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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.
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?
How short can a code for a data source be, if the code must still be uniquely decodable — if every string of concatenated codewords can be unambiguously split back into the original symbols? Shannon's 1948 source coding theorem answers this exactly: the entropy of the source is a hard lower bound on the average codeword length of any uniquely decodable code, and it is also achievable up to a one-symbol slack. Entropy is not just a measure of "average surprise" — it is the literal, tight answer to a combinatorial question about how densely symbols can be packed into strings without losing decodability. This is the theorem that gives Shannon's entropy its operational meaning, and it underlies every practical lossless compression scheme (Huffman coding, arithmetic coding, Lempel–Ziv) as the benchmark they approach.
Timeline.
A source has a finite alphabet of symbols , with at least two symbols, and probability distribution (, ). A code assigns to each symbol a codeword , a finite string over a -ary code alphabet (); the code is uniquely decodable if every finite sequence of codewords is determined by its concatenation. The entropy of in base is
The expected codeword length of under is .
(For a source with symbols — see Formalization scope for why the single-symbol case must be excluded.)
The result itself. This theorem is the reason entropy is called entropy in an information-theoretic sense at all: it converts a quantity defined by an abstract formula () into the exact answer to an operational question (minimum achievable expected code length), with a slack no worse than one symbol. It is the founding theorem of lossless source coding and the benchmark every practical compressor is measured against.
Formalizing it. Mathlib recently gained genuine information-theoretic coding content: InformationTheory.UniquelyDecodable and the necessity direction of the Kraft–McMillan inequality (McMillan's 1956 result: a uniquely decodable code's lengths satisfy ) are already proved, via a counting argument on concatenations of codewords. This mission builds directly on that foundation rather than duplicating it. What Mathlib does not have — and what this mission's milestones supply — is Kraft's original 1949 sufficiency direction (existence of a uniquely decodable code realizing any length assignment satisfying the Kraft sum bound), any notion of Shannon entropy for a general finite distribution, and the source coding theorem itself.
The lower bound () is the easier half: it follows from the Kraft–McMillan inequality (already in Mathlib) via Gibbs'/Jensen's inequality applied to the two probability-like sequences and (where is the Kraft sum) — a short, self-contained convexity argument.
The achievability half is the genuine construction. Given the ideal (generally non-integer) lengths , one rounds up to (Shannon–Fano–Elias lengths); a one-line estimate shows , so the Kraft sum of the rounded lengths is still , and the bound gives immediately once a code with exactly these lengths is shown to exist. Producing that code is Kraft's sufficiency direction, and it needs an explicit construction: order the lengths, and assign to symbol the first digits of the -ary expansion of the cumulative sum . Verifying this assignment is injective, has the prescribed lengths, and is uniquely decodable (indeed prefix-free) is a careful but standard combinatorial argument — the main open piece of this mission.
The source alphabet must have at least two symbols (), not merely be nonempty. A single-symbol source forces and entropy , so the achievability conjunct would demand a codeword of length — but a uniquely decodable code can never contain the empty codeword (InformationTheory.UniquelyDecodable.epsilon_not_mem, provable from the definition: the empty string decodes ambiguously as zero or two copies of itself), so no admissible code exists and the theorem would be false, not merely hard, at . The same defect breaks Kraft's sufficiency direction (Milestone 2) whenever any prescribed length is , independent of ; that milestone accordingly requires every length strictly positive. With and full support, every strictly, so the Shannon–Fano lengths are automatically all , and the achievability construction only ever needs Milestone 2 at positive lengths. The code alphabet is likewise an arbitrary finite type (matching Mathlib's own Fintype/Nonempty conventions for the Kraft–McMillan file), with required to keep Real.logb non-degenerate. The source distribution is required strictly positive () — the standard simplifying assumption (zero-probability symbols can always be dropped without loss). Unique decodability is stated exactly as Mathlib's InformationTheory.UniquelyDecodable, not re-derived from a "prefix code" definition, so the mission's results transport directly onto Mathlib's existing Kraft–McMillan file. A trivializing route to rule out: proving only the lower bound (citing Mathlib's inequality) while leaving the existential achievability half unaddressed would not be Shannon's theorem — the sandwich is the theorem's actual content, and the lower bound alone (already essentially free from Mathlib) is not a novel contribution on its own.
Reusable output: the Kraft sufficiency construction (Milestone 2) is directly reusable for any future formalization of Huffman coding optimality, arithmetic coding, or the general "Kraft-inequality-achieving code exists" fact used throughout coding theory. Contributions are welcome starting from Milestone 2 (the open construction) or Milestone 3 (the Gibbs'-inequality lower bound, which only needs Milestone 1, already available via Mathlib).
Mathlib.InformationTheory.Coding.UniquelyDecodable and Mathlib.InformationTheory.Coding.KraftMcMillan (2026).The separation theorem — certainty equivalence for linear-quadratic control with imperfect state information — is one of the celebrated structural results of stochastic control: the optimal controller splits into a least-squares estimator and the deterministic LQR actuator, designed independently. It underlies every LQG autopilot and Kalman-filter-based regulator. Section 5.2 of Bertsekas, Dynamic Programming and Optimal Control, Vol. I (3rd ed., 2005) proves it from the DP algorithm over information vectors, with Lemma 5.2.1 supplying the key fact that the estimation error is beyond the controller's influence. No formal analogue exists in Mathlib.
Linear dynamics and measurements
with quadratic cost , , . The initial state and the zero-mean disturbances/noises are independent with finite ranges; independence is structural — the sample space is the product of an initial-state coordinate and per-stage noise coordinates (BertsekasLQGModel, BertsekasLQGSample, BertsekasLQGProb). A policy maps the realized measurement history to ; the closed-loop process is BertsekasLQGTraj, the expected cost BertsekasLQGCost. The estimator is an explicit conditional average (BertsekasCondExpVec, BertsekasLQGEstimate); the gains come from the time-varying Riccati recursion (BertsekasLQGRiccati, BertsekasLQGGain).
— BertsekasDP.lqg_certainty_equivalence (goal). Milestone: Lemma 5.2.1 in pointwise form — the error is the same under any two policies, outcome by outcome (lqg_estimation_error_policy_independent).
This is the theorem that justifies designing estimator and controller separately — remove it and the entire LQG methodology loses its warrant. The formalization also yields the first machine-checked instance of the informational decomposition (control-dependent part + policy-independent error) that recurs throughout imperfect-information control. Notably the result needs no Gaussian assumption — only zero mean and independence — and the finite-support model makes that generality exact. The result is classical (Joseph–Tou 1961, Gunckel–Franklin 1963; the book's §5.2); the formal proof is new.
The heart is Lemma 5.2.1: showing the estimation error coincides, sample by sample, with the error of the control-free system — which requires proving that the observation-history σ-events under any policy coincide with those of the control-free system (controls are determined by the history, so they shift observations by a known amount). Then the DP argument over information histories must carry the quadratic decomposition through the backward recursion. Bookkeeping over histories-as-lists is the main formal burden; probability theory stays finite.
Finite-support randomness (all expectations are finite sums); conditional expectation with the explicit junk value 0 on zero-probability events — the goal's hypothesis is accordingly restricted to outcomes of positive probability. Policies are functions of the measurement list only (equivalent to the book's information vector for deterministic policies, since past controls are recoverable from past measurements). Matrices are time-varying; positive definiteness of makes every matrix inverse in the gains genuine. Measurement noise covariance is not assumed positive definite — the estimator is the abstract conditional expectation, not the Kalman filter (whose recursive form, §5.2.1, would be a natural follow-up mission).
Label correcting methods are the workhorse family of shortest-path algorithms — Dijkstra's method, Bellman–Ford, SLF/LLL variants and A* all fit the template analyzed in §2.3.1 of Bertsekas, Dynamic Programming and Optimal Control, Vol. I (3rd ed., 2005), where shortest paths appear as the purely deterministic face of dynamic programming. The correctness proof (Prop. 2.3.1) is short on paper but genuinely nondeterministic — any node may be removed from the candidate list, children processed in any order — so a formal proof certifies a whole family of concrete algorithms at once.
A finite directed graph with arc set , real arc lengths , origin and destination (BertsekasSPGraph). Walks are nonempty node lists whose consecutive pairs are arcs (BertsekasIsWalkFrom), with length the sum of arc lengths (BertsekasWalkLength); the shortest distance is the infimum of walk lengths in the extended reals, if no walk exists (BertsekasShortestDistance). The standing assumption of §2.3: every cycle has nonnegative length (negative arcs allowed).
The algorithm state (BertsekasLCState) carries labels , the scalar UPPER, and the candidate list OPEN. Initially , all other labels , UPPER , OPEN . One iteration (BertsekasLCStep, nondeterministic): remove any from OPEN; for each child of in any order, if set , and put in OPEN if , or update UPPER if . The algorithm terminates when OPEN is empty.
for every execution, under the nonnegative arc length assumption of §2.3 ( for every arc) — BertsekasDP.label_correcting_correctness_of_nonneg_arcs (goal). Milestones: termination — no infinite execution exists, which needs only the weaker nonnegative-cycle assumption (label_correcting_terminates) — and the workhorse invariant that every finite label is the length of an actual walk from , which needs neither (label_correcting_invariant).
The nonnegative-arc hypothesis is essential and not a formalization artifact: the algorithm prunes with the test , and with a negative arc a longer prefix can still reach more cheaply, so the pruned node is never entered into OPEN. An earlier version of this mission's goal carried only the nonnegative-cycle assumption of §2.1 and was disproved by the counterexample , , , , (a graph with no cycles at all), where the algorithm terminates with while the shortest distance is . Exercise 2.7 of the source treats the nonnegative-cycle case, which requires a modified algorithm.
Prop. 2.3.1 certifies simultaneously breadth-first search, Dijkstra (best-first), depth-first and small-label-first variants — every removal discipline is one refinement of the nondeterministic relation. Formally, the development contributes a reusable small-step framework for label-setting/correcting algorithms on which sharper results (Dijkstra's single-pass property, A* admissibility, §2.3.3) can later be built. The result is classical; the formal content is the induction along the nondeterministic step relation.
Termination is the subtle half: labels do not decrease monotonically along the run in an obvious well-founded way; the book's argument counts the finitely many distinct walk lengths below a bound — this needs the nonnegative-cycle assumption and a careful bound relating labels to simple-path lengths. The invariant proof must thread through the fold over children within a single step.
Finite node type with decidable equality; arcs as a Finset of ordered pairs; lengths total on (only arc values matter). The step relation is fully nondeterministic in pivot choice and child order (a permutation quantifier); correctness quantifies over all reachable terminal states — there is no fixed schedule to exploit. Distances live in EReal, so the no-path case is the honest empty infimum, not a sentinel. The trivializing risk of restricting to nonnegative arcs is avoided: only cycles are constrained.
The fundamental group is the first algebraic invariant a student of topology meets, and is the first computation of it that carries real content. Allen Hatcher's Algebraic Topology (Cambridge University Press, 2002; freely available at pi.math.cornell.edu/~hatcher/AT/AT.pdf) is the standard text on the subject. Its Chapter 1 opens with exactly this computation (Theorem 1.7, p. 29) and immediately draws three classical consequences from it: the Fundamental Theorem of Algebra (Theorem 1.8), the Brouwer fixed point theorem for the disk (Theorem 1.9), and the Borsuk–Ulam theorem for the sphere (Theorem 1.10).
This mission is the opening entry in a series that formalizes Hatcher's book capstone by capstone. It covers the subsection "The Fundamental Group of the Circle" of Section 1.1 (pp. 29–33): the covering-space lifting properties that drive the proof, the theorem itself, and its three applications. Later entries in the series (van Kampen's theorem, the classification of covering spaces, simplicial and singular homology) will build on the declarations introduced here, which all live in the shared Lean namespace Hatcher.
A path in a topological space is a continuous map , where . A homotopy of paths is a family , , such that the endpoints and are independent of and the associated map , , is continuous. A loop at a basepoint is a path with . The set of homotopy classes of loops at is the fundamental group ; its product is , where traverses and then , each at double speed (Hatcher, Proposition 1.3).
The circle is realised as the unit circle of , so the point is and the basepoint is . Hatcher's map
is Hatcher.circleCover. The loops
based at are Hatcher.omegaLoopN n, and is Hatcher.omegaLoop; its class is Hatcher.omegaClass.
A covering space of is a space together with a map such that every has an open neighbourhood for which is a disjoint union of open sets each mapped homeomorphically onto by (Hatcher's condition , p. 29; such a is evenly covered). A lift of a map is a map with .
is an infinite cyclic group generated by . In the form stated in Lean:
Surjectivity of says generates; uniqueness of says the group is infinite cyclic rather than finite.
The result itself. The computation assigns to every loop in the circle an integer, its winding number, and shows that this integer is the only homotopy invariant of the loop. It is the seed of degree theory, and in Hatcher's text it is the starting point for every later computation of fundamental groups (products, van Kampen, covering spaces). The three applications are the standard demonstration that a single algebraic invariant can settle purely geometric or algebraic existence questions.
Formalizing it. Mathlib (revision 0df444a) already contains the covering-space infrastructure: IsCoveringMap, path lifting (IsCoveringMap.liftPath, eq_liftPath_iff'), homotopy lifting (IsCoveringMap.liftHomotopy, eq_liftHomotopy_iff'), monodromy, and the fact that Circle.exp is a covering map (Circle.isCoveringMap_exp). It also has FundamentalGroup X x as the endomorphism group of the fundamental groupoid. It does not contain the computation , nor the two-dimensional Brouwer and Borsuk–Ulam theorems. The Fundamental Theorem of Algebra is in Mathlib as Complex.exists_root (proved by Liouville's theorem rather than by Hatcher's argument); it is kept as a milestone because it is one of the section's stated theorems, and a solver may close it directly from Mathlib. Milestones 2–4 are also within reach of the existing lifting API, but they are the lemmas Hatcher states and uses, and a faithful record of them in the mission's own namespace is what later entries in the series will import.
The obvious first idea for the goal is to define the winding number of a loop through the complex argument. That fails because is discontinuous on ; the integer has to be produced by lifting the loop through and reading off the endpoint of the lift, which is only well defined because of the uniqueness in the path lifting property. The second difficulty is uniqueness of : this needs lifting of homotopies (milestone 4), not just of paths, together with the observation that a lifted homotopy of paths has constant endpoints.
Connecting the concrete loops to Mathlib's abstract is its own obstacle. FundamentalGroup Circle 1 multiplies by composing morphisms of the fundamental groupoid, so identifying with the class of the explicit loop (milestone 6) requires reparametrization arguments for concatenated paths, for negative as well as positive.
For Theorem 1.9 the difficulty is the construction and continuity of the retraction from a fixed-point-free map, and then the non-existence of a retraction, which uses that . For Theorem 1.10 Hatcher's proof lifts a loop through and shows the lift changes by an odd integer over half a turn; making that parity argument rigorous in Lean is the substance of the milestone.
Circle (the unit circle in ) with basepoint 1; is Metric.closedBall (0 : EuclideanSpace ℝ (Fin 2)) 1; is Metric.sphere (0 : EuclideanSpace ℝ (Fin 3)) 1, with the antipodal point.IsCoveringMap p. This agrees with Hatcher's condition ; neither requires to be surjective.C(I, X) or Mathlib Paths; for homotopies of paths, the square is written I × I with Hatcher's coordinate order : the first coordinate is the path parameter, the second the homotopy parameter. In the general homotopy lifting property the domain is Y × I with an arbitrary topological space, as in Hatcher.FundamentalGroup Circle 1, and is FundamentalGroup.fromPath ⟦omegaLoop⟧. Because the goal quantifies over integer powers of a single element, the order of multiplication in FundamentalGroup is immaterial to its truth.FundamentalGroup Circle 1 ≃* Multiplicative ℤ sending to is an immediate corollary and a welcome contribution.0 < f.degree, which excludes both the zero polynomial and nonzero constants.Contributions welcome: proofs of the milestones from Mathlib's lifting API, a degree homomorphism packaged for reuse, and any lemma about concatenation and reparametrization of loops in Circle that later chapters of the series can import.
Mathlib/Topology/Homotopy/Lifting.lean (path and homotopy lifting for covering maps). https://github.com/leanprover-community/mathlib4/blob/0df444a360eaa60ab8c11dca51a86af692955474/Mathlib/Topology/Homotopy/Lifting.leanMathlib/AlgebraicTopology/FundamentalGroupoid/FundamentalGroup.lean (the fundamental group). https://github.com/leanprover-community/mathlib4/blob/0df444a360eaa60ab8c11dca51a86af692955474/Mathlib/AlgebraicTopology/FundamentalGroupoid/FundamentalGroup.leanThe strategic-form game is the basic object of noncooperative game theory, and the Nash equilibrium — a profile of randomized strategies from which no player benefits by deviating unilaterally — is its central solution concept. Nash proved in 1951 that every game with finitely many players and finite strategy sets has such an equilibrium (Nash, Non-cooperative games, Ann. Math. 54 (1951)); this single existence theorem is the reason the concept organizes the rest of the field, from the computational complexity of finding equilibria to the price of anarchy. The theorem is stated as Theorem 1.8 of Nisan–Roughgarden–Tardos–Vazirani (eds.), Algorithmic Game Theory (Cambridge, 2007), the source text of this mission series, whose first chapter (Tardos–Vazirani) also treats the two special cases that admit direct algorithmic proofs: two-person zero-sum games, where equilibria are exactly the optimal solutions of a dual pair of linear programs (von Neumann 1928; Theorem 1.11), and a simple linear market, where equilibrium prices are computed by an ascending tight-set algorithm (Theorem 1.17).
A timeline of the existence theorem: von Neumann (1928) proved the minimax theorem for two-person zero-sum games; Nash (1950, 1951) extended existence to arbitrary finite games, first via Kakutani's fixed-point theorem and then via Brouwer's. All known proofs of the general theorem pass through a fixed-point principle, and this is not an artifact: computing a Nash equilibrium is PPAD-complete (Daskalakis–Goldberg–Papadimitriou 2009; Chen–Deng–Teng 2009), and PPAD is precisely the complexity class of the fixed-point arguments.
A finite strategic-form game consists of a finite set of players, for each player a finite nonempty set of pure strategies, and for each player a payoff function ; all players are utility maximizers. A mixed strategy for player is a probability distribution on , represented as a weight function with and (a lottery). Players randomize independently, so a mixed profile induces the product distribution on pure strategy vectors, and player 's expected payoff is
A mixed profile is a (mixed) Nash equilibrium if for every player and every lottery on , replacing by does not increase .
A two-person zero-sum game is given by a matrix : the row player picks a row distribution , the column player a column distribution , and the column player pays the row player in expectation.
The market of §1.8.1 of the source has finitely many divisible goods, good in units, and finitely many buyers, buyer bringing budget and interested in a nonempty set of goods; utilities are linear 0/1, so a buyer wants any goods from her interest set and none other. Market-clearing prices are positive prices under which each buyer can spend her whole budget on cheapest goods in her interest set while every good sells out exactly.
Stated for an arbitrary finite family of finite nonempty strategy types; no bound on the number of players, no genericity assumptions.
Mathlib currently has no form of Brouwer's theorem; every known proof of Theorem 1.8 needs it (or an equivalent), so it enters the mission as an explicit milestone rather than an assumed library fact.
of the explicit two-player game with payoffs to the row player and to the column player. The source states the result as: optimal solutions of a dual pair of LPs form a Nash equilibrium of the zero-sum game; the first two conjuncts are the saddle point that LP optimality amounts to, and the third states the Nash-equilibrium clause against the mission's own game vocabulary, so "zero-sum" is formal (the two payoffs sum to zero) rather than implicit in the shape of the statement.
The source proves this by an ascending-price algorithm and also bounds its running time; the complexity half has no formal counterpart in this mission.
The capstone is the foundation of the whole mission series: correlated equilibria, price-of-anarchy bounds, and mechanism-design characterizations in later missions all quantify over or compare against Nash equilibria, and the series inherits its game vocabulary (IsLottery, IsMixedProfile, expectedPayoff, IsMixedNash) from this mission.
Formalizing it produces the first Brouwer fixed-point theorem in this environment — a well-known gap in mathlib with reuse value far beyond game theory (every degree-theoretic and equilibrium-existence argument needs it). The zero-sum milestone yields the minimax theorem, reusable for the learning-dynamics mission that follows. All results here are classical and proved on paper; the work requested is machine-checked proof, not new mathematics.
The central difficulty is Brouwer. The standard routes are (i) Sperner's lemma plus a limit argument, which needs a formal theory of simplicial subdivisions that does not exist in mathlib; (ii) algebraic topology (no retraction of the ball onto the sphere), for which mathlib has singular homology but not yet the homology of spheres in usable form; (iii) analytic proofs (Milnor–Rogers). None is short; the milestone is deliberately stated for a general nonempty compact convex set in a finite-dimensional normed space so that any route serves, and so the lemma lands in reusable generality.
Given Brouwer, Theorem 1.8 still requires Nash's gain-function construction on the product of simplices and the verification that fixed points are equilibria — bookkeeping-heavy but standard. Theorem 1.11 does not need Brouwer: mathlib's Sion minimax theorem (Mathlib.Topology.Sion) applies to the bilinear payoff on the product of standard simplices, or one can argue by LP duality directly. Theorem 1.17 needs the tight-set/max-flow argument of Lemmas 1.15–1.16 or any direct construction of the equilibrium.
Games are presented concretely: players form a finite index type, strategies a finite type per player, payoffs are functions into ; mixed strategies are weight functions with a IsLottery predicate, not measure-theoretic distributions. Deviations in the equilibrium definition range over all lotteries (not only pure strategies): the pure-deviation reduction is a lemma a solver may prove, not part of the definition. Strategy sets are assumed nonempty in the capstone; the player set need not be. In the zero-sum milestone both dimensions are positive (Fin (m+1), Fin (n+1)), payoffs flow from the column player to the row player, stdSimplex plays the role of the mixed-strategy space, and the Nash-equilibrium conjunct is stated for the Boolean-indexed two-player game built by matrixGameStrat/zeroSumPayoff/matrixGameProfile from the definitions bundle. In the market milestone all supplies and budgets are positive, every buyer's interest set is nonempty, and every good has an interested buyer, matching the standing assumptions of §1.8.1; allocations are recorded as money spent, so the clearing condition is with no division anywhere.
Trivializing readings are ruled out: the empty simplex has no lotteries, so nonemptiness hypotheses appear exactly where their absence would make an existence claim false (Brouwer on the empty set, games with an empty strategy set, zero-dimensional matrix games).
The matrix-multiplication exponent measures the asymptotic arithmetic cost of multiplying square matrices. An upper bound means that, over the field under consideration, matrices can be multiplied using arithmetic operations for every . Improvements to are a central benchmark in algebraic complexity because matrix multiplication is also a basic subroutine in linear algebra, graph algorithms, and symbolic computation.
The existing Prove2Me mission formalizes Schönhage's bound from a concrete two-summand tensor degeneration. The present mission advances the same formal development to the next clean historical construction. Pan and Winograd found a simultaneous approximate algorithm for three matrix products; Romani recorded its tensor form and the parameter choice , , which gives . Schönhage's 1981 paper reports the equivalent bound in the arbitrary-field setting. The exact formal target here is the slightly weaker rational inequality .
For a field , the matrix-multiplication tensor encodes multiplication of an matrix by a matrix:
A direct sum places several such tensors in disjoint coordinate blocks. A tensor has border rank at most when it is a polynomial degeneration of the diagonal tensor . In the Lean development this relation is Degenerates T (TensorObj.diagObj K 3 r). The argument order matters: the first tensor is the target and the diagonal tensor is the source.
The platform already defines ordinary tensor rank, asymptotic tensor rank, the tensor-rank exponent matMulExp K, the equivalent Strassen-preorder exponent matMulExp_strassen K, and Schönhage's asymptotic sum inequality. This mission reuses those declarations. No alternative definition of is introduced.
The goal has exactly the same quantified proposition as the existing mission, with only the rational endpoint changed:
The source construction to be formalized is
Each summand has volume :
The milestone chain records the degeneration, its asymptotic-rank consequence, the exact numerical implication
and the resulting Strassen-form exponent bound. The public goal then transfers the bound to matMulExp K through the already established equality of the two exponent definitions.
Mathematically, this construction improves the concrete exponent certified by the existing mission from to without changing the surrounding theory. It isolates the first genuinely new ingredient after the accepted Schönhage example: a larger simultaneous tensor degeneration rather than a sharper numerical estimate for the old witness.
For formalization, the mission tests whether the current polynomial-degeneration API can express a historically important trilinear aggregation at realistic scale. Once the explicit witness is available, the remaining declarations form a reusable template for later bounds: a source tensor degeneration, an asymptotic-rank bound, a specialization of the asymptotic sum inequality, and a final exponent transfer. This creates a trustworthy stepping stone toward the Coppersmith--Winograd tensor and later laser-method analyses.
The theorem is known mathematically; the open work is its machine-checked Lean formalization. The exact numerical endpoint and every downstream bridge from the degeneration have already been checked locally. The explicit Pan--Winograd degeneration remains the substantive open milestone.
The central difficulty is not the logarithmic comparison. It is constructing and verifying the polynomial family whose leading nonzero coefficient is exactly the tagged direct sum of the three matrix-multiplication tensors and whose earlier coefficients vanish. The family has diagonal source slots and many indexed target coordinates. A proof must account for all mixed-coordinate terms and all cancellations uniformly over an arbitrary field.
Romani's published summary states the approximate-rank inequality but does not spell out a Lean-ready map between its trilinear forms and the platform's TensorObj.bigAdd coordinate spaces. A solver must therefore recover the source indexing carefully and prove that the resulting modewise linear maps have the required coefficients. Reversing the degeneration direction, conflating tensor rank with asymptotic rank, or silently assuming a characteristic-zero scalar identity would invalidate the result.
All theorems quantify over an arbitrary type with [Field K], matching the existing Schönhage goal and the arbitrary-field statement of the source bound. Tensor spaces are finite-dimensional function spaces already packaged by MMObj; the three products are combined with TensorObj.bigAdd. Border rank is represented by the existing finitely supported polynomial-family predicate Degenerates. Because the source summary specifies approximate rank but not a leading order, the main degeneration milestone existentially quantifies that order instead of hard-coding one.
The mission includes no placeholder laser-value definition and makes no claim about the later analysis. It also excludes Schönhage's additional microscopic symmetrization improvement beyond . A valid solution must construct the stated degeneration itself; a vacuous hypothesis or a redefinition of matMulExp is outside scope.
Reusable contributions include coefficient lemmas for polynomial tensor families, finite-index equivalences for direct sums, and generic aggregation identities that specialize to the , witness. Contributions that merely restate the target under stronger field hypotheses do not close the arbitrary-field milestone.
The coupling method of Mission III asks for a coupling of two copies of a chain from every pair of starting states — often painful to construct globally. Chapter 14 of Levin–Peres–Wilmer, Markov Chains and Mixing Times (AMS, 2009) replaces that global demand by a local one. The path coupling technique of Bubley and Dyer says: put a connected graph structure on the state space, and couple one step of the chain only across edges; if each edge contracts in expectation, contraction propagates automatically along paths to arbitrary pairs of distributions. The bookkeeping runs through the transportation metric (Kantorovich distance) between distributions, whose theory — attainment by an optimal coupling, the triangle inequality — is developed on the way. The chapter's payoff is the sharpest elementary bound for sampling proper colorings (Theorem 14.8: the Glauber dynamics mixes in steps once ), and, through the sampling-to-counting reduction of Jerrum–Valiant–Vazirani, a polynomial-time approximation algorithm for counting colorings — the paradigm of the Markov chain Monte Carlo method as an algorithmic tool.
All chains live on a finite state space with a transition matrix ; is the time- distribution from , the total variation distance, the worst-case distance to the stationary distribution , and the mixing time. A coupling of distributions is a distribution on with marginals and .
Given a metric-like cost on pairs of states, the transportation metric between two distributions is the cheapest expected cost of moving one onto the other:
Given a connected graph structure on the state space with edge lengths , the path metric is the least total length of a -path from to .
For the colorings application: a -coloring of the vertices of a graph is proper when adjacent vertices receive distinct colors, and the Glauber dynamics on proper colorings picks a uniform vertex and re-samples its color uniformly among the colors legal there; its stationary distribution is uniform on the proper colorings. Throughout, is the number of vertices and the maximum degree of the graph being colored.
Theorem 14.8, the capstone of Chapter 14: for the Glauber dynamics on proper -colorings, if then
The results. Path coupling converted the coupling method from an art into a calculus: one bounds a single-edge contraction constant, and the machinery does the rest. It is the standard tool for Glauber dynamics on colorings, independent sets, and other constraint-satisfaction models, and the colorings bound is its flagship application. Theorem 14.12 is the discrete embodiment of the Jerrum–Valiant–Vazirani equivalence between approximate counting and sampling — the conceptual foundation of the entire MCMC approach to -hard counting problems.
Formalizing them. Mathlib has no transportation/Kantorovich metric in the finite setting, no path coupling, and nothing on approximate counting. The transportation-metric layer (optimal couplings, triangle inequality) is reusable far beyond this mission — it is the finite Wasserstein distance. The path-coupling theorem feeds directly into Mission IX (Ising) and is quoted throughout modern mixing literature.
The transportation metric asks for minimization over the (compact) polytope of couplings: attainment is a finite-dimensional compactness argument, and the triangle inequality requires gluing two optimal couplings along their common marginal — the classic construction that must be carried out with explicit finite sums here. Path coupling itself is an induction along geodesics of the path metric, with the subtlety that the composite coupling produced along a path need not be optimal, only admissible; the bookkeeping of the contraction constant through the induction is exactly the kind of argument Lean keeps honest. Theorem 14.8 instantiates the machinery: the single-edge coupling for colorings needs a careful case analysis of the proposed recolorings at the two endpoints (matching legal colors bijectively), and the contraction constant emerges from counting disagreeing proposals. Theorem 14.12 layers a probabilistic-amplification argument (medians of means over independent runs) on top of the mixing bound; its combinatorial core — expressing as a telescoping product of marginal probabilities — is elementary but notation-heavy, and the formal statement quantifies over explicit seed spaces, so the whole estimator is a finite object.
The transportation metric is an sInf over coupling costs (the coupling polytope is nonempty for genuine distributions, and attainment is part of the milestone, so the junk value never propagates); the path metric is an sInf over walk lengths in a connected graph. The Glauber dynamics on colorings is the restriction to proper colorings of the single-site heat-bath chain of Mission II, matching §3.3 of the book; its state space is the subtype of proper colorings, nonempty whenever (a fact the hypotheses of the goal supply). Mixing-time upper bounds are stated with the book's explicit ceilings, so no rounding slack is hidden. In Theorem 14.12 the estimator is presented concretely as a function of finitely many uniform seeds, and "with probability " is a counting inequality over the seed space — no measure theory enters. Contributions of intermediate lemmas (optimal-coupling gluing, geodesic decompositions, colorings edge-coupling) are welcome and will be reused by Mission IX.
A reversible Markov chain is an electrical network: states are nodes, and the conductance turns hitting probabilities into voltages and hitting times into resistances. This dictionary, going back to Kakutani and popularized by Doyle and Snell, converts probabilistic estimates into the physical laws of circuits — series/parallel reduction, energy minimization, monotonicity under edge removal. Chapters 9–11 of Levin–Peres–Wilmer, Markov Chains and Mixing Times (AMS, 2009) develop the dictionary and its two crown results: the commute time identity of Chandra–Raghavan–Ruzzo–Smolensky–Tiwari, , and the Matthews method bounding cover times by hitting times with harmonic-number precision.
A network is a symmetric nonnegative conductance function on pairs of vertices; the associated walk moves with probabilities
where , and . A function is harmonic at if . The voltage with boundary values at and at is , the harmonic extension of its boundary data; the current flowing out of has strength , and the effective resistance is . A flow from to is an antisymmetric edge function obeying the node law off ; its energy is . Hitting times , their expectations, the Green's function , the maximal hitting time , and the cover time (expected time to visit every state, maximized over starts) all use the trajectory calculus of Mission I.
This is Proposition 10.6, the commute time identity — the exact bridge between the probabilistic and electrical sides, and the engine of the transience/recurrence theory of Mission XII.
Reversibility and stationarity of the network walk with (§9.1); Proposition 9.1 (existence and uniqueness of harmonic extensions with given boundary values, ); Lemma 9.6 (the Green's function identity ); Theorem 9.10 (Thomson's principle: is the minimal energy of a unit flow, attained); Theorem 9.12 (Rayleigh monotonicity: lowering conductances raises effective resistance); Lemma 10.1 (the random target lemma: does not depend on ); Corollary 10.8 (the resistance triangle inequality); Theorem 11.2 (Matthews: ); Proposition 11.4 (the matching Matthews lower bound over subsets).
The results. The commute time identity computes hitting times from circuit reductions — this is how hitting times on trees, tori and glued graphs are actually evaluated — and, through Thomson and Rayleigh, makes them monotone under graph operations, something invisible probabilistically. The Matthews bounds pin cover times up to a factor in complete generality; they are the tool behind cover-time results for lamplighter groups in Mission XI's sequel. Green's function identities feed Mission XII's recurrence theory, where decides transience.
Formalizing them. Mathlib has graph Laplacians but no electrical network theory: no effective resistance, no flows, no energy, no Thomson/Rayleigh, no hitting or cover times. This mission publishes that layer over the trajectory calculus of Mission I. It is the most reusable single block of the series outside Missions I–II: effective resistance on finite networks is of independent interest to combinatorics (spanning trees, spectral sparsification) well beyond mixing times.
The identity chain behind the goal runs: Green's function of the stopped walk escape probability (via harmonic uniqueness) the Aldous–Fill occupation identity for stopping times with — each step is a manipulation of infinite series of trajectory sums whose exchange steps (splitting a path at its first visit, last-exit decompositions) need summability from Mission I's Lemma 1.13. Thomson's principle is a finite-dimensional convex minimization: existence of the minimizer needs a compactness or completing-the-square argument, and the identification of the minimizer with the current flow needs the cycle law; the naive "differentiate the energy" route must be made exact. Matthews' method is a clean but genuinely clever argument — a uniformly random ordering of targets and the harmonic-number telescoping; the formal cost is the exchangeability of the randomized order against the chain, handled combinatorially.
Networks are functions with a symmetry-and-nonnegativity predicate; loops are permitted; connectivity enters as irreducibility of the induced walk. The voltage is defined probabilistically as (the book's harmonic characterization is Proposition 9.1); is the reciprocal of the explicit current strength, with total division junk when are disconnected — statements carry irreducibility so this does not arise. Energy counts each undirected edge once, formalized as half the ordered double sum, and handles absent edges. Cover times are tail sums of the explicit event "some state unvisited". The Matthews lower bound is stated with an arbitrary lower bound for the pairwise hitting times of the subset — equivalent to the book's min over pairs and easier to instantiate.
Welcome contributions: series/parallel reduction laws, the cycle and node law API for flows, escape-probability lemmas — all reused in Mission XII's infinite-network arguments.
Modern A/B tests must infer lifetime treatment effects — e.g. customer lifetime value under a new feature — from short-horizon experiment data. Chen, Simchi-Levi and Wang (arXiv:2407.19618) model the experiment as a Markov decision process and exploit a structural fact of many practical interventions: the treatment is local, modifying the system at a single crucial state only. This mission formalizes the core asymptotic theory of the paper: for any differentiable estimator built from the experiment's transition and reward statistics, information sharing — pooling across test arms the samples collected away from the treated state — keeps the estimator asymptotically normal with the same asymptotic bias and never increases its asymptotic variance (Theorem 9), and is asymptotically efficient among unbiased estimators (Theorem 5). The route runs through a Markov chain central limit theorem with the asymptotic variance identified as the autocovariance series, and the linearization/delta method for functionals of chain statistics.
Finite Markov chains are the basic model for memoryless random dynamics: card shuffles, random walks on graphs and groups, Monte Carlo samplers, and queueing systems are all chains on a finite state space. The single most used fact about them is that an irreducible chain has exactly one stationary distribution — a probability vector with — and that this is strictly positive and encodes the long-run behaviour of the chain through the return-time identity . Every later result in the theory of mixing times (convergence theorems, coupling bounds, spectral methods, cutoff) is a statement about the distance of the chain from this , so nothing in the subject can be formalized before this mission is.
This mission is the first in a series formalizing D. A. Levin, Y. Peres and E. L. Wilmer, Markov Chains and Mixing Times (AMS, 2009), covering Chapters 1–2: the basic vocabulary of finite chains (stochastic matrices, irreducibility, period, reversibility, time reversal, random walks on graphs and groups) and the classical examples of Chapter 2 (gambler's ruin, coupon collecting, the reflection principle for simple random walk on ). Later missions in the series build on the definitions published here.
A chain on a finite state space is presented by its transition matrix, a matrix with nonnegative entries whose rows sum to . A distribution is a row vector with nonnegative entries summing to ; one step of the chain carries to , and the -step transition probabilities are the entries of the matrix power .
The chain is irreducible if for all states there is a with . The period of a state is where , and the chain is aperiodic if every state has period . A distribution is stationary if , and and are in detailed balance (the chain is reversible) if for all .
Trajectory events over a finite horizon are finite sums of path weights: a length- trajectory is a function , with weight conditional on its starting state. The tail probability of the first hitting time is the sum of the weights of the trajectories from that avoid at times , and expectations of hitting times are recovered by the tail-sum formula , formalized as a tsum over .
This is Corollary 1.17 of the book. It asserts only existence and uniqueness, leaving the finer structure of to the milestones; it is the weakest statement on which the rest of the series can stand, which is why it is the goal.
The milestone list follows the book's own route: well-definedness of the period (Lemma 1.6), positivity of some matrix power for irreducible aperiodic chains (Proposition 1.7), finiteness of expected hitting times (Lemma 1.13), existence of a positive stationary distribution together with
(Proposition 1.14), constancy of harmonic functions (Lemma 1.16), stationarity from detailed balance (Proposition 1.19), the stationary and reversible measure of simple random walk on a graph (Examples 1.12 and 1.20), the time reversal and its path-reversal identity (Proposition 1.22), and the random walks on finite groups of Section 2.6 (Propositions 2.12–2.14). From Chapter 2 the list adds the gambler's ruin formulas and (Proposition 2.1), the coupon collector expectation and tail bound (Propositions 2.3 and 2.4), and the reflection principle and the bound for simple random walk on (Lemma 2.18 and Theorem 2.17).
The result itself. Existence and uniqueness of is the pivot on which the entire quantitative theory turns: it defines the target of convergence, and the identity ties the stationary measure to return times, which later missions use for hitting-time and cover-time results. Detailed balance is the practical tool by which stationary measures of graph and group walks are computed, and the Chapter 2 examples (gambler's ruin, coupon collecting, reflection) are the standard building blocks reused throughout the book — the coupon collector bound, for instance, is exactly the estimate behind the analysis of the top-to-random shuffle in a later mission of this series.
Formalizing it. Mathlib currently has no theory of finite Markov chains: no stochastic-matrix predicate, no stationary distribution, no periodicity, no hitting times. Everything proved in this mission is new formal mathematics, and the definition layer published here (mm_basic, mm_path, mm_classical) is the shared foundation that all twelve subsequent missions of the series import. All results are classical and have textbook proofs; none has a machine-checked proof.
The delicate point is the existence proof. The natural first idea — extract from an eigenvector of for eigenvalue , or invoke a fixed-point theorem — either does not give positivity and nonnegativity without further work, or uses compactness machinery (Brouwer) that is unavailable. The book's proof instead builds and verifies by reindexing trajectory sums; formalizing it requires managing infinite series of path sums (summability from the geometric tail bound of Lemma 1.13, exchanging tsum with finite sums, splitting a trajectory at its last step). The uniqueness half is linear algebra via constancy of harmonic functions (Lemma 1.16), which is elementary but requires a maximum-principle argument over a finite state space. The reflection principle and Theorem 2.17 are finite combinatorics on paths — the bijection is easy to describe and fiddly to implement.
States form a Fintype with decidable equality; chains are Matrix V V ℝ with the row-stochasticity predicate IsStochastic; distributions are functions V → ℝ with the predicate IsDist. Everything is distribution-side: no probability space or measure theory is used. The period is formalized as , which equals when and takes the junk value otherwise. Expectations of hitting times are tsums of tail probabilities, with the usual junk value for non-summable families — the statements are arranged (e.g. multiplicatively, ) so that junk values cannot make them vacuously true. Existence statements carry a Nonempty V hypothesis; irreducibility on the empty space is vacuous, and without nonemptiness the goal would be false, not trivial. The coupon collector and the walk on are presented directly by their driving randomness (uniform draws Fin t → Fin n, uniform sign strings Fin r → Bool), so those probabilities are elementary counting; in particular the reflection principle is stated as an equality of cardinalities of sets of sign strings — this is equivalent to the probabilistic statement because all strings are equally likely.
Contributions welcome beyond the milestone list: simp lemmas for the definition layer, the taboo-matrix representation of avoidance probabilities (useful for Lemma 1.13), and any interface lemmas connecting pathWeight sums to matrix powers — these will be reused by every later mission in the series.
The classical convergence theory of smooth convex minimization. For a function that is -strongly convex and -smooth (), gradient descent converges linearly, while Newton's method exhibits its famous two phases: a damped phase in which every backtracking step decreases the objective by a fixed amount , and a quadratically convergent phase in which the scaled gradient norm squares at each step, . Together they give the iteration count of B&V (9.36),
with the Lipschitz constant of the Hessian and the backtracking parameters. This mission formalizes Chapters 9–10 of Boyd & Vandenberghe with every constant exactly as printed — a quantitative theory entirely absent from Mathlib.
The Karush–Kuhn–Tucker conditions are the central result of convex optimization: for a convex differentiable problem satisfying Slater's condition, a point is optimal exactly when primal feasibility, dual feasibility, complementary slackness and Lagrangian stationarity hold. This mission formalizes Chapters 4–5 of Boyd & Vandenberghe end to end — the first-order optimality criterion, concavity of the Lagrange dual, weak duality, Slater's strong-duality theorem with dual attainment (via the separating-hyperplane argument of §5.3.2), the saddle-point characterization, sensitivity bounds and Pareto scalarization — culminating in the full KKT characterization.
Linear programming has a complete duality theory; integer programming does not — and the Lagrangean dual measures exactly how far duality reaches. This mission formalizes the duality theory of integer programming from Section 11.4 of Bertsimas–Tsitsiklis, built on the general linear programming duality of Section 4.10. For the integer program
with integer data, the complicating constraints are dualized with multipliers over the tractable set : the dual function is
and the Lagrangean dual is . Weak duality (Theorem 11.2) always holds, but strong duality can fail. The convex hull of the integer points of a polyhedron with integer data is itself a polyhedron (Theorem 11.3, Meyer's theorem), and the capstone — Theorem 11.4, the central result of Section 11.4 — identifies the Lagrangean dual exactly: equals the optimal cost of the linear program
. This is the geometric explanation of the strength of Lagrangean relaxation, yields the bound ordering , and Corollary 11.1 characterizes exactly when the bounds collapse. The polyhedral engine is the general weak/strong duality pair (Theorems 4.17/4.18) over a primal s.t. , , and the formulation-strength comparison of Theorem 10.1 supplies the motivating principle that tighter relaxations of the same integer set give sharper bounds.
How can an unbounded polyhedron be described by finitely many geometric objects? Sections 4.8-4.9 of Bertsimas-Tsitsiklis build the cone machinery: recession cones and their rays, extreme rays (defined, like basic solutions, by linearly independent active constraints), the pointedness criterion (Theorem 4.12: is an extreme point of a polyhedral cone iff the cone contains no line iff of the constraint vectors are linearly independent), and the characterization of unbounded linear programs (Theorems 4.13-4.14: over a pointed polyhedral cone, and then over any polyhedron with an extreme point, the optimal cost is iff some extreme ray has ). The capstone is the resolution theorem (Theorem 4.15): a nonempty polyhedron with at least one extreme point equals — the convex hull of its extreme points plus the cone generated by a complete set of its extreme rays. It specializes to Theorem 2.9 / Corollary 4.4 (a nonempty bounded polyhedron is the convex hull of its extreme points) and Corollary 4.5 (a pointed polyhedral cone is generated by its extreme rays). The converse, Theorem 4.16, states that every finitely generated set is a polyhedron — in particular the convex hull of finitely many vectors is a polyhedron. Together these form the Minkowski-Weyl equivalence of the two representations of polyhedra, verified absent from Mathlib and the genuine content of this mission.
Matrix completion — recovering a low-rank matrix from a small random subset of its entries — powers recommender systems and collaborative filtering. In practice it is solved by running (stochastic) gradient descent on the non-convex objective
where and is a certain regularizer. from a random starting point, and it just works.
Ge, Lee and Ma (NeurIPS 2016 Best student paper award) explained why: the regularized objective has no spurious local minima — every local minimum is global and exactly recovers . This mission formalizes that landmark theorem in Lean 4, in its strongest known form and along its simplest known proof: the unified landscape analysis of Ge–Jin–Zheng (ICML 2017) and an improved sampling bound in Chen–Li (JMLR 2019). Conditional on an explicit good-sample predicate (which holds with high probability under Bernoulli sampling), every local minimum of satisfies .
Bandit feedback is only one point on a spectrum: a learner might see more than its own loss (full information) or less (a spam filter never learns what happened to mail it deleted). Chapter 37 of Lattimore–Szepesvári studies finite adversarial games where the loss matrix and the feedback matrix are decoupled. The goal theorem is the celebrated classification theorem: every finite partial-monitoring game has minimax regret exactly , , or — determined by two purely combinatorial conditions, global and local observability, on the game's neighbourhood structure. A single geometric dichotomy thus governs the price of information in every online decision problem with finite actions and feedback.
When actions are feature vectors and the mean reward is linear — — a bandit can generalize across arms: pulling one arm reveals information about all of them. Chapter 19 of Lattimore–Szepesvári carries the optimism principle into this setting: LinUCB (a.k.a. OFUL) plays the action maximizing over the confidence ellipsoid of Mission IX. The goal theorem: with probability , — regret independent of the number of actions. The combinatorial engine is the elliptical potential lemma, bounding how many times adaptively chosen directions can be surprising. Chapter 22's phased elimination with G-optimal design (Mission IX) sharpens this to for finite action sets.
Least-squares estimation from adaptively collected data is the statistical heart of linear bandits: the actions depend on past noise, so classical fixed-design theory does not apply. Chapter 20 of Lattimore–Szepesvári resolves this with the method of mixtures: the process is a supermartingale, and integrating over a Gaussian mixture yields the self-normalized bound — the goal theorem — , valid uniformly over all times. The resulting confidence ellipsoids for the regularized least-squares estimator (Abbasi-Yadkori et al.) calibrate every algorithm of Mission X. The mission also formalizes the Kiefer–Wolfowitz theorem of Chapter 21: G-optimal and D-optimal experimental designs coincide, with optimal value exactly — the classical equivalence theorem of optimal design theory.
Hassler Whitney introduced matroids in On the Abstract Properties of Linear Dependence (Amer. J. Math. 57, 1935, doi:10.2307/2371182) to capture what linear dependence among the columns of a matrix and the circuit structure of a graph have in common. One of the central constructions of the paper is duality. For graphs, duality exists only for planar graphs (Whitney, Non-separable and planar graphs, Trans. AMS 34, 1932); for matroids Whitney showed that every matroid has a dual, and that duality has a direct linear-algebra model: the matroid of a subspace of and the matroid of its orthogonal complement are duals.
Matroid duality underlies much of combinatorial optimization: the duality between cycles and cuts in graphs, the relation between a linear code and its dual code, the exchange of rank and corank in matroid intersection and partition, and the treatment of network flows as duals of potential problems. Whitney's §§11–13 are where this structure is first defined and where its linear-algebra meaning (Theorem 28) is established.
A matroid is a finite set of elements with a rank function on its subsets; equivalently, a family of independent sets, or of bases (maximal independent sets). For a subset write for its number of elements and
for its nullity; and are the rank and nullity of the whole set of elements.
Duality (§11). Let and be matroids and a one-to-one correspondence between their elements. is a dual of (via ) if, for every subset of , with the complement in of ,
The matroid of a subspace (§12). Let be -dimensional Euclidean space with coordinates , and let be a hyperplane through the origin, which in Whitney's usage is a linear subspace of any dimension. For a set of coordinates, project onto the coordinate subspace spanned by the axes , . The matroid associated with has elements , one per coordinate, and gives the set the rank
If is the row space of a matrix , then is the rank of the columns of indexed by , so the associated matroid is the column matroid of .
In the Lean development these objects are WhitneyMatroid.Components.nullity (shared), IsDualVia M M' σ, IsDual M M', subspaceRank H S and IsAssociated M H.
Let be a subspace of and its orthogonal complement. If and are the matroids associated with and , then and are duals under the correspondence of equal coordinates:
The result. Theorem 28 identifies abstract duality with orthogonal complementation. With Theorem 23 it says that the column matroid of a real matrix whose rows span and the column matroid of a matrix whose rows span have complementary bases. This is the basis of the standard representation of the dual of a represented matroid ( and ), of the cycle/cocycle duality of graphs viewed through incidence matrices, and of the fact that a matroid representable over a field has a dual representable over the same field. Theorems 20–23 are the basic facts every later treatment of matroid duality starts from: duality exchanges rank and nullity, is symmetric, always exists, and is characterized by base complements.
Formalizing it. All of these results are classical and proved in Whitney's paper; what this mission adds is a machine-checked development of Whitney's own definition of duality, the rank identity (11.1), rather than the base-complement definition used in modern libraries. Mathlib defines the dual matroid by base complements and proves that it is a matroid; it does not, at the pinned revision, contain the rank formula (11.1) for , the matroid of a real subspace, or Theorem 28. Theorem 6 is already proved on the platform and enters as a reference. Theorems 7 and 8 are close to Mathlib lemmas and are footholds rather than new content.
The difficulty of the goal is not in the combinatorics but in the passage between the two descriptions of a subspace. The natural first idea is to compare the ranks and directly by counting dimensions of and . This does not close: is the dimension of a projection of onto coordinates, while only controls as a whole, and the rank of a complementary coordinate set in is not determined by the dimensions of and alone. Some relation between coordinate projections of and the structure of on the complementary coordinates has to be established. Theorem 27 is itself nontrivial in Lean: the rank function has to be shown to satisfy the matroid axioms, which needs a careful treatment of coordinate projections and of submodularity of dimension. On the abstract side, Theorem 23 needs both directions of the passage between the rank identity and base complements, including the counting step .
Matroid α on a finite type α whose ground set is all of α (Whitney's matroid is its finite set of elements). This finiteness and the ground set convention are part of the definitions; all theorems are stated under them.eRk, finite on a finite type and converted to integers; the nullity and the identity (11.1) are computed in . No truncated subtraction appears.σ : α ≃ β (IsDualVia) or some bijection (IsDual). It is deliberately not defined as Mathlib's M✶: with that definition Theorem 23 would be nearly definitional and Theorem 28 would lose the rank identity. A formalization that replaces (11.1) by the base-complement condition, or states Theorem 28 only for bases, is not this mission's goal.EuclideanSpace ℝ (Fin n) with its standard inner product; the orthogonal hyperplane is Hᗮ. The field is , as in the paper; the analogous statement over other fields with the dot-product annihilator is a generalization and not part of this mission.IsAssociated M H): ground set Fin n, and the rank of every set of coordinates equals the finrank of the image of under the coordinate projection onto (not of , a different set function). Existence and uniqueness is Theorem 27.M✶, and supporting lemmas on coordinate projections are welcome.Mathlib.Combinatorics.Matroid (Dual, Rank). https://github.com/leanprover-community/mathlib4Chapter 4 of F. P. Kelly's Reversibility and Stochastic Networks (Wiley, 1979) applies the product-form theory of Chapter 3 to concrete systems. Two of its results have numbered statements, and they answer two practical questions.
The first comes from the design of store-and-forward communication networks (telegraph and packet-switched data networks). Messages queue at channels; the designer chooses each channel's capacity subject to a budget, and wants to minimize the delay messages suffer. Because the equilibrium law at every channel is geometric under several different modelling assumptions (§4.1, pp. 95–96), the mean delay has a closed form, and the budget allocation problem becomes a small convex program with an explicit solution. This square-root capacity assignment goes back to L. Kleinrock's work on communication nets (Communication Nets, McGraw-Hill, 1964) and remains the textbook example of optimal design for a network of queues.
The second comes from compartmental models in biology, birth–illness–death processes and manpower planning (§4.5, pp. 113–115). Individuals enter a system as a Poisson stream and move through it independently. Equilibrium results follow from Chapter 3; Theorem 4.2 describes the transient behaviour exactly, starting from an empty system.
Capacity allocation (§4.1). A network has channels. Channel receives traffic at average rate and is given capacity . In equilibrium the number of messages at channel has the geometric law (4.1),
which requires . Its mean is . Capacity on channel costs per unit and the total budget is , giving the cost constraint (4.2)
The mean number of customers in the network is
and the feasible set is the set of with for every that satisfy (4.2). In Lean these are meanNumberInNetwork a φ and FeasibleCapacities a f F. The proof works with the Lagrangian lagrangian a f F y φ .
Compartmental model (§4.5). Individuals arrive in a Poisson stream of rate at a system of compartments that is empty at time . Let be the probability that an individual is in compartment a time after its arrival; and , since individuals may leave. Let be the number of individuals in compartment at time , and
In Lean the model is the predicate IsCompartmentModel P ν t p M T Loc, the counts are compartmentCount M Loc j, and is alpha p j t.
If , , and , then
is feasible and minimizes over the feasible set, and every other feasible has .
The proof's generating-function identity, for ,
and the theorem itself: are independent and is Poisson with mean .
Theorem 4.1 is a closed-form design rule. Every channel first receives the capacity needed to carry its traffic; the remaining budget is shared in proportion to , not to the traffic . Sizing capacity in proportion to the load, which is the obvious rule, is therefore not optimal. The same calculation applies to any network whose stations have the geometric law (4.1), for example a manufacturing job shop (p. 97). The mean number in the network and the mean time a customer spends in it are minimized together.
Theorem 4.2 is the exact transient law of a network of infinite-server queues started empty. It holds however complicated the motion of an individual is, provided individuals move independently, and letting it recovers the equilibrium Poisson law of §4.5.
Both results are classical and proved in the book. Neither is formalized on Prove2Me. The platform has the geometric equilibrium law of the M/M/1 queue (KellyStochasticNetworks.mm1_equilibrium) but not its mean, and no result on Poisson thinning or marking by independent random locations. The formal work for Theorem 4.1 is a strict-convexity and Lagrangian-sufficiency argument in . For Theorem 4.2 it is a Poisson marking theorem in measure-theoretic probability. Both pieces are reusable.
For Theorem 4.1, the book's proof sets the partial derivatives of the Lagrangian to zero. A stationary point is not a global minimizer in general, so the formal proof must show that is (strictly) convex on the open region , and it must use Lagrangian sufficiency, not first-order conditions alone. The region is open and the objective is unbounded near its boundary. Feasibility of needs , which the book leaves implicit.
For Theorem 4.2, the steps of the proof that read "conditional on " have to be carried out with measure-theoretic independence. One step averages a product over independent uniform instants. Another sums the Poisson mixture into an exponential. The last turns a factorized generating function into mutual independence of counts with Poisson marginals. Mathlib has the Poisson distribution (ProbabilityTheory.poissonMeasure) but no marking or thinning theorem, and no uniqueness theorem for multivariate probability generating functions.
Channels and compartments are indexed by Fin J; all rates, costs and capacities are real numbers.
Theorem 4.1. The statement carries the book's implicit hypotheses explicitly: , , , . Stability is part of the feasible set. The conclusion is global optimality over the feasible set (IsMinOn) together with feasibility of , plus strict optimality against every other feasible point. Uniqueness is a slight strengthening of the book's "the optimal allocation is", and it holds by strict convexity. A statement that satisfies (4.2), or that it is a stationary point of the Lagrangian, is not the theorem: those are one-line computations or the proof method, and the goal is stated as global optimality to rule them out.
Theorem 4.2. The model is pinned down as in the proof on p. 115:
"Individuals move independently" is formalized as this conditional independence. The are measurable sub-probabilities, not assumed to sum to one. The conclusion is mutual independence of the counts (iIndepFun) together with the Poisson probability mass function of each. Infinite time horizons and the point-process description of the system are out of scope.
Useful contributions: a reusable Lagrangian-sufficiency lemma for separable convex objectives under one linear constraint; a Poisson marking (colouring) theorem for finitely many colours; the multivariate generating-function uniqueness lemma for -valued random vectors.
Many problems in optimization, image reconstruction and statistics reduce to the convex feasibility problem: given closed convex sets , , find a point of their intersection . A classical approach is the relaxation method: from the current point, move to the nearest point of one of the sets, and repeat. For Euclidean distance and half-spaces this is the method of Agmon and of Motzkin and Schoenberg (1954); for hyperplanes it is Kaczmarz's method.
L. M. Bregman's 1967 paper replaced the Euclidean distance by an abstract function satisfying six conditions. The resulting D-projections include what are now called Bregman projections, and the paper is the origin of the Bregman divergence , used today in mirror descent, entropy maximization and the theory of row-action methods (Censor and Zenios, 1997).
The paper proves convergence for two rules for choosing which set to project onto. This mission covers the second one, Theorem 2: always project onto the set that is farthest from the current point in the sense of (the "remotest-set" or maximal-distance control).
Let be a real linear topological space and a family of closed convex subsets of ; the index set is arbitrary and may be infinite. Let be convex with , and let satisfy:
A relaxation sequence starts at and sets ; the sequence of indices is the control. The control is remotest-set if at every step realizes
In the Lean development these objects are DConditions A S D P (conditions I–IV and VI), CondV S D ((⋂ j, A j) ∩ S) (condition V), IsRelaxSeq S P i x (in the series' shared namespace BregmanRelax.Cyclic) and IsRemotestControl D P i x (in BregmanRelax.Remotest).
For every remotest-set control and every relaxation sequence it generates, every limiting point is a common point:
Theorem 2 is the first convergence result for greedy (most-violated-constraint) selection in projection methods with a non-Euclidean distance. Unlike the cyclic Theorem 1 of the same paper, it applies to infinite families of sets, which covers semi-infinite systems of convex inequalities. Lemma 1, the generalized Pythagorean inequality, is the basic estimate for Bregman projections and recurs throughout mirror-descent and row-action analyses; Lemma 2 records the Fejér-type monotonicity of the iterates with respect to .
The results are classical and proved in the paper. To our knowledge they are not formalized in Lean or another proof assistant at this level of generality. The mission produces a machine-checked version of the abstract D-projection framework, with the conditions stated so that the Bregman divergence of §2 of the paper, and the Euclidean distance, are instances.
The obvious argument for Euclidean projections uses Fejér monotonicity and the fact that a bounded sequence in has convergent subsequences. Here neither the triangle inequality nor symmetry of is available, and need not be normed or finite-dimensional. The remotest-set rule controls only the D-distance from the iterate to each projection, in that argument order; turning "these distances tend to zero along a subsequence" into "the limit lies in every " requires the interplay of conditions V and VI, and it must hold uniformly over a possibly infinite index set.
Conventions committed to in Lean:
A trivializing formalization is ruled out: the hypotheses are jointly satisfiable with a genuine run (in with , , , and ), checked locally without sorry, so neither the goal nor the lemmas holds vacuously.
A complete development needs subsequence extraction from sequentially compact sets, one-sided limits of difference quotients, and monotone convergence of real sequences — all available in Mathlib. The D-projection framework, Lemma 1 and Lemma 2 are shared with the cyclic-control mission of the same paper and are reusable for any Bregman-projection algorithm. Proofs of the lemmas, of Theorem 2, and of the instance showing the Euclidean distance satisfies conditions I–VI are welcome.
Discounted dynamic programming with bounded cost per stage is the standard setting in which infinite-horizon sequential decision problems are well posed: the optimal cost exists, satisfies Bellman's equation, and can be computed by iterating the DP operator. Shapley proved this for stochastic games in 1953 (Shapley 1953), Blackwell for discounted Markov decision processes in 1965 (Blackwell 1965), and Denardo observed in 1967 that the arguments use only two properties of the DP operator: monotonicity and contraction in the supremum norm (Denardo 1967). Bertsekas (1975, 1977) and Bertsekas and Shreve (1978) turned this observation into an abstract dynamic programming framework, in which a single mapping encodes stochastic, deterministic, minimax and multiplicative-cost problems at once (Bertsekas 1977).
Chapter 4 of Bertsekas and Shreve, Stochastic Optimal Control: The Discrete-Time Case, is the contraction part of that framework. Its results are the abstract form of what every course on Markov decision processes proves for the discounted case, and they are what later work on abstract DP (Bertsekas, Abstract Dynamic Programming, 2022) and on robust and regularized MDPs builds on.
A model consists of a state space , a control space , a nonempty constraint set for each , a mapping , where is the set of functions , and a function with . is monotone: implies .
A selector is a function with ; is the set of selectors, and a policy is a sequence in . The operators are
The cost of is , the optimal cost is , and is the cost of the stationary policy .
is the Banach space of bounded real functions on with . Assumption C asks for a closed set containing and invariant under and every ; that every limit defining exist and be real; and that for some integer and scalars , ,
Under Assumption C,
implies and implies for ; each is the unique fixed point of in ; and for every
The statement carries no constants beyond those of Assumption C.
Proposition 4.2 is the existence-and-uniqueness theorem for Bellman's equation in the contraction regime, together with the convergence of value iteration from an arbitrary start in . Propositions 4.3 to 4.5 turn it into statements about policies: when a stationary optimal policy exists, how one is found from the DP algorithm, and how much is lost when Bellman's equation is solved only approximately. Proposition 4.11 shows the assumption is met by a concrete class of problems, discounted minimax control, and so certifies that the abstract theorems are not vacuous.
The results are classical and have been proved in print since 1978; none of them is open. What this mission adds is a machine-checked version of the abstract theory itself, rather than of a single model. Mathlib has the Banach fixed point theorem for a contracting map of a complete space (ContractingWith) and a lemma for contracting iterates, but not the version on a closed subset with norm convergence of every orbit, and nothing on abstract DP. The platform has proved the finite-state discounted case for a concrete model (BertsekasDP.discounted_main_theorem); the abstract statements here cover infinite state spaces, minimax problems and -step contractions, and are reused by the later missions of this series (generalized models, Chapter 6) and by papers that cite the book.
The first idea is to apply the contraction mapping principle to and read off as its fixed point. That gives a fixed point of but says nothing about , which is defined as an infimum over all, generally nonstationary, policies of limits of compositions. The identification of the fixed point with is the content of the proposition, and it is where the Lipschitz condition (2) on all of , not only on , enters.
Two further features block a direct appeal to Mathlib. The contraction is only -step, so neither nor need be a contraction. And takes extended-real values, so every passage between and the Banach space must be justified by the invariance of .
The state and control spaces are arbitrary types. is S → EReal; is Mathlib's lp (fun _ : S => ℝ) ⊤, whose norm is the supremum norm, and toF embeds into . is an arbitrary closed subset of , not itself, and uniqueness of fixed points is asserted within . Policies are sequences ℕ → M; applies first. is the pointwise limit (limUnder), which exists and is real under Assumption C; is the infimum over all policies.
The book computes in with , whereas Mathlib's EReal has . No statement adds infinities of opposite sign. A norm bound between functions of is the predicate SupDistLe: both functions are real at every point and differ by at most , which is what the bound means under the book's arithmetic. Condition (2) is imposed on all of , as on p. 53. The scalars of Assumption C are explicit parameters, so the constant of Proposition 4.5 is the book's exact expression. The Fixed Point Theorem assumes nonempty, which the page leaves implicit and without which the statement is false.
Defining as the fixed point of , or replacing it by the infimum over stationary policies, would make the goal trivial. Neither is done here: is the infimum of the policy costs, exactly as in Eq. (8) of Chapter 2.
A complete development needs the -step fixed point theorem on closed subsets of a Banach space, which can be reused well beyond dynamic programming; the elementary calculus of SupDistLe and of the embedding of into S → EReal; and the monotone-operator inequalities of Section 2.1. Contributions of any of these, or alternative proofs of the milestones, are welcome.
A matching of a graph is a set of edges no two of which share a vertex. A matching with as many edges as possible is a basic object of combinatorial optimization. Assignment problems, pairing and scheduling problems, and the Chinese postman problem reduce to it, and it is the standard example of a problem with a polynomial-time algorithm that is not an instance of linear programming over a bipartite structure.
For bipartite graphs the problem was settled by the 1950s, through the theorems of König and Hall and the Hungarian method of Kuhn, whose correctness rests on linear programming duality. As Berge notes on p. 842, that duality "no longer subsists when the graph is not bipartite". C. Berge's three-page note of 1957 (doi:10.1073/pnas.43.9.842) gave the criterion that works for every graph: a matching is maximum exactly when it admits no augmenting chain.
Timeline.
Let be a finite graph without loops or multiple edges, with vertex set and edge set . A matching is a set of edges no two of which have a vertex in common. Its size is its number of edges. A matching is maximum if no matching of has more edges. This is a statement about cardinality: a matching to which no edge can be added (maximal under inclusion) need not be maximum.
Fix a matching . Its edges are strong, all other edges weak. A vertex is neutral if no strong edge contains it, and is the set of neutral points. An alternating chain is a walk in that does not use the same edge twice and in which, of any two consecutive edges, one is strong and the other weak. Vertices may repeat.
For the milestones, Berge adds a new vertex joined by strong edges to every neutral point, which gives a graph . Whenever an alternating chain of runs from to a vertex , its last edge carries an arrow from to . The non-neutral vertices then fall into four classes:
The mission's Lean development uses the same names: IsNeutral, IsAlternatingChain, IsMaximumMatching, barGraph, Arrow, IsInaccessible, IsWeakPt, IsStrongPt, IsMedium.
The statement fixes nothing beyond finiteness: the graph need not be connected, and no bound on its size is assumed.
Theorem 1 reduces the optimality of a matching, a statement about all matchings of the graph, to the absence of one kind of local structure. Its consequences include:
A matching that is not maximum can be certified by exhibiting the chain, and a maximum matching is certified by the labelling of Lemmas 1–4.
Theorem 1 has been proved since 1957 and appears in every textbook on matching theory. It is not formalized in Mathlib at the revision this mission uses. Mathlib has matchings as subgraphs, perfect matchings, alternating cycles and Tutte's theorem, but no augmenting-path characterization. This mission adds a formal statement of the theorem, together with the labelling of Berge's proof in a form that later missions on Edmonds' algorithm can reuse.
The "only if" direction is a local computation: the symmetric difference along an augmenting chain is again a matching, with one more edge. Even this needs care, because an alternating chain is only a trail and may a priori revisit vertices, while the augmentation needs a path.
The converse is the substance. In bipartite graphs, a search from the neutral points that alternates weak and strong edges labels each vertex at most one way, and its failure yields a cover of the same size as the matching. In general graphs an odd cycle lets a vertex be reached both through a strong and through a weak edge (the medium points). The bipartite labelling then fails, and no cover of size need exist: in a triangle the maximum matching has one edge and the minimum cover two. The proof has to treat these odd structures separately, and that is what Lemmas 1 and 4 do.
SimpleGraph V on a Fintype V. The paper's "unoriented graph (or 1-dimensional regular complex)" may have parallel edges, but a matching uses at most one edge of a parallel class, so nothing is lost. The matching is a subgraph M with M.IsMatching, and its size is M.edgeSet.ncard. Maximum means cardinality-maximum over all matching subgraphs. Internally stable sets and covers are Mathlib's IsIndepSet and IsVertexCover.Option V, with = none. An arrow needs an alternating chain of positive length from .IsMedium in Lean, because M names the matching.Any proof of the goal is welcome, whether it follows Berge's induction, uses the symmetric difference of two matchings, or goes through the Tutte–Berge formula. Reusable infrastructure is especially welcome: augmentation along a path, the symmetric difference of two matchings as a union of paths and cycles, and trails that alternate with respect to a matching.
A single machine must process a set of jobs, each with a processing time and a due date, and the cost of a schedule depends on how late the jobs finish. Total tardiness is the classical criterion, but in many applications lateness is penalised more than proportionally: a job one week late costs more than twice a job half a week late, and a quadratic or other convex penalty describes this better. Hamilton Emmons's 1969 paper in Operations Research (DOI 10.1287/opre.17.4.701) derives dominance rules for total tardiness and then asks which of them survive when total tardiness is replaced by for an arbitrary convex nondecreasing loss .
Timeline of the relevant results:
A finite set of jobs is to be sequenced on one machine. Job has a processing time and a due date . All jobs are available at time , and the machine processes them one after another without idle time. A schedule is an ordering of the jobs of . The completion time of in is the sum of the processing times of and of every job before it; its waiting (starting) time is , and its tardiness is
A loss function , convex and nondecreasing on , is fixed, the same for every job. The objective is , and a schedule is optimal if no schedule of has a smaller objective. With this is total tardiness.
Where the paper uses job indices, the jobs are indexed in SPT order: implies , or and . The notation means that some optimal schedule has before ; for a set of jobs, means that some optimal schedule has before every job of , and is the set of jobs of not in . An EDD schedule sequences the jobs in nondecreasing order of due dates.
If an EDD schedule of satisfies
then minimises over all schedules of , for every convex and nondecreasing on .
The goal fixes no constant and no particular : it is a statement about the whole class of convex nondecreasing penalties.
Two further statements of the same section are included as items: Corollary 1.3* (the SPT schedule is optimal if it coincides with the EDD schedule) and Theorem 3 for the generalised objective.
The goal says that EDD is optimal for every convex nondecreasing tardiness penalty as long as no job starts after its due date. The classical sufficient condition, that at most one job is tardy, follows from Jackson's rule; Emmons's condition allows any or all jobs to be tardy, provided each is tardy by at most its own processing time. Because the conclusion holds for the whole class of penalties at once, an instance satisfying it needs no knowledge of : total tardiness, total squared tardiness, and any other convex nondecreasing cost are minimised by the same sequence. Theorems 1* and 2* are the dominance rules that the paper's ordering procedure applies pairwise; they reduce the search space of branch-and-bound methods for convex tardiness objectives.
The results are proved in the paper (for the proofs are said to be "easily established" and omitted). None of them has, to our knowledge, a machine-checked proof. The mission produces formal statements and proofs of the generalised results, including the omitted ones, on top of a reusable single-machine model.
The total-tardiness proofs compare changes in tardiness additively: an interchange is good if the decrease in one job's tardiness is at least the increase in another's. For a convex this comparison is not enough, because a unit of tardiness costs more at higher tardiness levels; the changes must also occur at the right height on the curve, and part (b) of the proof of Theorem 1 fails for this reason. Each generalised argument therefore has to check, for every job whose tardiness changes, both the size and the location of the change, including jobs whose tardiness changes from zero to positive. Ties for the latest due date are a further obstacle: the printed proof of Corollary 2.1 cites Theorem 2, whose hypothesis is strict, so a job sharing the maximum due date is not covered by the argument as written, although the corollary is stated without excluding ties.
Jobs are elements of a type ; the job set is a Finset , processing times and due dates are real functions . Where the paper's index matters, is linearly ordered and its order is the job index, together with the SPT-indexing hypothesis. A schedule is a duplicate-free list whose elements are exactly , and completion times are the published single-machine definition MooreLateJobs.Shared.completionTime (Moore 1968), which starts the machine at time with no idle time. Optimality is against every schedule of . The relation is formalised as the existence of an optimal schedule with before (keeping the premise in the conclusion where the theorem has one); the paper's cumulative reading of the notation is not formalised.
Standing assumptions and deviations:
A trivializing formalization is ruled out: the goal requires optimality of the given EDD list against every schedule of , not of some EDD list, and a sorry-free check shows its hypotheses hold on a two-job instance in which both jobs are tardy.
A complete development needs list lemmas for moving one job to a later position, the effect of such moves on completion times, and slope inequalities for convex functions on . The schedule-manipulation lemmas are reusable for other single-machine sequencing results. Proofs of any milestone, of the two further items, and of general interchange lemmas are welcome.
The gradient method minimizes a differentiable function on by moving from the current point along the negative gradient, . Every practical implementation has to choose the step . A fixed step requires knowing a Lipschitz constant of ; an exact line search is expensive. Larry Armijo's four-page note (Pacific J. Math. 16 (1966) 1–3) proved that any step achieving a sufficient decrease proportional to yields convergence, and gave the halving rule that is now called the Armijo rule or backtracking line search. The rule is the default step-size strategy of gradient, quasi-Newton and Newton-type methods in textbooks such as Nocedal–Wright and Bertsekas, and its sufficient-decrease test reappears in essentially every later line-search analysis.
Timeline:
Let be real Euclidean -space with norm , and let be continuous and bounded below. For a fixed the level set is . Write for the gradient of at . " on " means that is differentiable at every point of and is continuous there.
with when the set is empty. Condition IV forces to be the unique minimizer and the only stationary point in .
For and , the sufficient-decrease set of display (1) is
The steepest descent algorithm uses . The modified steepest descent algorithm fixes , sets , and takes with the smallest positive integer such that
Assume is continuous and bounded below and Conditions III and IV hold at . If , then for every
and every sequence with first term and for all satisfies
The theorem separates the analysis of a gradient method from the choice of its step: whatever rule produces steps in converges, and the two corollaries show that both a fixed step and a halving search started from an arbitrary qualify. The halving rule needs no knowledge of , which is why it became the standard globalization device for descent methods. The hypotheses are local to the level set: need only be Lipschitz on , not on all of , so functions whose gradient is only locally Lipschitz, such as polynomials of degree above two with bounded level sets, satisfy Condition III.
The result is classical and proved on the page; it has not, to our knowledge, been machine-checked in this form. Formalizing it provides a verified sufficient-decrease convergence theorem with level-set-local hypotheses, a verified Armijo backtracking rule with explicit constants , and , and reusable definitions (sufficient-decrease set, Armijo test with halving steps) on which later line-search results can build.
The mean value step in the proof of the descent inequality silently uses the Lipschitz bound along the whole segment from to , but Condition III gives it only on . One has to show that the segment cannot leave the closed set for ; replacing the hypothesis by a global Lipschitz condition is not acceptable, because it changes the theorem. The final step needs Condition IV in its exact form: gradients tending to zero along the iterates do not by themselves give convergence of the iterates, and the positive lower bound away from is what turns stationarity into convergence. Corollary 2 additionally requires a termination argument for the halving search and a uniform lower bound on accepted steps.
is EuclideanSpace ℝ (Fin n), is the norm and is Mathlib's gradient f. All declarations live in the namespace ArmijoGrad.Conv, with one definition file ArmijoGrad.Conv.Setting.
Committed conventions:
Continuous f, BddBelow (Set.range f), ConditionIII f x0 K, ConditionIV f x0 xstar. Milestones keep only the assumptions their claim uses, which makes them more general; nothing is added.gradient f on . The Lipschitz bound of Condition III holds on only; is part of Condition III.ℕ → EuclideanSpace ℝ (Fin n) with first term equal to , the base point of , as in the paper's notation.A trivializing formalization is ruled out: the goal assumes nothing about , the descent inequality or , Condition IV cannot hold vacuously because it names a minimizer, and all hypotheses are satisfiable by nontrivial data: satisfies Conditions III and IV with and , and both algorithms have runs for it.
A complete development needs the one-dimensional mean value inequality along a segment, a continuity argument keeping the segment in , the telescoping bound for monotone sequences bounded below, and the halving-search termination. The definitions file and the descent and Armijo-index lemmas are reusable for later line-search missions. Contributions are welcome on every milestone; the descent inequality is the natural first target.