uniform expansion rather than for a privileged position in space, and items 4–6 are the statements through which cosmological observations (the sign of , the approach of to in CDM) are turned into claims about the past and future behaviour of and .
Formalizing it produces a small, reusable Lean layer for expansion kinematics: the scale factor, the Hubble and deceleration parameters, proper position and proper distance, with the differentiation lemmas that connect them. The platform already carries Friedmann-equation missions that fix the dynamics of ; this mission is the kinematic complement, and its Hubble parameter is the same function that those developments use. Nothing here is an open research problem: every statement is a known textbook fact, and the work is the formalization.
The mathematical content is a few lines of calculus, so the difficulty is entirely in the encoding. Three places are easy to get wrong. First, proper distance is defined through a norm, so the identity needs positivity of , and differentiating it needs positivity on a neighbourhood, not just at the point. Second, is defined by a quotient with in the denominator: in Lean division by zero returns zero, so a statement about that forgets silently changes meaning. Third, milestone 5 propagates a hypothesis stated on down to the endpoint , where the Big Bang condition lives; the continuity argument at the endpoint is the only step with any technical content.
Time is and space is EuclideanSpace ℝ (Fin 3); derivatives are Mathlib's
deriv / HasDerivAt, so "velocity" is always a derivative at a point rather than a
difference quotient. Smoothness is assumed exactly where it is used: differentiability at a
single time for the first-order statements, ContDiff ℝ 2 for the statements involving
. Positivity of the scale factor is stated explicitly wherever it is needed, as is
in the statements mentioning .
Degenerate readings are excluded: the goal's hypotheses are satisfiable (any pair of comoving points in an expanding universe satisfies them, as milestone 2 shows), and the milestones are non-vacuous for concrete scale factors such as and . The goal theorem's second clause is a norm identity that is true for every ; it carries the "speed proportional to distance" half of the source's statement, while the first clause carries the "moving away from each other along the separation" half.
The definition layer is a single self-contained file (scale factor derived quantities, proper position, proper distance); it is reusable by any mission about expansion kinematics. Contributions of the milestone proofs, and of variants such as the redshift relation derived from null geodesics rather than assumed, are welcome.
Battery-powered and energy-harvesting wireless devices cannot treat transmission power as free: a control algorithm that maximizes throughput without regard to energy will drain a device long before the network's other resources are exhausted. Chapter 6 of Georgiadis, Neely & Tassiulas's survey Resource Allocation and Cross-Layer Control in Wireless Networks (Foundations and Trends in Networking, 2006) extends the drift-plus-penalty framework of Chapter 5 to problems with an explicit average-resource-budget constraint, and works out the Energy Constrained Control Algorithm (ECCA) of Neely [116] as its flagship example: a joint flow-control and power-allocation policy that keeps every data queue and a virtual "excess energy" queue bounded by an explicit, finite constant on every single time slot — not merely in expectation or in the long run.
A multi-user wireless downlink serves users over channels with a time-varying collective topology state and link rate function under power allocation , subject to a per-slot power budget and a target average power constraint . Exogenous arrivals are entirely admitted or entirely dropped each slot (no transport-layer storage). The ECCA algorithm runs, every slot: flow control — admit into queue if (a fixed parameter), otherwise drop it entirely; power allocation — choose to maximize subject to the power budget, where is a virtual power queue tracking accumulated excess energy expenditure, updated by .
where is the constant in the rate function's marginal-benefit inequality ( being with its -th entry zeroed). These bounds hold for every topology state process and every admissible arrival process , on every sample path — the weakest, most general level, with no distributional assumption whatsoever on either process.
Theorem 6.3 gives a hard, deterministic worst-case guarantee — every queue in the system, actual or virtual, is bounded by an explicit closed-form constant on every single slot, not merely on average or asymptotically — which is exactly the kind of guarantee a resource-constrained embedded or battery-powered device needs: a device can be provisioned with buffer and battery-reserve capacity of the theorem's own , and know, with certainty rather than in expectation, that it will never overflow. The corollary that no -slot interval spends more than in energy translates directly into a battery-lifetime guarantee. This is also, methodologically, the one point in the whole book's drift-plus-penalty method where an elementary induction — not a probabilistic Lyapunov drift argument — suffices, because ECCA's flow control rule directly caps regardless of what the channel or arrivals do.
Formalizing it. No result in this mission has a machine-checked proof anywhere; nothing adjacent exists on the platform (searched for virtual queue, energy-constrained control, power allocation — no faithful hits; one unrelated coincidental keyword match in a neural-coding combinatorics item was checked and is not relevant). This mission is the first formalization of the ECCA performance guarantee.
The obvious first idea is to bound and by unrolling their recursions and applying a probabilistic drift argument, as in every other capstone in this series. This is unnecessary and would in fact obscure the actual mechanism: follows from a two-case induction that needs no probability at all — if , the flow control rule admits at most more, giving ; if , flow control drops everything, so by the induction hypothesis. The harder part is : it requires connecting the virtual queue's own accumulation to the fact that the power-allocation optimization (6.14) will stop spending power on link once grows past — a consequence of 's optimality for (6.14) together with the -inequality on , not a property one can read off the -recursion alone.
The power-allocation rule is stated as an explicit optimality hypothesis (hPopt): for every
competing power vector respecting the budget, the objective (6.14) at the chosen is at
least as large — the faithful rendering of " is chosen to maximize (6.14)" without needing
Lean's argmax/IsMaxOn machinery. The rate function's -inequality is stated exactly as
the book gives it, using Function.update P i 0 for . Out of scope for this mission:
the throughput conclusion under i.i.d. (an expectation/liminf statement, unlike the
rest of this theorem, needing the same conditional-expectation machinery as the other chapters'
capstones) and Theorem 6.2 (the general GCLC framework this specializes, which needs Assumptions
1-4's four simultaneous existential clauses over an "S-only" policy) — both left for a future
mission. A trivializing formalization to rule out: weakening the two sample-path conclusions to a
limsup/expectation-style bound (the convention used elsewhere in this series) would misstate
this theorem, whose entire distinguishing content is that the bounds hold for every time slot
and every sample path, not merely in the long run.
Chapters 3-4 of Georgiadis, Neely & Tassiulas's survey Resource Allocation and Cross-Layer Control in Wireless Networks (Foundations and Trends in Networking, 2006) answer "can the network stay stable?" — the backpressure algorithm's throughput optimality (Theorem 4.5) says yes, for any arrival rate inside the capacity region. But networks are usually operated with plenty of unused capacity precisely so that they can also serve a second goal: maximizing some concave utility (throughput, fairness, revenue) of the rates actually delivered. Answering "how close to utility-optimal can a stable policy get, and at what congestion cost?" needs a single theorem that treats stability and utility optimization in one drift analysis. Theorem 5.4, adapted from Neely, Modiano & Li [108], Georgiadis, Neely & Tassiulas [115] and Neely [116], is that theorem, and it is the technique — "Lyapunov optimization" or "drift-plus-penalty" — behind a large fraction of the cross-layer control literature that followed this book.
A network of queues has backlog vector , and a -dimensional control process influencing the system's dynamics (e.g. admitted data rates). For any nonnegative function of the backlog vector, the one-step Lyapunov drift is , the expected one-slot change in conditioned on the current backlog. Given any scalar-valued concave utility function of a -dimensional rate vector and an arbitrary target value , the goal is to stabilize while making the time-average utility of close to . The time-average rate vector is (5.18), and the achieved long-run utility is .
The most abstract drift lemma in the whole book: are arbitrary scalar processes, not necessarily linear or even a function of .
This is the weakest, most general level — a drift-minus-utility condition, checked against an arbitrary target , with no reference to any specific control policy.
Theorem 5.4 is the abstract engine that every concrete utility-maximizing algorithm in the rest of the chapter (the CLC1 joint flow-control/routing/scheduling algorithm of Theorem 5.1, and its robustness variant under approximate scheduling, Corollary 5.2) instantiates by exhibiting a control rule whose per-slot decision satisfies (5.19) for some — the drift condition does the work of turning a per-slot optimization rule into a global performance guarantee. Its tradeoff (utility gap shrinks like , congestion grows like ) is the quantitative signature of every "Lyapunov optimization"/"drift-plus-penalty" algorithm in the cross-layer control literature that followed this book, making Theorem 5.4 the single result a reader needs to understand that entire family of algorithms at once, independent of which specific control problem it is applied to.
Formalizing it. No result in this mission has a machine-checked proof anywhere; nothing adjacent exists on the platform (searched for Lyapunov optimization, drift-plus-penalty, utility fairness, concave utility — no hits at all, not even an adjacent object). This mission is the first formalization of the book's utility-optimization engine.
The central subtlety, flagged explicitly by the book's own notation, is that (5.20) and (5.21) compare two different kinds of quantity that are easy to conflate: is a limsup of an expectation of utility, , while in (5.21) is the utility evaluated at the time-averaged expected rate. Jensen's inequality (using 's concavity) is exactly what would relate to — and it is not proved or needed by this theorem's own statement, only by later results that connect the two more tightly. A formalization that silently merges these into one "utility" quantity would be proving something the book does not claim. The second trap is the target : it is a free parameter of the hypothesis, not the true optimal utility — the theorem's strength is exactly that it says nothing special about how was chosen, so a formalization must not add a hidden side condition forcing to equal a genuine optimum.
∆(U(t)) is restated locally in this chunk's own sub-namespace (drafts cannot import chunk
04-backpressure's copy), via Mathlib's condExp on the σ-algebra generated by the current
backlog vector, exactly as in Chapter 4. Explicit Measurable/Integrable guards on U, R,
g∘R and every drift term prevent condExp/∫ from silently defaulting to 0 (a trivializing
formalization this mission rules out: without these guards, a non-integrable process would satisfy
the hypotheses vacuously and "prove" a stability/utility conclusion for a process that is not
actually controlled). r(t) and \bar g are written inline as their defining Cesàro averages
rather than as separate named definitions, since each is used exactly once. Out of scope for
this mission: Theorem 5.1 (the CLC1 algorithm's concrete instantiation), Corollary 5.2 (the
robustness-to-suboptimal-scheduling variant), and Lemma 5.5 (continuity of near-optimal solutions,
which needs the capacity region Λ as a hypothesis object). All three need substantially more
setup (a named algorithm, or the capacity-region machinery of Chapter 3) than this session's time
budget allowed without approximating either — left for a future mission rather than approximated.
Every stability result in Georgiadis, Neely & Tassiulas's survey Resource Allocation and Cross-Layer Control in Wireless Networks (Foundations and Trends in Networking, 2006) — the throughput optimality of the dynamic backpressure algorithm (Theorem 4.5), the utility-optimal Lyapunov drift bound of Chapter 5, the energy-constrained control algorithm of Chapter 6 — is proved the same way: exhibit a drift condition on a quadratic Lyapunov function of the queue backlog vector, then invoke one fixed abstract lemma to conclude stability and an explicit congestion bound. That fixed lemma, Lyapunov drift stability (Lemma 4.1, adapted from Kingman [67], Loynes [87] and Neely, Modiano & Li [110]), is the engine this mission formalizes — the single result the rest of the book's algorithm-specific theorems are instances of.
A network of queues has backlog vector process on slots , on a common probability space . The quadratic Lyapunov function is . A single queue's backlog sequence (read as ) is strongly stable if ; a network is strongly stable if every one of its queues is. The conditional drift of given the current backlog vector, , measures how the aggregate squared backlog is expected to change over one slot, conditioned on where the network currently is — formalized via Mathlib's conditional expectation on the -algebra generated by .
This is the weakest, most general level: a bounded-outside-a-compact-region drift condition implies both a qualitative stability conclusion and a quantitative congestion bound, with no mention of any particular algorithm, arrival distribution, or network topology.
Lyapunov drift stability is the one lemma every later result in this book's method reduces to: Chapter 4's own Theorem 4.5 (backpressure throughput optimality) applies it directly to the dynamic backpressure algorithm's per-slot optimality property; the book explicitly notes ("This drift inequality is in the exact form for application of the Lyapunov drift lemma... proving the result", p. 57) that once a drift bound of this shape is established, the stability conclusion is free. The T-slot version (Lemma 4.2) extends the reach of the method to settings — Markov-modulated channels, non-i.i.d. arrivals — where no single slot need have negative drift.
Formalizing it. No result in this mission has a machine-checked proof anywhere; nothing adjacent exists on the platform (searched for Lyapunov drift, max-weight, differential backlog, backpressure — the one Lyapunov-drift hit, a Foster-Lyapunov hitting-time bound for a finite-state MDP with an absorbing target state, is a different mathematical object from a vector queue backlog with no absorbing state, and is not reused). This mission is the first formalization of the book's stability engine.
The obvious first idea is to bound directly via the one-step queueing recursion and telescope. This fails for the same reason it fails in Chapter 3: expectation does not commute with . Lemma 4.1's actual content is a genuine Foster–Lyapunov drift argument on the aggregate quadratic quantity , not a per-queue linear one — the quadratic form is what turns a one-sided drift condition into a bound on itself (via the elementary inequality of Lemma 4.3, which converts the linear queueing recursion into a quadratic one that the drift condition directly controls). A second trap is conditioning: the drift bound is conditioned on the current backlog vector , not on the full history — a weaker, memoryless form of conditioning that is exactly what makes the lemma apply uniformly to i.i.d., Markov-modulated, and adversarial channel processes alike, provided the algorithm itself is memoryless in the state.
The Lyapunov-drift hypothesis conditions on the σ-algebra generated by the current backlog vector,
MeasurableSpace.comap (U t) inferInstance, via Mathlib's condExp; explicit Measurable/
Integrable guards on U t and on the drift term prevent condExp/∫ from silently defaulting
to 0 for a non-measurable or non-integrable process (a trivializing formalization this mission
rules out: without those guards, an arbitrary non-integrable backlog process would satisfy the
drift hypothesis vacuously and "prove" the theorem for a process that manifestly is not stable).
The limsup ≤ B/ε conclusion is stated as ∀ t, (Cesàro average at t) ≤ B/ε rather than via
Mathlib's Filter.limsup, matching the standard telescoping proof (which bounds the average for
every t, not merely eventually) and avoiding Filter.limsup's junk value on an unbounded
sequence in the non-complete lattice ℝ. Out of scope for this mission: the algorithm-specific
Theorem 4.5 (dynamic backpressure throughput optimality) and Lemma 4.4 (the single-queue corollary
via admissible processes) — the former needs the book's named algorithm plus Chapter 3's capacity
region restated as a local hypothesis and the explicit constant B of Eq. (4.12); the latter needs
this chunk's own restatement of Chapter 3's admissibility structures. Both are left for a future
mission with a larger time budget rather than approximated.
Search-engine ad-auctions sell items (ad slots) to buyers who arrive with per-item bids and a fixed daily budget, online: each item must be allocated the moment it appears, with no ability to revisit past allocations, and no buyer may ever be charged more than its budget. Buchbinder and Naor's survey [1] models this as a generalization of online bipartite matching and derives an allocation algorithm through the same online primal-dual recipe formalized elsewhere in this series (Chapter 4's framework), applied to a genuinely different LP shape — a maximization (revenue) problem whose "packing" role and "covering" role are the reverse of Chapters 4, 5 and 7 — and to a different constraint structure (budget caps on a per-buyer accumulated sum, not a per-constraint 0/1 covering requirement). This mission covers Section 10.1, "The Basic Algorithm": the single-slot allocation algorithm and its -competitive analysis (Theorem 10.1). Sections 10.2-10.3 (multiple ad-slots via strong duality; stochastic per-buyer spending guarantees) are out of scope — see Formalization scope.
Fix a finite set of buyers, each with budget , and a finite set of items; buyer bids for item , revealed one item at a time in the order enumerated by . Let , carried as an explicit positive parameter. The Allocation algorithm (p. 212), upon each item 's arrival, allocates it to the buyer maximizing (where is buyer 's current primal value); if already, nothing happens (the buyer is "full"). Otherwise it charges buyer the minimum of and its remaining budget, sets the dual allocation variable , sets , and updates for a constant fixed by the analysis. The revenue actually collected from buyer is — the buyer is never charged more than its budget. Unlike Chapters 4/7, this LP's dual (the ad-auctions revenue objective, Fig. 10.1) is the maximization problem being solved online; the "primal" covering LP (, variables) exists only as a duality certificate.
Theorem 10.1 (the goal, p. 212), given the milestone's dual near-feasibility bound and the fact that each buyer's total accrued bids exceed its budget by at most a factor of (Claim (3)'s consequence):
with taken verbatim from the theorem's own statement — the exact formula, not an instantiation. Inequality (10.1) (p. 213), the milestone, is the book's own induction-proved lower bound on a buyer's primal value in terms of its accrued bids: .
Theorem 10.1 is the entry point to a short but influential sub-line of the online primal-dual method — Section 10.2 extends it to multiple ad-slots via strong duality for maximum-weight matching (rather than the weak duality this framework otherwise relies on throughout), and Section 10.3 incorporates stochastic per-buyer spending guarantees, both reusing this section's constant and its limit as (recovering the classic -competitive ratio for the unweighted, unbudgeted case). It is also the first mission in this series to apply the online primal-dual method to a genuine revenue-maximization problem rather than a covering/packing pair with matching roles. No formal development of ad-auctions, budgeted online matching, or this constant was found on the platform as of 2026-09-20 (searches below); this mission is the first.
As with 04-framework's Algorithm 3 and 07-generalized-caching's Fractional Caching algorithm,
the central obstacle is characterizing an online process by its own final output. Unlike those
two chapters, however, step (3)'s update increment varies per allocation (bd, the specific bid
of the item just won), so no closed-form solution of the recurrence exists in general; buyerX is
instead defined as an explicit List.foldl realizing the exact per-step update, over the
(temporally ordered) list of bids a buyer actually won — a faithful, if less immediately readable,
transcription of "the algorithm's output as a function of its own trajectory," in the same spirit
as this series' other closed-form definitions. A second difficulty specific to this chapter: the
book's derivation of inequality (10.1) is itself an induction on iterations (not a single algebraic
step, unlike Eq. (7.2) in 07-generalized-caching), and the theorem's final bound further combines
it with a separate "at most one undercharged iteration" argument (Claim (3)'s conclusion,
p. 214-215) turning the raw accrued-bid bound into one about the actually collected (budget-capped)
revenue — both are stated here as explicit hypotheses (the milestone, and
h_at_most_one_undercharge) rather than derived, since the goal is a faithful statement, not a
proof; both sorrys are documented, not silently discharged.
AdAuctionsInstance I M bundles b : I → M → ℝ, B : I → ℝ (hB_pos), and Rmax : ℝ with
hRmax_pos : 0 < Rmax and hRmax_bound : ∀ i j, b i j ≤ Rmax * B i — the last two as explicit
hypotheses, never derived via Finset.sup, matching 04-framework's d and
07-generalized-caching's k. cParam inst := (1+Rmax)^(1/Rmax) uses Real.rpow (ℝ^ℝ).
buyerX inst i bids folds step (3)'s update over a list of won bids; revenue/actualCharge
realize and its budget-capped charge. Reals throughout. Explicitly out of
scope: Section 10.2's multiple-slot generalization (Theorem 10.2), which requires strong duality
for maximum-weight bipartite matching as an explicit premise (the book: "our analysis... crucially
relies on strong duality") — a substantially different LP structure (an integral matching LP, not
this section's per-buyer budget LP) that this mission's AdAuctionsInstance does not model, and
whose applicability of PrimalDualOnline.LP.strong_duality_adapter (this book's own Theorem 2.2)
was not verified in the time available. Section 10.3's stochastic guarantee (Theorem 10.3) is
likewise out of scope, and its own BRIEF.md-flagged ambiguity (whether its scalar g is
or another aggregate of the per-buyer vector ) was not resolved. Both are natural
follow-on missions, not attempted here. Welcome contributions: completing the two sorrys, and the
Section 10.2-10.3 follow-on mission(s).
Many online problems — renting vs. buying equipment, routing traffic without knowing future
demand, allocating advertising budget as bids arrive — share a common linear-programming shape:
a covering (minimization) problem whose constraints appear one at a time, or its dual packing
(maximization) problem whose variables appear one at a time, with no ability to revisit past
decisions. Buchbinder and Naor's survey [1] recasts the online primal-dual method, originally
developed for offline approximation in Section 2 of the same survey (already covered on this
platform via the PrimalDualOnline namespace, citing Buchbinder's thesis [2]), as a general
recipe for such online problems, unifying earlier ad hoc analyses of the ski-rental problem
(Chapter 3) and paving the way for the online set-cover, routing, caching, and ad-auction
algorithms formalized elsewhere in this series. This mission covers Chapter 4, "The Basic
Approach": the framework itself and its three founding algorithms.
Fix a finite index set of primal variables with non-negative cost coefficients , and a finite index set of covering constraints, revealed one at a time in the order enumerated by . Each constraint is given by a set (the book's simplified setting, in which every non-zero coefficient equals and every right-hand side equals ; Chapter 14 removes this restriction) and asserts . An online covering algorithm may only increase the , never decrease them, and upon a constraint's arrival must eventually make it hold. The online covering problem is to minimize subject to every revealed constraint, online. Its Lagrangian dual is the online packing problem: a dual variable arrives together with constraint , may only be increased while is being processed, and the objective is to maximize subject to for every — the packing constraint on becomes fully known only once every with has arrived, so it, too, is revealed gradually. , the largest constraint size, is carried as an explicit parameter throughout.
Section 4.2 gives three algorithms solving both problems simultaneously — the same run produces a covering solution and a packing solution — with the same worst-case guarantee but different flavors: Algorithm 1 is a discrete process (each processing round performs a whole number of identical multiplicative-plus-additive updates until its constraint is satisfied); Algorithm 2 is the continuous limit of Algorithm 1 (dual variables increase continuously and follows an explicit exponential of the accumulated dual sum); Algorithm 3 replaces the continuous update with one triggered by an approximate complementary-slackness condition, at the cost of a mild dual infeasibility. All three make essential use of the online order: an algorithm that saw the whole instance up front would trivially solve the offline LP.
Theorem 4.3 (Algorithm 3 — the goal, p. 124):
Theorem 4.1 (Algorithm 1) and Theorem 4.2 (Algorithm 2, p. 118 and p. 121) are the same three-part guarantee for the other two algorithms, with in place of for Algorithm 1 (and its packing solution genuinely integral), and with in place of both and for Algorithm 2 (whose packing solution is exactly feasible, not merely approximately so). The competitive ratios are stated against an arbitrary offline-feasible comparison solution on each side (covering and packing) rather than against an unconstructed LP optimum — the standard weak-duality reformulation of "-competitive", and the one the book's own proofs (which invoke weak duality directly, never LP optimality) actually establish.
The framework converts three qualitatively different design intuitions — the ski-rental-style discrete doubling, the continuous primal-dual differential equation, and complementary slackness — into three algorithms with an identical asymptotic guarantee, , matching the (packing) and (covering) lower bounds the book proves in Section 4.3 (Lemmas 4.5-4.6, not part of this mission). This is the load-bearing substrate for the rest of the survey: Chapter 5's online set-cover algorithm, Chapter 13's bounded-allocation problem, and Chapter 14's general packing-covering constraints all restate this chapter's framework locally rather than re-deriving it, and are formalized as separate missions in this series. Formalizing it here, once, with the exact constants each proof establishes, is what lets those missions cite a single faithful statement instead of three independently-drifting restatements. No formal development of this framework was found on the platform as of 2026-09-20 (searches below); this mission is the first.
The central obstacle is not the algebra of any single algorithm's proof — each is a short, self-contained argument — but stating the guarantee for an online process using only its final output. An algorithm is characterized here by the closed-form relation its update rule establishes between the accumulated dual sum and the primal value (e.g., Algorithm 3's once activated, before), together with primal feasibility as the hypothesis that a run has completed; a formalization that instead handed the algorithm the whole instance in advance, or dropped primal feasibility as a hypothesis, would either trivialize the online promise or make the stated bound simply false. A second difficulty specific to this formalization: Theorem 4.3's own proof bounds the primal cost by splitting it into a piece controlled by the final-state complementary-slackness conditions (immediate from the closed form and the -bound) and a piece controlled by a derivative/telescoping argument over the continuous accumulation process itself — the latter is a genuinely dynamic fact about the trajectory, not just its endpoint, and is recorded as a documented simplification below rather than folded into the hypotheses, since the goal is a faithful statement, not a proof.
CoveringInstance I J bundles S : J → Finset I, c : I → ℝ, and d : ℝ with 0 < d and
∀ j, (S j).card ≤ d as explicit hypotheses (never derived as d := ⨆ j, (S j).card, matching
the book's own presentation and avoiding a vacuous formalization in which d is chosen after
the fact to make the bound trivial). dualSum inst y i := ∑_{j \mid i \in S(j)} y_j. Each
algorithm's output is a noncomputable def from the accumulated dual data to ℝ (alg1X,
alg2X, alg3X), so that "the algorithm's output" is genuinely a function of its dual
trajectory rather than an independently-constrained free variable — ruling out the trivializing
formalization in which and are unrelated variables merely required to satisfy the
conclusion's own inequalities. Reals throughout (ℝ, not ℝ≥0 or ENNReal); Finset.filter
realizes " such that "; Real.logb 2 and Real.log (natural log) match the
book's own log₂ and ln. Reusable beyond this mission: CoveringInstance, dualSum, and the
weak-duality-style "competitive against any feasible comparison solution" pattern, which
05-online-set-cover, 13-bounded-allocation, and 14-general-packing-covering are expected to
restate locally (per this series' own rule against cross-draft imports between concurrent
missions) rather than import directly. Welcome contributions: completing any of the three
sorrys, and formalizing the Section 4.3 lower bounds (Lemmas 4.5-4.6) as a follow-on mission.
Perturbative quantum field theory computes correlation functions of a field by expanding around a Gaussian (free) theory. Every term of that expansion is a Feynman diagram, and the rule that turns a diagram into a number is Wick's theorem: the expectation of a product of Gaussian field modes is the sum, over all ways of pairing the modes up, of the product of the two-point functions of the pairs. The same identity is known in probability and statistics as the Isserlis theorem (L. Isserlis, 1918) and is the standard tool for computing moments of Gaussian vectors; in random-matrix theory the counting of pairings it produces is the origin of the Catalan-number asymptotics of Wigner's semicircle law.
The uploaded source is the Wikipedia article Feynman diagram, which states Wick's theorem for the free scalar field and then, in the section Higher Gaussian moments — completing Wick's theorem, verifies the one-variable case by direct Gaussian integration. This mission formalizes that content: the combinatorics of pairings, the one-dimensional Gaussian moment formulas, and the multivariate identity itself.
Fix and work on with coordinates . Let be a centered Gaussian measure on : a Gaussian probability measure all of whose coordinate means vanish, for every . Its covariance (in the physics reading, the propagator) is
A pairing of the labels is a partition of these labels into unordered pairs; equivalently, a permutation of the labels with and for all (a fixed-point-free involution). The set of pairings is written . For a weight indexed by labels, the Wick sum is
the inner product ranging over the pairs of , each counted once through its smaller element.
In the article's field-theory notation the labels are momenta , the coordinates are the field modes , and the two-point function carries the momentum-conserving delta function, . This mission works with the finite-dimensional Gaussian vector rather than the field, so the delta functions are absorbed into the covariance matrix .
For a centered Gaussian measure on and any labels ,
No hypothesis is imposed on the covariance: it may be singular and the labels may repeat, which is exactly the situation the article's "completing Wick's theorem" section addresses.
Targets 1–3 are the article's displayed Gaussian integrals, target 4 is its pairing count, targets 5–6 are the two explicit consequences it records for the field correlators.
Wick's theorem is the computational content of every Feynman-diagram expansion: once it is available, a perturbative term is a finite sum over diagrams, and the symmetry factors of diagrams are bookkeeping on the pairing set . On the probabilistic side it gives all moments of a Gaussian vector in closed form, which is the entry point to Gaussian chaos expansions, Wiener–Itô integrals, and moment methods for random matrices.
Mathlib (revision 0df444a) has real Gaussian measures gaussianReal, the general class
IsGaussian of Gaussian measures on a topological vector space, the Gaussian integral
, and the double factorial Nat.doubleFactorial, but no
higher-moment formula for Gaussian measures and no Isserlis/Wick statement. The mission therefore
produces new library-level content, not a re-derivation of existing formal results; the result
itself has been classical since 1918 (Isserlis) and 1950 (Wick).
The obvious route — expand the characteristic function and differentiate times at — requires differentiating under an integral sign times and identifying the resulting combinatorial sum with a sum over pairings; both steps are where the formal work lies. Integrability is not automatic from the statement and has to be established (Gaussian measures have moments of all orders, but the product must be shown integrable before any manipulation). The naive attempt to reduce to the independent case by diagonalizing meets a second difficulty: the change of variables must be tracked through the pairing sum, and may be singular, so no invertible whitening transform exists in general. The one-variable case (milestone 3) is not a special case to be waved through either: it is the statement the article singles out, because a naive "each mode pairs with a distinct partner" argument fails when all labels coincide.
The ambient space is EuclideanSpace ℝ (Fin d); measures are Mathlib Measures and Gaussianity
is the Mathlib class IsGaussian, which is defined by every continuous linear functional pushing
forward to a real Gaussian law. Centering is stated as an explicit hypothesis on the coordinate
means, so the measure is not assumed standard and the covariance is unconstrained (in particular
degenerate covariances, and repeated labels , are included). Integrals are Bochner
integrals, which return for non-integrable functions; the statements are nonetheless
non-vacuous because Gaussian measures integrate all polynomials.
Pairings are formalized as fixed-point-free involutions of Fin (2 * n) and the pair product
ranges over , so each pair contributes once. The case is included:
the empty product is , the unique pairing of the empty label set is the identity, and both
sides of the goal equal . The double factorial is Mathlib's Nat.doubleFactorial, evaluated at
2 * n - 1 in truncated natural subtraction, so the value is .
Contributions welcome: the Gaussian moment lemmas (milestones 1–3) as standalone Mathlib-style results, the pairing count (milestone 4) as pure combinatorics independent of the analysis, and any reduction of the goal to the independent-coordinate case.
Since the 2019 revision of the SI, seven constants of nature have fixed, exact numerical values: the caesium hyperfine frequency , the speed of light , the Planck constant , the elementary charge , the Boltzmann constant , the Avogadro constant and the luminous efficacy . Every other constant is either measured, and then carries an uncertainty, or is a closed-form expression in the seven, and is then known exactly. The CODATA 2022 adjustment (Mohr, Newell, Taylor and Tiesinga, Rev. Mod. Phys. 97, 025002, 2025) is the authoritative statement of which constants fall on which side of that line, and of the statistical procedure that produces the recommended values of the measured ones.
The exactly known side of the line is pure mathematics: the tabulated value of the Stefan-Boltzmann constant, of the radiation constants, of the molar gas constant and of the Faraday constant are theorems about , an integral, and rational arithmetic - not measurements. The measured side rests on one statistical object, the weighted linear least-squares fit of input data to adjusted constants, together with the diagnostics (, the Birge ratio, normalized residuals, expansion factors) the task group uses to decide that the fit is acceptable. This mission formalizes both.
Write , , , , for the numerical values of the defining constants in coherent SI units, and . The spectral radiance of a blackbody at thermodynamic temperature , per unit frequency, is Planck's function
and per unit wavelength
The derived constants named in Tables XXXII and XXXIII of the paper are (Stefan-Boltzmann), and (radiation constants), (molar gas), (Faraday), (Josephson) and (von Klitzing).
For the adjustment, let be the design matrix of the linearized observational equations, the input data and the weight matrix - the inverse of the covariance matrix of the data, symmetric and positive definite. The fit minimizes
with degrees of freedom, Birge ratio and normalized residuals .
The goal fixes no digits: it asserts that the constant appearing in the tabulated closed form is exactly the one produced by integrating Planck's law over the spectrum and over a Lambertian hemisphere. The numerical enclosure of is a separate, weaker milestone, so that the goal survives any change of tabulated digits.
together with the numerical enclosures of , of Wien's displacement constants and (where , solve and ), and the identity .
On the statistical side:
the equivalence of minimality with the normal equations, the closed form when is invertible, and the behaviour of , and under an expansion factor applied to the uncertainties.
The exact-constant statements are what makes the tabulated values checkable rather than authoritative: a reader who accepts the SI definitions can, in principle, re-derive and by arithmetic, and from Planck's law and . Formalizing them turns "in principle" into a machine-checked derivation from the seven fixed numbers.
The least-squares statements make precise what the adjustment is, independently of the 133 data of this particular round: that the recommended values are the unique minimizer of a quadratic form, that the normal equations characterize it, and that multiplying a group of uncertainties by an expansion factor - the device used in 2022 with , and - rescales by , the Birge ratio by and the residuals by , without moving the fitted values. None of these statements is new mathematics; none of them, as far as the catalog shows, is currently available as a machine-checked statement about the CODATA quantities.
Two hard points. First, : the obvious route expands as a geometric series and integrates term by term, which needs a dominated convergence or monotone convergence argument plus the value ; interchanging the sum and the integral is the step that must be justified, not assumed. The goal then needs this together with a change of variables on and an integrability argument for the original integrand.
Second, the numerical enclosures. requires certified rational bounds on ; the Wien constants require certified bounds on a transcendental root, obtained from interval evaluation of at rational points plus monotonicity of . Floating-point evaluation proves nothing here.
The least-squares milestones are routine linear algebra, with one subtlety: the decomposition identity needs symmetry of , and the characterization of minimizers needs positive definiteness, so both hypotheses are carried explicitly.
All physical quantities are plain real numbers: each constant is the numerical value of the quantity in the coherent SI unit named in its docstring, and the code carries no units. The defining constants are given by the exact decimal literals of Table XXXII, so they are exact rationals; is defined as and is therefore irrational.
Planck's function is a total function of two real arguments. Outside the physical regime it returns junk rather than failing - at the exponent is , the denominator , and Lean's division returns - so every statement about it carries and a positivity hypothesis on the spectral variable. The goal's integral is the Bochner integral over with respect to Lebesgue measure; it is not trivially true by non-integrability, since the integrand is integrable for .
The adjustment is formalized over and with an arbitrary design matrix and an arbitrary weight matrix , hypotheses on (Hermitian, positive definite) stated where they are used rather than baked into the definition; is an integer, not a truncated natural number.
Contributions welcome: the integral and the interval-arithmetic enclosures are reusable well beyond this mission, as are the least-squares statements.
Linear systems are the inner loop of scientific computing: discretized differential equations, least-squares fitting, network flow balances and equilibrium models all end in . Direct elimination solves the system exactly in operations, but for the large sparse systems produced by discretization the cost and the round-off growth make iterative methods preferable: start from an arbitrary vector and apply a cheap update until the residual is small. The question such a method raises is when the iteration converges, and to that the chapter gives a clean sufficient answer: diagonal dominance.
This mission is the third in a series formalizing the lecture notes Métodos Numéricos by Sergio Roberto de Freitas (UFMS, 2000). It covers the iterative part of Chapter 5, Solução de Sistemas Lineares.
Let be a real matrix and . Assume for all .
The Jacobi method computes every coordinate of the new iterate from the old one:
The Gauss-Seidel method updates the coordinates in order and uses the values already updated in the same sweep:
Both are instances of an affine iteration associated with an equivalent rewriting .
The matrix is diagonally dominant when each diagonal entry dominates its row:
The goal theorem is Proposição 5.10.1: if is diagonally dominant and x^\\star solves Ax^\\star = b, then the Gauss-Seidel iterates converge to x^\\star from any starting vector.
The milestones are the general facts the source uses to get there: that the limit of a convergent affine iteration is a fixed point of it and hence a solution of the system (Proposição 5.5.1), that a contraction condition with forces convergence to the solution (Proposição 5.5.3), and that the Jacobi sweep has exactly the solutions of as its fixed points (Proposição 5.5.2).
Diagonal dominance is the hypothesis a practitioner can check by inspection, and it is satisfied by the matrices that come from standard finite-difference stencils, from strictly diagonally dominant collocation systems and from many equilibrium models. The theorem says that for those systems Gauss-Seidel needs no spectral analysis and no preconditioner to be safe: convergence holds from any starting vector. The supporting milestones isolate the two halves of the argument — a fixed-point identification and a contraction estimate — in a form reusable for other splittings (Jacobi, SOR, block variants).
Mathlib has Banach's fixed point theorem and the theory of matrix norms, but not the Gauss-Seidel sweep, the notion of diagonal dominance as used here, or the convergence statement, which is what this mission adds.
Gauss-Seidel is not a plain affine map applied coordinatewise: within one sweep the coordinates are updated sequentially, so the new value of coordinate depends on the new values of coordinates . Formalizing the sweep therefore requires a recursion over the coordinate index before the recursion over the iteration counter, and the contraction estimate has to be propagated along that inner recursion. The classical proof compares \\max_i |x_i^{(k+1)} - x_i^\\star| with \\max_i |x_i^{(k)} - x_i^\\star| and needs, for each , a bound that already uses the improved bounds for ; getting that induction right is the substance of the mission.
Vectors are functions from a finite index type with elements to , and convergence is convergence in that finite product space (equivalently, coordinatewise). The Gauss-Seidel sweep is defined through an auxiliary partial sweep: after inner steps the first coordinates carry their new values and the remaining ones their old values, and the full sweep is the partial sweep after steps. The Jacobi sweep is defined directly. Diagonal dominance is the strict inequality above, with the sum taken over the row with the diagonal index removed; for every statement is vacuous, and the goal theorem is then trivially true because the space has a single point. Division by the diagonal entry is total division, so the definitions make sense even when ; diagonal dominance rules that out, because the right-hand side of the dominance inequality is nonnegative. The goal theorem assumes a solution x^\\star is given rather than asserting its existence, and it asserts convergence for every starting vector, generalizing the source's choice . The contraction milestone states the consistency of the norms as the hypothesis in the supremum norm rather than fixing a particular matrix norm.
Most equations that arise in applications cannot be solved in closed form: the age of the Moon from a radioactive-decay balance, the deflection of a clamped beam, the equilibrium of a catenary, all reduce to solving for a function with no algebraic inverse. A first course in numerical methods therefore opens with root finding: constructive procedures that produce a sequence of approximations together with a theorem saying that the sequence converges to a root and how fast.
This mission is the first in a series formalizing the lecture notes Métodos Numéricos by Sergio Roberto de Freitas (Departamento de Computação e Estatística, UFMS, 2000). It covers Chapter 3, Zeros de Funções, whose capstone is the local convergence of Newton's method.
Let and let be a zero of , i.e. . The chapter studies three constructions.
Bisection. Starting from an interval with , put and keep the half of on whose endpoints still changes sign. The width of the bracketing interval is halved at every step.
Linear iteration (MIL). Rewrite as a fixed-point equation and iterate from an arbitrary .
Newton's method. Take the particular iteration function
obtained by truncating the Taylor expansion of at after the linear term.
A sequence has order of convergence when tends to a finite constant, where ; is linear and quadratic convergence.
The goal theorem is the local convergence of Newton's method at a simple zero. If is twice differentiable on an open interval containing , its second derivative is continuous there, and never vanishes on , then there is such that
where is the Newton sequence started at . This is the formal content of the chapter's statement that Newton's method converges provided the initial guess is chosen close enough to the root.
The milestones are the supporting results of the chapter: the error bound and convergence of bisection, the convergence of the linear iterative method under a derivative bound , the a posteriori estimate , and the quadratic order of Newton's method at a simple zero.
Bisection, fixed-point iteration and Newton's method are the three root finders every numerical-analysis course starts with, and the three convergence theorems above are what justifies using them. The a posteriori estimate is what turns the iteration into an algorithm with a stopping criterion: it bounds the distance to the root by a quantity the program can measure. The quadratic order statement explains the observed doubling of correct digits per step at a simple zero and its loss at a multiple zero.
On the formalization side, Mathlib already has a fixed-point theorem for contractions on complete spaces and the mean value theorem, but not the statements in the form used in numerical analysis: bisection with its explicit bracket, the a posteriori bound, or Newton's local convergence and quadratic rate stated for the concrete iteration sequence. This mission asks for those.
The delicate point in all three theorems is that the iterates must be known to stay in the region where the hypotheses hold; the informal proofs assume this silently. For the linear iterative method the mission therefore states the invariance hypothesis explicitly ( maps the closed interval into itself). For Newton's method, no such hypothesis is given: the existence of a neighbourhood of that the iteration preserves is part of what must be proved, and it comes from the continuity of together with . The quadratic-order milestone also has to handle the degenerate possibility , which would put a zero in the denominator; it is excluded by hypothesis.
Everything is over the real numbers. The iterations are given as explicit recursive sequences: the bisection construction returns the bracketing pair and its midpoint, and there are separate sequences for the fixed-point and Newton iterations. Newton's method takes the derivative as a separate function argument , tied to by a hypothesis of the form " has derivative at every of the interval"; this avoids relying on any junk value of a derivative operator where fails to be differentiable. Division is total, so a step at a point where vanishes would leave the iterate unchanged; the hypotheses exclude this inside the interval.
Two deliberate deviations from the source are worth flagging for the auditor. First, Proposição 3.5.1 assumes only for , but its proof divides by ; the formal statement assumes on the whole interval, so is a simple zero. Second, the convergence statement for the linear iterative method adds the hypothesis that maps the closed interval into itself, without which the iterates may leave the region where the derivative bound is assumed.
A table of values is all one has of many functions: measurements, tabulated physical constants, the output of an expensive simulation. Interpolation reconstructs a function between tabulated points by passing a polynomial through them, and it underlies much of the rest of numerical analysis — quadrature rules, finite-difference formulas and predictor-corrector schemes for differential equations are all obtained by integrating or differentiating an interpolating polynomial. What makes the reconstruction trustworthy is a formula for the error committed away from the nodes.
This mission is the fifth in a series formalizing the lecture notes Métodos Numéricos by Sergio Roberto de Freitas (UFMS, 2000). It covers Chapter 7, Interpolação.
Let be distinct nodes and the tabulated values of a function . The Lagrange form of the interpolating polynomial is
so that for every , and is the unique polynomial of degree at most with this property.
The interpolation error at a point that is not a node is the difference .
The goal theorem is Proposição 7.3.1: if is times differentiable on and all nodes lie in , then for every different from all nodes there exists with
The milestones are the other results of the chapter needed to make sense of it: the existence and uniqueness of the interpolating polynomial of degree at most through points with distinct abscissas (Proposição 7.2.1), the fact that the Lagrange formula does interpolate the data, and the practical error bound of Proposição 7.3.2, when .
The error formula is the reason interpolation is a numerical method and not just a curve-drawing device: it shows that the error is governed by two independent factors, the smoothness of through and the geometry of the nodes through the product . Everything downstream follows from it — the and error terms of the trapezoidal and Simpson rules in the next mission are obtained by integrating exactly this expression, and the choice of Chebyshev nodes is the attempt to make the product factor small.
The standard proof introduces the auxiliary function with chosen so that , and then applies Rolle's theorem times to conclude that vanishes somewhere. Formalizing the repeated application of Rolle's theorem, keeping track of the distinct zeros and the nested intervals they generate, is the substance of the work; it is an induction that has to be organized carefully rather than a computation.
The nodes are given as a strictly increasing family of reals lying in the open interval , and the interpolating polynomial is the explicit Lagrange sum rather than an abstract polynomial: at a node it is defined by the same formula, whose factors then include contributions unless the nodes are distinct, which is why the distinctness hypothesis appears in the interpolation milestone. Smoothness is expressed as continuous differentiability of order on the open interval , and the derivative appearing in the error term is the iterated derivative computed within that set; since the set is open this agrees with the ordinary -st derivative. The evaluation point is assumed to lie in and to differ from every node; the point is asserted to exist in , with no claim of uniqueness or of any relation to beyond membership in the interval. The uniqueness milestone is stated with Mathlib's polynomial type and the degree bound , which includes the zero polynomial.