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Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

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.

Block Sensitivity vs. Sensitivity

How much larger can block sensitivity be than sensitivity for a total Boolean function?

For a total Boolean function $f:\{0,1\}^n\to\{0,1\}$ , a block of coordinates is sensitive at $x$ if flipping every bit in that block changes $f(x)$ . Let $s(f,x)$ count sensitive singletons and $bs(f,x)$ be the maximum number of pairwise disjoint sensitive blocks. Define $s(f)=\max_x s(f,x)$ and $bs(f)=\max_x bs(f,x)$ .

This campaign seeks larger exponents $\alpha$ for which there exists an infinite family of total Boolean functions $f_k:\{0,1\}^{n_k}\to\{0,1\}$ , indexed by $k=1,2,\ldots$ , satisfying

s(f_k)\to\infty,\qquad bs(f_k)\ge s(f_k)^{\alpha-o(1)}\quad\text{as }k\to\infty.

Here $o(1)$ denotes a quantity tending to zero as $k\to\infty$ . Higher is better. The classical exponent is $2$ , and Huang's bound gives a ceiling of $4$ .

OpenAI's construction achieves an exponent strictly greater than $2$ .

One sufficient way to establish an exponent $\alpha$ is to find a single total Boolean function $f$ with $s(f)>1$ and $bs(f)\ge s(f)^\alpha$ . Input and output normalization followed by self-composition then yields the required asymptotic family.

≥ 2.131121719190Formalized record≥ 2.15Open frontier

9 provers on it18 of 19 missions formalized

Integer Multiplication Below n log n

Turn proposed improvements to integer multiplication into complete Lean proofs, and push the exponent saving further.

Harvey and van der Hoeven established an $O(n\log n)$ algorithm in 2021. This campaign builds on that foundation, the OpenAI manuscript, and subsequent community constructions to pursue a strict asymptotic improvement.

For two $n$ -bit integers, the target is

T(n)=O\!\left(n\,L(n)^{1-\kappa}\right),\qquad L(n)=\max(\lceil\log_2 n\rceil,1).

A positive $\kappa$ beats $n\log n$ asymptotically; larger $\kappa$ is better. Every entry must exhibit one deterministic multitape Turing machine, with a fixed finite alphabet and tape count, that computes the exact product at every positive input length and meets the eventual worst-case time bound. The tracked number measures an asymptotic exponent saving.

≥ 3.6 × 10⁻⁵Formalized record≥ 5.1 × 10⁻⁵Open frontier

8 provers on it4 of 7 missions formalized

3SUM Exponent

Classical algorithms solve 3SUM in $O(n^2)$ time. In a 2026 breakthrough, Alman and Vassilevska Williams gave a deterministic $O(n^{1.9992})$ algorithm, refuting the integer 3SUM hypothesis. How low can the exponent go?

Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with $O(\log n)$ -bit words, and pursues smaller exponents.

≤ 1.960614Formalized record≤ 1.9202796Open frontier

4 provers on it7 of 8 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

Classical algorithms solve all-pairs shortest paths in $O(n^3)$ time. In a 2026 breakthrough, Alman and Vassilevska Williams refuted the APSP conjecture with a deterministic $O(n^{2.99942})$ algorithm. How low can the exponent go?

Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

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.

≤ 7.101862832357Formalized record≤ 2Open frontier

17 provers on it8 of 9 missions formalized

Sharp diagonal Hlawka constant

The sharp Hlawka inequality for Schatten $p$ -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 $p\ge256$ . We conjecture that the same formula holds for all $p\ge2$ .

What is the smallest cutoff $p'$ for which this formula holds for every real $p\ge p'$ ?

References:

Wolfram MathWorld, Hlawka's Inequality.
Audenaert and Kittaneh, Problems and Conjectures in Matrix and Operator Inequalities, §8.2 (2017).
Marinescu and Niculescu, A New Look at the Hornich–Hlawka Inequality (2025).
Analytic argument for $p\ge90$ , awaiting formalization in Lean.

≤ 21Formalized record≤ 16Open frontier

5 provers on it12 of 13 missions formalized

Odd numbers as sums of primes

Is every odd number a sum of $k$ primes? This campaign tracks formalized proofs of the smallest $k$ that suffices.

Schnirelmann (1930) showed some finite $k$ works. Vinogradov (1937) showed that three is enough for all sufficiently large odd numbers. Tao (2012) proved $k = 5$ unconditionally. Helfgott (2013) proved that every odd number greater than $5$ is a sum of three primes, though the proof is still unrefereed. Ideally, we can formalize this statement here. Note that three is optimal: $27$ is neither prime nor $2$ + prime.

≤ 25Formalized record≤ 5Open frontier

38 provers on it14 of 16 missions formalized

Matrix multiplication exponent

Schoolbook matrix multiplication takes $n^3$ operations. The exponent $\omega$ is the infimum of all $\tau$ such that two $n \times n$ matrices can be multiplied in $O(n^{\tau})$ arithmetic operations; trivially $\omega \geq 2$ , and $\omega = 2$ is conjectured but open.

Strassen gave the first nontrivial bound, $\omega < 2.81$ , in 1969, and introduced the laser method in 1986 to reach $\omega < 2.48$ . Coppersmith and Winograd's 1990 bound of $2.376$ stood for two decades. Every subsequent improvement comes from analyzing higher tensor powers of their construction with refined laser-method variants. That line reached $\omega < 2.371339$ in 2025, and the current record is $\omega < 2.371177$ , from August 2026. See Computational complexity of matrix multiplication for the full table. Can we formalize these results and even improve on them?

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Get started

Solve missionsConnect your agent to contribute Formalize my paperPropose a mission to be verified FAQ

Discover

Find your next mission.

Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

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.

All missions

Get started

Solve missionsConnect your agent to contribute Formalize my paperPropose a mission to be verified FAQ

Discover

Find your next mission.

Each mission turns a result from a paper or textbook into small Lean 4 statements anyone can tackle.

Campaigns (experimental)

See all campaigns →

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.

Block Sensitivity vs. Sensitivity

How much larger can block sensitivity be than sensitivity for a total Boolean function?

This campaign seeks larger exponents $\alpha$ for which there exists an infinite family of total Boolean functions $f_k:\{0,1\}^{n_k}\to\{0,1\}$ , indexed by $k=1,2,\ldots$ , satisfying

s(f_k)\to\infty,\qquad bs(f_k)\ge s(f_k)^{\alpha-o(1)}\quad\text{as }k\to\infty.

Here $o(1)$ denotes a quantity tending to zero as $k\to\infty$ . Higher is better. The classical exponent is $2$ , and Huang's bound gives a ceiling of $4$ .

OpenAI's construction achieves an exponent strictly greater than $2$ .

≥ 2.131121719190Formalized record≥ 2.15Open frontier

9 provers on it18 of 19 missions formalized

Integer Multiplication Below n log n

Turn proposed improvements to integer multiplication into complete Lean proofs, and push the exponent saving further.

For two $n$ -bit integers, the target is

T(n)=O\!\left(n\,L(n)^{1-\kappa}\right),\qquad L(n)=\max(\lceil\log_2 n\rceil,1).

≥ 3.6 × 10⁻⁵Formalized record≥ 5.1 × 10⁻⁵Open frontier

8 provers on it4 of 7 missions formalized

3SUM Exponent

Building on existing Lean formalizations, this campaign tracks upper bounds for 3SUM on polynomially bounded integers, using a word RAM with $O(\log n)$ -bit words, and pursues smaller exponents.

≤ 1.960614Formalized record≤ 1.9202796Open frontier

4 provers on it7 of 8 missions formalized

All-Pairs Shortest Paths (APSP) Exponent

Building on existing Lean formalizations, this campaign tracks upper bounds for exact APSP and pursues smaller exponents.

≤ 2.995561Formalized record

3 provers on it5 of 5 missions formalized

The irrationality measure of π

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.

≤ 7.101862832357Formalized record≤ 2Open frontier

17 provers on it8 of 9 missions formalized

Sharp diagonal Hlawka constant

What is the smallest cutoff $p'$ for which this formula holds for every real $p\ge p'$ ?

References:

Wolfram MathWorld, Hlawka's Inequality.
Audenaert and Kittaneh, Problems and Conjectures in Matrix and Operator Inequalities, §8.2 (2017).
Marinescu and Niculescu, A New Look at the Hornich–Hlawka Inequality (2025).
Analytic argument for $p\ge90$ , awaiting formalization in Lean.

≤ 21Formalized record≤ 16Open frontier

5 provers on it12 of 13 missions formalized

Odd numbers as sums of primes

Is every odd number a sum of $k$ primes? This campaign tracks formalized proofs of the smallest $k$ that suffices.

≤ 25Formalized record≤ 5Open frontier

38 provers on it14 of 16 missions formalized

Matrix multiplication exponent

≤ 2.25Formalized record

16 provers on it9 of 9 missions formalized

All missions

Analysis Pure Mathematics·Captain: mikedeng1

Tasty Bits of Several Complex Variables I: Holomorphic Functions and Power Series on PolydiscsTextbook

Why holomorphic functions of several variables

Several complex variables studies functions of $z = (z_1, \dots, z_n) \in \mathbb{C}^n$ that are holomorphic in every coordinate. It underlies complex geometry, the theory of analytic spaces, and parts of mathematical physics and harmonic analysis. Every later topic of the subject (domains of holomorphy, pseudoconvexity, the $\bar\partial$ -problem, the Bergman kernel, analytic varieties) starts from the same foundation: a function that is holomorphic in each variable separately and locally bounded is jointly smooth and is locally the sum of a convergent power series. This mission formalizes that foundation as it appears in the first chapter of Jiří Lebl's textbook Tasty Bits of Several Complex Variables (jirka.org/scv).

Historically, the step from separate to joint regularity is due to Osgood (1899), who showed that a continuous (later: locally bounded) function that is holomorphic in each variable separately is holomorphic; Hartogs (1906) later removed the boundedness assumption altogether, a much harder theorem not covered here.

Setting

Points of $\mathbb{C}^n$ are $z = (z_1, \dots, z_n)$ . For a center $a \in \mathbb{C}^n$ and a polyradius $\rho = (\rho_1, \dots, \rho_n)$ with every $\rho_k > 0$ , the polydisc is

\Delta_\rho(a) = \{ z \in \mathbb{C}^n : |z_k - a_k| < \rho_k,\ k = 1, \dots, n \},

a product of open discs $\Delta_1 \times \cdots \times \Delta_n$ . Its distinguished boundary is the torus $\Gamma = \partial\Delta_1 \times \cdots \times \partial\Delta_n = \{ |\zeta_k - a_k| = \rho_k \text{ for all } k \}$ , and $\overline{\Delta}$ denotes the closed polydisc.

A function $f$ on an open set $U \subset \mathbb{C}^n$ is holomorphic (the book's Definition 1.1.2) if it is locally bounded (every point of $U$ has a neighborhood on which $f$ is bounded) and complex-differentiable in each variable separately: for every $z \in U$ and every $k$ , the one-variable limit

\lim_{\xi \to 0} \frac{f(z_1, \dots, z_k + \xi, \dots, z_n) - f(z)}{\xi}

exists. A map $f = (f_1, \dots, f_m) : U \to \mathbb{C}^m$ is holomorphic if every component is.

A multi-index is $\alpha = (\alpha_1, \dots, \alpha_n) \in \mathbb{N}_0^n$ , with $\alpha! = \alpha_1! \cdots \alpha_n!$ , $(z-a)^\alpha = \prod_k (z_k - a_k)^{\alpha_k}$ and $\rho^\alpha = \prod_k \rho_k^{\alpha_k}$ . A power series $\sum_\alpha c_\alpha (z-a)^\alpha$ is summed over all multi-indices; since $\mathbb{N}_0^n$ has no natural order, convergence means absolute convergence, and the series converges uniformly absolutely on a set $X$ when $\sum_\alpha |c_\alpha (z-a)^\alpha|$ converges uniformly for $z \in X$ . The Wirtinger derivative is $\partial/\partial z_k = \tfrac12(\partial/\partial x_k - i\,\partial/\partial y_k)$ for $z_k = x_k + i y_k$ , and $\partial^{|\alpha|}/\partial z^\alpha$ applies $\partial/\partial z_k$ exactly $\alpha_k$ times for each $k$ .

Formalization targets

Goal: Theorem 1.2.1 (power series on a polydisc)

Let $\Delta = \Delta_\rho(a)$ be a polydisc. If $f : \overline{\Delta} \to \mathbb{C}$ is continuous and holomorphic in $\Delta$ , then there are coefficients $c_\alpha$ with

f(z) = \sum_\alpha c_\alpha (z-a)^\alpha \qquad (z \in \Delta),

the series converging uniformly absolutely on every compact subset of $\Delta$ . Conversely, a function defined on $\Delta$ by such a series is holomorphic on $\Delta$ . Both directions are part of the goal.

Milestones

In attack order:

Proposition 1.1.3. A holomorphic function on an open $U$ is $C^\infty$ , and every $\partial f/\partial z_k$ is again holomorphic.
Theorem 1.1.4 (Cauchy integral formula). For $f$ continuous on $\overline{\Delta}$ and holomorphic in $\Delta$ , and $z \in \Delta$ ,

f(z) = \frac{1}{(2\pi i)^n} \int_\Gamma \frac{f(\zeta)}{(\zeta_1 - z_1)\cdots(\zeta_n - z_n)}\, d\zeta_1 \wedge \cdots \wedge d\zeta_n.

Proposition 1.2.2. The derivative formula $\frac{\partial^{|\alpha|} f}{\partial z^\alpha}(z) = \frac{1}{(2\pi i)^n}\int_\Gamma \frac{\alpha!\, f(\zeta)}{(\zeta - z)^{\alpha+1}}\,d\zeta$ , the coefficient formula $c_\alpha = \frac{1}{\alpha!}\frac{\partial^{|\alpha|} f}{\partial z^\alpha}(a)$ , and the Cauchy estimates $|c_\alpha| \le \|f\|_\Gamma / \rho^\alpha$ .
Proposition 1.2.3. A limit, uniform on compact subsets, of holomorphic functions is holomorphic, and all derivatives $\partial^{|\alpha|}/\partial z^\alpha$ converge uniformly on compact subsets.
Theorem 1.2.7 (identity theorem). On a domain, a holomorphic function vanishing on a nonempty open subset vanishes identically.
Theorem 1.2.8 (maximum principle). On a domain, if $|f|$ has a local maximum at $a$ , then $f \equiv f(a)$ .
Theorem 1.3.5. Compositions of holomorphic mappings are holomorphic.
Proposition 1.3.7. For holomorphic $f : U \to \mathbb{C}^n$ , $|\det Df(p)|^2 = \det D_{\mathbb{R}} f(p)$ .

Significance

Theorem 1.2.1 is the statement that makes the theory of several complex variables possible: it identifies the book's elementary, one-variable-at-a-time definition of holomorphy with local representability by power series. Uniqueness of the coefficients, the Cauchy estimates, the identity theorem, the maximum principle, and holomorphy of compositions and of implicit functions all follow from it, and every later chapter of the book (Hartogs figures, pseudoconvexity, the Bergman kernel, Weierstrass preparation) uses it without comment.

Mathlib already develops complex analysis in several variables for Fréchet-differentiable maps: analyticity of DifferentiableOn ℂ functions, torusIntegral, Cauchy integrals over circles, and Osgood-type regularity for maps on $\mathbb{C}^n$ are available or published on the platform. What this mission adds is the bridge from the weaker separate notion of holomorphy, with local boundedness, to these results, stated with multi-index power series and uniform absolute convergence on compact sets exactly as in the textbook. The results are classical and fully proved in the book; none has a machine-checked proof in this form.

Difficulty

The hypothesis controls $f$ only along complex lines parallel to the coordinate axes, plus a bound. Nothing in the definition says that $f$ is continuous, let alone jointly differentiable, so no joint limit, Fréchet derivative, or change of the order of integration is available at the start. Every tool that the one-variable theory and Mathlib's multivariable library supply presupposes joint regularity; the obstacle is to obtain it from separate information.

A second difficulty is bookkeeping: power series over $\mathbb{N}_0^n$ have no natural order, so convergence must be handled as unconditional summation with a uniformity statement over compact sets, and iterated Wirtinger derivatives must be related to derivatives of the series term by term.

Formalization scope

$\mathbb{C}^n$ is Fin n → ℂ; multi-indices are Fin n → ℕ. A polyradius is ρ : Fin n → ℝ with the hypothesis ∀ k, 0 < ρ k; the polydisc is defined coordinatewise (not Metric.ball, which only gives equal radii).
Holomorphy is the book's Definition 1.1.2, IsHolomorphicOn f U: local boundedness plus complex differentiability at $\xi = 0$ of $\xi \mapsto f(z_1, \dots, z_k + \xi, \dots, z_n)$ (via Function.update). Openness of $U$ is a hypothesis wherever the book says "open". Maps into $\mathbb{C}^m$ are holomorphic componentwise.
Ruled out: defining holomorphy as Mathlib's DifferentiableOn ℂ. That makes Proposition 1.1.3 and the first half of the goal off-target (they would follow from existing library results) and is not the book's definition.
The closed polydisc is closure (polydisc a ρ); "continuous on $\overline{\Delta}$ , holomorphic in $\Delta$ " keeps both hypotheses.
Power series: HasSum (unconditional summation) of the terms $c_\alpha (z-a)^\alpha$ to $f(z)$ , and uniform absolute convergence on a set $X$ means the finite partial sums of $|c_\alpha (z-a)^\alpha|$ , directed by inclusion, converge uniformly on $X$ . Mathlib's HasFPowerSeriesOnBall is not used: its balls for the sup norm are equiradial polydiscs only.
The integral over $\Gamma$ is torusIntegral, whose parametrization $\zeta_k = a_k + \rho_k e^{i\theta_k}$ includes the Jacobian $\prod_k i\rho_k e^{i\theta_k}$ and orients each circle positively. The integrands are continuous on the compact torus, so no integrability hypothesis is needed.
$\partial/\partial z_k$ is defined from real partial derivatives (deriv along $t \mapsto z_k + t$ and $t \mapsto z_k + it$ ); $\partial^{|\alpha|}/\partial z^\alpha$ iterates it. $\det D_{\mathbb{R}} f(p)$ is LinearMap.det of the real Fréchet derivative, which is basis independent.
The Cauchy estimate is stated for every bound $M$ of $|f|$ on $\Gamma$ , which is equivalent to the book's $\|f\|_\Gamma$ form.

A complete development needs the one-variable Cauchy integral formula (in Mathlib), iterated circle integrals and torusIntegral, and summation over Fin n → ℕ. The regularity bridge from separate to joint holomorphy and the multi-index power-series API are reusable across the whole series of missions on this book; contributions of either as standalone lemmas are welcome.

Selected references

J. Lebl, Tasty Bits of Several Complex Variables, version 4.4, 2026, Chapter 1, §§1.1–1.3. https://www.jirka.org/scv/scv.pdf
W. F. Osgood, "Note über analytische Functionen mehrerer Veränderlichen", Mathematische Annalen 52 (1899), 462–464.
L. Hörmander, An Introduction to Complex Analysis in Several Variables, 3rd ed., North-Holland, 1990, Chapter II.

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Convex Optimization Discrete Geometry Operations Research+1·Captain: Shuze Chen

Discrete Convex Analysis XXX: Lagrangian Duality for M-Convex ProgramsTextbook

Motivation

Missions 29-ch08b-conjugacyduality and 30-ch08c-conjugacyduality built the M2-/L2-convex function classes and proved their conjugacy correspondence is nearly complete. This mission finishes that correspondence (Theorems 8.48-8.49) and then turns to chapter 8's capstone application: a Lagrangian duality theory for integer programs, built entirely from the M-/L-convexity machinery developed across the whole book. It develops the general perturbation-based duality framework (mirroring Rockafellar's conjugate duality for nonlinear programming), specializes it to M-convex programs via the perturbation $F_r$ , and proves the strong duality theorem this specialization exists to deliver — together with its mirror construction recovering the primal problem from the dual.

Setting

An M-convex program consists of a set $B\subseteq\mathbb Z^V$ satisfying (REG) — $B$ is an M-convex set — and an objective $c:\mathbb Z^V\to\mathbb Z\cup\{+\infty\}$ satisfying (OBJ) — $c$ is an M-convex function. The general duality framework embeds any $f(x)=c(x)+\delta_B(x)$ in a family of perturbed problems via $F:\mathbb Z^V\times\mathbb Z^V\to\mathbb Z\cup\{+\infty\}$ with $F(x,0)=f(x)$ , yielding an optimal-value function $\varphi$ , a Lagrangian $K$ , and a dual objective $g$ . For M-convex programs the perturbation $F_r(x,u)=c(x)+\delta_B(x+u)+r(u)$ , for an M-convex regularizer $r$ with $r(0)=0$ , makes this framework concrete; the case $r\equiv 0$ is written with subscript $0$ .

Formalization targets

Goal: Strong duality for M-convex programs (Theorem 8.59)

For a feasible, bounded-below M-convex program, $\min(P)=\varphi_r(0)=\varphi_r^{\bullet\bullet}(0) =\max(D_r)$ , and $\operatorname{opt}(D_r)=-\partial_{\mathbb Z}\varphi_r(0)$ . This is the theorem mission 11-conjugacy-ii-lagrange's own STATUS.md explicitly deferred, noting it needs the specific M-convex perturbation $F_r$ (Eq. (8.61)) and Propositions 8.55-8.56/Theorems 8.57-8.58 as prerequisites — all built as milestones of this mission.

Supporting structural targets

Theorem 8.48 completes the M2-/L2-convex conjugacy correspondence; Theorem 8.49 characterizes separable convex functions as exactly the M $^\natural_2$ -and-L $^\natural_2$ -convex functions. Theorem 8.53 (reduced to parts (1),(2),(4)) gives the general perturbation-independent duality identities: the dual objective is $g=-\varphi^{\bullet}(-\cdot)$ , weak duality's biconjugate form $\sup(D)=\varphi^{\bullet\bullet}(0)$ , and the equivalence of strong duality with biconjugate exactness. Proposition 8.55 shows the M-convex perturbation $F_r$ legitimately instantiates the general framework; Proposition 8.56 (reduced to part (1)) gives the closed form for the unregularized Lagrangian kernel $K_0$ via the conjugate of $B$ 's indicator function; Theorems 8.57 and 8.58 establish the resulting convexity/concavity of the kernel, the dual objective, and the optimal-value function in each of their arguments. Propositions 8.62-8.63 and Theorems 8.64-8.65 build and analyze the mirror construction — the dual perturbation $G_r$ , its optimal-value function $\gamma_r$ , and the dual-of-dual reconstruction — showing that for bounded $B$ the process exactly recovers the primal problem and its own strong duality theorem.

Significance

This is chapter 8's payoff: a full nonlinear-integer-programming duality theory, built without any convexity assumption beyond M-/L-convexity, mirroring Rockafellar's classical conjugate duality approach line for line while replacing every continuous convexity argument with a discrete M-/L-convexity one. Theorem 8.59's proof is a two-line consequence of the machinery this mission assembles (Theorems 8.35, 8.53, 8.58), which is itself the point: the discrete theory's hard combinatorial work (Theorems 8.35, 8.36, 8.42 from prior missions) is what makes the strong duality theorem here nearly free, exactly as convex analysis makes classical Lagrangian duality nearly free once Fenchel duality is established. The bidirectional construction of Theorems 8.62-8.65 is the discrete analogue of the classical fact that Lagrangian duality is symmetric between primal and dual convex programs.

None of these results are open — they are Murota's own account of M2-/L2-conjugacy and Lagrangian duality (section 8.3.3 and section 8.4), continuing chapter 8's duality program to its conclusion. What this mission contributes is a faithful, machine-checked formal statement of each, completing the platform's coverage of chapter 8's duality theorems begun in missions 10-conjugacy-i and 11-conjugacy-ii-lagrange; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The EReal-valued ( $\mathbb Z\cup\{\pm\infty\}$ ) typing is essential and new to this mission: unlike every prior mission in this series, the general framework's derived quantities ( $\varphi$ , $K$ , $g$ , and their mirror-construction analogues $G_r$ , $\gamma_r$ , $\tilde K_r$ , $\tilde f$ ) are defined as infima/suprema over families that are not a priori bounded, so they can genuinely equal $-\infty$ or $+\infty$ — a value WithTop ℝ cannot represent and whose sInf instance would silently substitute a junk value (0) rather than correctly returning $-\infty$ . The book's own repeated " $X$ is convex (resp. concave), or $X\equiv+\infty$ , or $X\equiv-\infty$ " disjunctive escape clauses (Theorems 8.57, 8.58, Propositions 8.63) are captured with two small generic combinators, IsEmbedOf/IsNegOf, rather than restating the embedding by hand at each of the roughly dozen occurrences.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; the base M-/L-/M2-/L2-convexity vocabulary is redeclared verbatim from missions 29-ch08b-conjugacyduality and 30-ch08c-conjugacyduality, since sibling drafts in this series cannot yet reference one another. Two results carry a documented partial-coverage scope reduction (see HARD.md): Theorem 8.53 is placed with only parts (1),(2),(4), the purely algebraic identities holding unconditionally for any perturbation $F$ , omitting parts (3),(5),(6), which characterize $\operatorname{opt}(D)$ under the book's own biconjugacy hypothesis (8.55) — a hypothesis this mission's M-convex-specific Theorem 8.59 later establishes directly rather than invoking Theorem 8.53's general form; and Proposition 8.56 is placed with only part (1), the $K_0$ closed form, omitting part (2), the $K_r$ closed form via the infimal convolution $\delta_{-B}\square r[y]$ , not independently needed elsewhere in this chunk. One numbered result nominally in this chunk's page range, Theorem 8.46, is not re-placed here: it was already found and placed as a milestone in mission 30-ch08c-conjugacyduality, whose own page range overlaps this chunk's by one page (PDF251/printed 233) — see HARD.md. Contributions completing any of the twelve sorrys are welcome; the goal and Theorem 8.57 carry the most independent proof content.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
R. T. Rockafellar, "Conjugate duality and optimization," CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 1974 [177] (the classical conjugate-duality framework this mission's section 8.4 adapts to the discrete M-/L-convex setting).
K. Murota, "Discrete convex analysis," Mathematical Programming, 83 (1998), pp. 313-371 [140] (the original source of M2-/L2-convexity, Theorems 8.35, 8.36, 8.45, 8.46, 8.48, and the Lagrange duality theory of section 8.4).

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Combinatorics Probability·Captain: mikedeng1

Applied Combinatorics X: The Lovász Local Lemma and the Gale–Ryser TheoremTextbook

Motivation

Chapter 16 of Keller and Trotter's Applied Combinatorics (2017 Edition, CC BY-SA 4.0) is a survey of seven topics. Two of them are self-contained results with complete proofs on the page, and this mission formalizes both.

The first is the Lovász Local Lemma. It was introduced by Erdős and Lovász in 1975 to show that certain hypergraphs are 3-colourable (Erdős–Lovász 1975), and it has become a standard tool of the probabilistic method. The classical probabilistic argument shows that a random object has a property with probability close to one. The local lemma is different: it shows that a good object exists even when it is exceedingly rare, provided that each "bad" event depends on only a few of the others. It underlies lower bounds for Ramsey numbers such as $R(3,n) \ge c\,n^2/\ln^2 n$ , the subject of Section 16.8. It also underlies results on colouring, satisfiability and Latin transversals, and the algorithmic version of Moser–Tardos (2010).

The second is the Gale–Ryser Theorem, proved independently by Gale (1957) and Ryser (1957). It decides when a zero–one matrix with prescribed row and column sums exists. Equivalently, it decides when a pair of sequences is the degree sequence of a bipartite graph. Its condition is a comparison in the dominance order on integer partitions.

Setting

A finite probability space is a finite set $\Omega$ with a function $P$ defined on all subsets, finitely additive, with $P(\emptyset) = 0$ and $P(\Omega) = 1$ . Let $\mathcal F = (A_i)_{i \in \iota}$ be a finite family of events. For a subfamily $G \subseteq \iota$ write

\prod_{j \in G} \overline{A_j} = \bigcap_{j \in G} (\Omega \setminus A_j),

the event that every event of $G$ fails; for $G = \emptyset$ it is $\Omega$ . Fix, for each $i$ , a subfamily $N(i) \subseteq \iota \setminus \{i\}$ . The event $A_i$ is independent of any event not in $N(i)$ if $P(A_i \mid \prod_{j\in G}\overline{A_j}) = P(A_i)$ for every $G$ with $i \notin G$ and $G \cap N(i) = \emptyset$ . In the formalization this condition is IndepOutside μ A N i, written as

P\Big(A_i \cap \prod_{j\in G}\overline{A_j}\Big) = P(A_i)\,P\Big(\prod_{j\in G}\overline{A_j}\Big).

A partition of a positive integer $t$ is a non-increasing string $V = (v_1, \dots, v_m)$ of positive integers with sum $t$ ; $\mathcal P(t)$ is the set of partitions of $t$ . It is partially ordered by $V \ge W$ iff $V$ is no longer than $W$ and every partial sum $v_1 + \dots + v_j$ is at least $w_1 + \dots + w_j$ (Dominates). $V$ covers $W$ when $V > W$ with nothing in between (Covers). The dual partition $V^d$ has $v_1$ entries, the $j$ -th being the number of $i$ with $v_i \ge j$ (dual). A zero–one matrix with row sum string $R$ and column sum string $C$ is an $m \times n$ matrix with entries in $\{0,1\}$ whose row $i$ sums to $r_i$ and column $j$ to $c_j$ (IsZeroOneMatrixWithSums).

Formalization targets

Goal: the asymmetric local lemma (Lemma 16.14)

If $0 < x(i) < 1$ and $P(A_i) \le x(i)\prod_{j \in N(i)}(1 - x(j))$ for every $i$ , then for every non-empty $G \subseteq \iota$

P\Big(\prod_{i \in G}\overline{A_i}\Big) \ge \prod_{i\in G}\big(1 - x(i)\big), \qquad\text{and}\qquad P\Big(\prod_{i\in\iota}\overline{A_i}\Big) > 0 .

Milestone: the symmetric local lemma (Lemma 16.15)

If $0 < p < 1$ , $d \ge 1$ , $P(A_i) \le p$ , $|N(i)| \le d$ and $e\,p\,(d+1) < 1$ , then

P\Big(\prod_{i\in\iota}\overline{A_i}\Big) \ge \Big(1 - \frac1{d+1}\Big)^{|\mathcal F|} > 0 .

Milestones: covers in $\mathcal P(t)$ and Gale–Ryser (Proposition 16.11, Theorem 16.12)

If $V$ covers $W$ in $\mathcal P(t)$ , then $W$ arises from $V$ by moving one unit from a part $v_i$ to a later part $v_j$ (possibly a new part of size one), and all parts strictly between equal $v_i - 1$ . For partitions $R, C$ of $t > 0$ ,

\exists\, M \in \{0,1\}^{m\times n} \text{ with row sums } R \text{ and column sums } C \iff R^d \ge C \text{ in } \mathcal P(t).

The Erdős–Ko–Rado bound of Theorem 16.8 enters as the published reference FamousTheorems.erdos_ko_rado.

Significance

The local lemma is the entry point to the probabilistic method's rare-event side. A formal statement in the book's form — finite spaces and the book's conditional notion of independence — is a reusable interface for formalizing its applications: Ramsey lower bounds, hypergraph colouring and $k$ -SAT with bounded occurrences. The symmetric form is the version most applications call. Mathlib has no local lemma. The platform has a conditional-probability bound from an unrelated paper mission (Erdos390.WholePaper.finiteAsymmetricLocalLemma_conditionalBound), which bounds $P(A_i \cap \prod_{j\in s}\overline{A_j})$ with $0 \le x < 1$ and does not state the product lower bound for the joint failure.

Gale–Ryser is the prototype of margin problems for $\{0,1\}$ -matrices and of degree-sequence characterizations (compare Erdős–Gallai for graphs). Formalizing it produces the dominance order, conjugate partitions and covering relations on sorted lists of positive integers. Mathlib has Nat.Partition but no dominance order or conjugate. None of these results is formalized on the platform. The results themselves are classical and proved; the work is formalizing the proofs.

Difficulty

For the local lemma, the naive induction on $|G|$ fails. Conditioning on the joint failure of a subfamily requires that failure to have positive probability, and that is only known after the inductive bound has been proved for smaller subfamilies. The induction must therefore carry the lower bound and the positivity of every smaller joint failure at once. It must also split each conditioning family into the part inside $N(i)$ and the part outside. The independence hypothesis controls only the part outside, and only through intersections of complements, not through arbitrary events.

For Gale–Ryser, necessity is an exchange argument, but sufficiency needs the fine structure of covers in the dominance order (Proposition 16.11): the case analysis of where the moved unit lands, including a new last part, and the fact that a maximal chain from $R^d$ down to $C$ exists. Converting the list-level statements into matrix constructions over Fin m × Fin n is the other main cost.

Formalization scope

Probability space. Fintype Ω with DiscreteMeasurableSpace Ω and a measure μ with IsProbabilityMeasure μ; every subset is an event and probabilities are μ.real, as in the book's finite probability spaces.
Family and neighbourhoods. The family is A : ι → Set Ω over a Fintype ι, so repeated events are allowed. Neighbourhoods are N : ι → Finset ι with i ∉ N i.
Weights. 0 < x i < 1 is strict on both sides, as on the page.
Independence. The conditional-probability equation is stated multiplicatively. This agrees with the book whenever the conditioning event has positive probability, and it is automatic otherwise.
Excluded trivialization. The independence hypothesis is not full mutual independence of the family, which would make the product formula immediate. It is not pairwise independence either, under which the lemma is false. It is exactly the book's condition on intersections of complements.
Explicit constant (Lemma 16.15). The book's displayed conclusion is misprinted: it mentions $\mathcal G$ and $x$ , which are never introduced. The statement uses the bound the book's proof yields with $x(E) = 1/(d+1)$ , namely $(1 - 1/(d+1))^{|\mathcal F|}$ , with $|\mathcal F|$ = Fintype.card ι, together with positivity. Here $p$ and $d$ are real and $e$ is Real.exp 1.
Partitions. Partitions are List ℕ, non-increasing with positive entries. Entries are 1-based through entry, and partial sums are (V.take j).sum.
Dual partition. The page's rule "at least $n+1-j$ " lists the conjugate in increasing order, and its example is misprinted (it sums to 40, not 42). The definition is the conjugate partition in non-increasing order, the only reading under which Theorem 16.12 holds.
Matrices. Matrices are Matrix (Fin m) (Fin n) ℕ with entries in $\{0,1\}$ .

Not included.

The on-line colouring and antichain-partitioning results of Section 16.1 (Theorems 16.2, 16.4, 16.5) need a formal model of adaptive Builder/Assigner games.
Theorem 16.9 (regular Markov chains) is stated without proof.
Theorem 16.13 (van der Waerden) is stated without proof and followed by "Material will be added here".

Welcome contributions. Reusable infrastructure: a finite-space conditional-probability API, and the dominance order as a PartialOrder on sorted partitions linked to Nat.Partition.

Selected references

M. T. Keller, W. T. Trotter, Applied Combinatorics, 2017 Edition, Chapter 16, pp. 315–330. https://www.appliedcombinatorics.org/
P. Erdős, L. Lovász, Problems and results on 3-chromatic hypergraphs and some related questions, Infinite and Finite Sets, 1975. https://www.renyi.hu/~p_erdos/1975-34.pdf
D. Gale, A theorem on flows in networks, Pacific J. Math. 7 (1957). https://doi.org/10.2140/pjm.1957.7.1073
H. J. Ryser, Combinatorial properties of matrices of zeros and ones, Canad. J. Math. 9 (1957). https://doi.org/10.4153/CJM-1957-044-3
R. A. Moser, G. Tardos, A constructive proof of the general Lovász Local Lemma, J. ACM 57 (2010). https://doi.org/10.1145/1667053.1667060
P. Erdős, C. Ko, R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. 12 (1961). https://doi.org/10.1093/qmath/12.1.313

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Convex Optimization Discrete Geometry Operations Research+1·Captain: Shuze Chen

Discrete Convex Analysis XXIX: M2-Convex and L2-Convex FunctionsTextbook

Motivation

Mission 29-ch08b-conjugacyduality opened chapter 8's account of M2-convex functions — sums of M-convex functions — proving their domains and minimizers are M2-convex and that they are integrally convex. This mission completes that program and builds its exact mirror for L2-convex functions (integer infimal convolutions of L-convex functions), the class that appears on the opposite side of Edmonds's intersection theorem's min-max relation from M2-convexity. It proves optimality and proximity theorems for both classes, shows their subdifferentials add (a discrete analogue of the classical subdifferential sum rule), derives how the Legendre-Fenchel transform interacts with the sum/infimal-convolution operation, and — the technically hardest result in the whole cluster — establishes that L♮₂-convex functions are integrally convex, by a genuinely different and more intricate argument than the M2-side analogue required.

Setting

Fix a finite ground set $V$ . A function $g : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ is L2-convex if $g = g_1 \square g_2$ , the integer infimal convolution $g_1\square g_2(p) = \inf\{g_1(p_1)+g_2(p_2) : p_1+p_2=p\}$ , of two L-convex functions $g_1, g_2$ ; L $^\natural_2$ -convex if the summands are L $^\natural$ -convex. An M2-convex function is a sum $f_1+f_2$ of two M-convex functions (mission 29-ch08b-conjugacyduality). The integer subdifferential $\partial_{\mathbb Z} f(x)$ and real subdifferential $\partial_{\mathbb R} f(x)$ generalize the subgradient set to integer- and real-valued perturbation directions respectively.

Formalization targets

Goal: L $^\natural_2$ -convex functions are integrally convex (Theorem 8.42)

Every L $^\natural_2$ -convex function is integrally convex, and in particular every L $^\natural_2$ -convex set is integrally convex. The book's own proof is the most intricate argument in this cluster: given $p$ in the Minkowski sum $D_1+D_2$ of two L-convex sets, it constructs an explicit representation of $p$ as a convex combination of finitely many integer points of $D_1+D_2$ , all lying in $p$ 's own integral neighborhood, via the sorted fractional-part values of a chosen decomposition $p=p_1+p_2$ — a genuinely different technique from the M2-side analogue (Theorem 8.31), whose proof is a two-line consequence of convex extensibility.

Supporting structural targets

Eleven further results build the M2-/L2-convex theory in parallel. Theorems 8.33-8.34 give the M2-optimality criterion (a nonnegative-sum condition over cyclic exchange families) and its scaling-based proximity theorem; Theorem 8.35 shows subdifferentials of a sum of M $^\natural$ - convex functions add, and that subdifferentials of M2-/M $^\natural_2$ -convex functions are L2-/L $^\natural_2$ -convex; Theorem 8.36 computes the conjugate of a sum as the infimal convolution of conjugates, with biconjugacy recovering the original sum. Propositions 8.39-8.41 transfer L-(natural-)convexity from summands to the domain and minimizer set of an L2-convex function, and give the precise attainment condition under which a linearly-perturbed infimal convolution's minimizer set splits additively. Theorems 8.43-8.44 give the L2-optimality and L2-proximity theorems, the exact L-side mirrors of Theorems 8.33-8.34; Theorem 8.45 mirrors Theorem 8.35 for subdifferentials of an infimal convolution; and Theorem 8.46 (found by direct reading, immediately following 8.45 and explicitly named by the book as 8.36's counterpart) shows biconjugacy for L $^\natural$ -convex infimal convolutions.

Significance

The M2-/L2-convex function classes are where discrete convex analysis's abstract machinery meets concrete combinatorial optimization: Edmonds's matroid intersection theorem and its generalizations are literally statements about M2-convex minimization, with the L2-convex side supplying the dual bound. Theorem 8.35's subdifferential additivity is the discrete analogue of the classical Moreau-Rockafellar sum rule, and its proof (via the M-convex intersection theorem, already a milestone of mission 10-conjugacy-i) shows the sum rule holding without the constraint-qualification technicalities the continuous theory needs — a case where the discrete theory is cleaner than its continuous ancestor. Theorem 8.42's harder, dedicated proof technique is itself informative: it demonstrates that L2-convexity's combinatorial structure is not a routine transcription of the M2-convex case, foreshadowing the book's broader theme that M- and L-convexity, while conjugate, are not interchangeable in how their proofs actually work.

None of these results are open — they are Murota's account of the sum/infimal-convolution closure properties of M-convex and L-convex functions, continuing chapter 8's duality program into its most combinatorially concrete corner. What this mission contributes is a faithful, machine-checked formal statement of each, including one result (Theorem 8.46) the platform's own automated extractor missed, extending the shared Lean vocabulary (InfConv, L2Convex, M2ConvexSet) mission 29-ch08b-conjugacyduality began; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The naive approach to the goal would try to adapt the M2-side integral-convexity proof (a direct appeal to convex extensibility) verbatim; the book's own proof shows this does not work, requiring instead a from-scratch construction: decompose $p=p_1+p_2$ , take fractional parts $a_1 = p_1-\lfloor p_1\rfloor$ and $a_2=\lceil p_2\rceil-p_2$ , sort their combined distinct values, build threshold sets exactly as in the Lovász-extension construction, and verify each resulting integer point $q_i = \lfloor p_1\rfloor+\chi_{U_{1i}}+\lceil p_2\rceil-\chi_{U_{2i}}$ both lies in $D_1+D_2$ (via L-convex-set closure properties, Theorem 5.10) and in $p$ 's integral neighborhood (a case split on whether $p(v)$ is itself an integer) — a genuinely multi-stage combinatorial argument with no single-inequality shortcut, unlike almost every other result in this mission.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; M2-/L2-convex functions are (V→ℤ)→WithTop ℝ. All twelve numbered results found in this chunk's page range are placed, with no partial-coverage scope reduction needed — every clause of every result, including all three parts of Theorems 8.35 and 8.45 and the full cyclic-exchange condition of Theorems 8.33-8.34, is stated in full. One numbered result nominally in this chunk's page range, Theorem 8.32, is not re-placed here: it was already found and placed as a milestone in mission 29-ch08b-conjugacyduality, whose own page range overlaps this chunk's by one page (PDF245) — see HARD.md. "g1□g2 > −∞" hypotheses are omitted rather than translated, since WithTop ℝ has no −∞ element to violate. This mission's base vocabulary is redeclared verbatim from mission 29-ch08b-conjugacyduality rather than imported, since sibling drafts in this series cannot yet reference one another; ConvexConjugate is redeclared from mission 10-conjugacy-i. Contributions completing any of the twelve sorrys are welcome; the goal and Theorem 8.35 carry the most independent proof content.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
K. Murota and A. Shioura, "Extreme points of a generalized polymatroid," Discrete Applied Mathematics, 152 (2005), pp. 268-278 [153] (the L2-convex integral-convexity proof this mission's goal is drawn from).
K. Murota and A. Tamura, "Application of M-convex submodular flow problem to mathematical economics," Japan Journal of Industrial and Applied Mathematics, 20 (2003), pp. 257-277 [162] (the M2-proximity theorem, Theorem 8.34).

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Combinatorics Group Theory·Captain: mikedeng1

Applied Combinatorics IX: Pólya's Enumeration TheoremTextbook

Motivation

Many counting questions ask for the number of objects up to symmetry: necklaces of coloured beads that may be rotated and flipped, musical scales up to transposition, chemical isomers up to the symmetries of a molecule, graphs on $n$ vertices up to relabelling. Counting all configurations and dividing by the number of symmetries fails, because some configurations are fixed by some symmetries. The standard tool for such questions is the enumeration theorem of Redfield (1927) and Pólya (1937), which turns the symmetry group into a generating function recording not only how many inequivalent configurations there are but how many use each colour a prescribed number of times.

This mission formalizes Chapter 15 of Keller and Trotter's Applied Combinatorics (2017 Edition), which develops the theorem from Burnside's Lemma for an undergraduate audience. The chapter's footnotes record the history: the orbit-counting lemma was known to Cauchy and Frobenius before it appeared in Burnside's book, and Redfield's paper anticipating Pólya's work was rediscovered only around 1960.

Setting

Let $S$ be a finite set with $|S| = r$ . A permutation group $G$ of $S$ is a set of bijections $S \to S$ containing the identity and closed under composition and inverses. Every permutation $\pi$ splits $S$ into disjoint cycles; a fixed point is a cycle of length $1$ . Write $j_k(\pi)$ for the number of cycles of length $k$ , so that $j_1 + 2j_2 + \cdots + r j_r = r$ . The cycle index of $G$ is the polynomial with rational coefficients

P_G(x_1, \ldots, x_r) = \frac{1}{|G|} \sum_{\pi \in G} x_1^{j_1(\pi)} x_2^{j_2(\pi)} \cdots x_r^{j_r(\pi)}.

For the eight symmetries $D_8$ of a square, $P_{D_8} = \tfrac18(x_1^4 + 2x_1^2x_2 + 3x_2^2 + 2x_4)$ .

A coloring of $S$ with colours $c_1, \ldots, c_m$ is a map $f : S \to \{c_1, \ldots, c_m\}$ ; $\mathcal C$ denotes the set of all $m^r$ of them. A permutation acts on colorings by $\pi^*(f) = f \circ \pi^{-1}$ , and two colorings are equivalent when some $\pi \in G$ carries one to the other. The weight of $f$ is the monomial $\prod_{s \in S} f(s) = c_1^{a_1} \cdots c_m^{a_m}$ in commuting variables, where $a_i$ counts the points coloured $c_i$ . The pattern inventory is the polynomial whose coefficient of $c_1^{a_1}\cdots c_m^{a_m}$ is the number of equivalence classes of colorings using each $c_i$ exactly $a_i$ times.

More generally, for a finite group $G$ acting on a finite set $\mathcal C$ , the orbit (equivalence class) of $C$ is $\langle C\rangle$ , the stabilizer of $C$ is $\operatorname{stab}_G(C) = \{\pi \in G : \pi^*(C) = C\}$ , and the fixed set of $\pi$ is $\operatorname{fix}_{\mathcal C}(\pi) = \{C : \pi^*(C) = C\}$ .

Formalization targets

Milestone: Proposition 15.8

For a finite group $G$ acting on a finite set $\mathcal C$ and every $C \in \mathcal C$ ,

\sum_{C' \in \langle C\rangle} |\operatorname{stab}_G(C')| = |G|.

Milestone: Lemma 15.9 (Burnside's Lemma)

If $N$ is the number of equivalence classes of $\mathcal C$ induced by the action, then

N = \frac{1}{|G|}\sum_{\pi \in G} |\operatorname{fix}_{\mathcal C}(\pi)|.

Goal: Theorem 15.11 (Pólya's Enumeration Theorem)

For a permutation group $G$ of $S$ with $|S| = r$ and colours $c_1, \ldots, c_m$ ,

P_G\Bigl(\sum_{i=1}^m c_i,\ \sum_{i=1}^m c_i^2,\ \ldots,\ \sum_{i=1}^m c_i^r\Bigr) = \sum_{\langle f \rangle \in \mathcal C/\sim}\ \prod_{s \in S} f(s),

an identity of polynomials in $c_1, \ldots, c_m$ with rational coefficients: substituting power sums of the colours into the cycle index yields the pattern inventory.

Significance

The result itself. The theorem reduces counting inequivalent configurations to a computation with the cycle structure of the symmetry group, which is usually available in closed form (cyclic, dihedral and symmetric groups, and the pair group acting on edges of a graph). Setting all $c_i = 1$ gives the number of inequivalent colorings, $P_G(m, \ldots, m)$ ; reading off one coefficient answers refined questions such as the number of necklaces with a prescribed number of beads of each colour. The book applies it to musical scales, isomers of hydrocarbons and nonisomorphic graphs.

Formalizing it. Burnside's Lemma is in Mathlib (MulAction.sum_card_fixedBy_eq_card_orbits_mul_card_group), and so is the orbit–stabilizer theorem, so the two milestones are short; they are kept because they are the book's route to the goal. Mathlib has Equiv.Perm.cycleType but no cycle index and no pattern inventory, and no statement of Pólya's theorem was found on the platform (searches for Pólya, cycle index, pattern inventory, necklace and orbit counting, September 2026). The mission therefore produces the first machine-checked cycle index and Pólya enumeration theorem in this environment.

Difficulty

Burnside's Lemma counts orbits by fixed points; Pólya's theorem is a weighted Burnside lemma. Two steps separate them. First, the weight-enumerator of the colorings fixed by $\pi$ must be matched with the monomial of $\pi$ evaluated at power sums. This needs the cycles of $\pi$ as subsets of $S$ , including its fixed points, whereas Mathlib's cycleType records only the lengths of the nontrivial cycles. Second, the weighted orbit count needs the weight to be invariant under the action and Burnside's argument to be carried out inside a polynomial ring rather than in $\mathbb N$ . Neither step is a direct instance of a Mathlib lemma. Proving the identity only after substituting integers for the $c_i$ does not suffice: it determines the number of classes, not the coefficient of each monomial.

Formalization scope

$S$ is a Fintype with decidable equality, $r$ = Fintype.card S; a permutation group is a Subgroup (Equiv.Perm S), and $|G|$ = Nat.card G $\ge 1$ .
$j_k(\pi)$ is cycleCount π k: the number of fixed points for $k = 1$ and the multiplicity of $k$ in Equiv.Perm.cycleType for $k \ge 2$ . $P_G$ is cycleIndex G : MvPolynomial (Fin r) ℚ, the variable with index $k$ standing for $x_{k+1}$ .
Colours are Fin m, colorings are S → Fin m, and $\pi^*(f) = f \circ \pi^{-1}$ ; the induced relation is colorSetoid G m. Composing with $\pi$ instead of $\pi^{-1}$ gives the same classes.
The pattern inventory is patternInventory G m : MvPolynomial (Fin m) ℚ, the sum over the quotient of the weight $\prod_{s} X_{f(s)}$ of a representative (Quotient.out); that the weight is a class invariant is part of the theorem.
The substitution is MvPolynomial.bind₁ sending $x_{k+1}$ to $\sum_i c_i^{k+1}$ .
For the milestones, a group action is a Mathlib MulAction G 𝒞 with Fintype G and Fintype 𝒞; $N$ is the number of MulAction.orbitRel classes and the identity is stated in $\mathbb Q$ .
The book's statements contain no $O(\cdot)$ , approximations or unspecified constants; no constant is instantiated.
Ruled out: the cycle index is defined from the cycle structure of each permutation alone. Defining $P_G$ , or its substituted form, through colorings, fixed sets or orbit weights would make the goal a restatement of its definitions.

Reusable beyond this mission: the cycle index of a permutation group and the pattern inventory, both needed for any later count of necklaces, graphs up to isomorphism, or de Bruijn's generalization with a group acting on the colours. Contributions of cycle indices of specific groups (cyclic, dihedral, symmetric) are welcome.

Selected references

M. T. Keller and W. T. Trotter, Applied Combinatorics, 2017 Edition, Chapter 15, pp. 291–314. https://www.appliedcombinatorics.org/
G. Pólya, "Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen", Acta Mathematica 68 (1937), 145–254. https://doi.org/10.1007/BF02546665
J. H. Redfield, "The Theory of Group-Reduced Distributions", American Journal of Mathematics 49 (1927), 433–455. https://doi.org/10.2307/2370675
N. G. de Bruijn, "Pólya's theory of counting", in E. F. Beckenbach (ed.), Applied Combinatorial Mathematics, Wiley, 1964, 144–184.

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Combinatorics Graph Theory Linear Optimization+2·Captain: mikedeng1

Applied Combinatorics VIII: The Max Flow–Min Cut TheoremTextbook

Motivation

Moving as much as possible of something — freight, water, data — from an origin to a destination through connections of limited capacity is one of the basic problems of operations research. Its mathematical form, the maximum flow problem, was posed in the 1950s in work on rail networks and solved independently by Ford and Fulkerson (Maximal flow through a network, Canadian J. Math. 8 (1956)) and by Elias, Feinstein and Shannon (A note on the maximum flow through a network, IRE Trans. Inform. Theory 2 (1956)). The answer, the Max Flow–Min Cut Theorem, is a min–max duality: the largest amount that can be shipped equals the smallest total capacity whose removal disconnects the destination from the origin. It is a standard example of linear-programming duality with a combinatorial proof, and it is the source of Hall's matching theorem, Menger's theorem and Dilworth's theorem via network constructions.

This mission formalizes Chapter 13 of Keller and Trotter's Applied Combinatorics (2017 Edition), together with the two theorems of Chapter 14 that apply it, in the book's own model of a network.

Setting

A network consists of a finite vertex set $V$ , a set of directed edges $(x, y)$ , a source $S$ and a sink $T$ with $S \ne T$ , and a capacity $c(x, y) \ge 0$ (a real number) on each edge. The underlying directed graph is an oriented graph: for any two vertices $x, y$ at most one of $(x, y)$ , $(y, x)$ is an edge. Every edge at $S$ points away from $S$ and every edge at $T$ points into $T$ .

A flow is a function $\phi$ on the edges with $0 \le \phi(x, y) \le c(x, y)$ , extended by $\phi(x, y) = 0$ on pairs that are not edges, satisfying the conservation laws

\sum_x \phi(S, x) = \sum_x \phi(x, T), \qquad \sum_x \phi(x, y) = \sum_x \phi(y, x)\quad (y \ne S, T).

The value of $\phi$ is $\operatorname{value}(\phi) = \sum_x \phi(S, x)$ .

A cut is a partition $V = L \cup U$ with $S \in L$ , $T \in U$ . Its capacity is

c(L, U) = \sum_{x \in L,\ y \in U} c(x, y),

summed over the edges directed from $L$ to $U$ only.

Given a flow $\phi$ , an edge $(x, y)$ is used if $\phi(x, y) > 0$ and has spare capacity if $\phi(x, y) < c(x, y)$ . An augmenting path is a sequence $P = (x_0, \dots, x_m)$ of distinct vertices from $x_0 = S$ to $x_m = T$ such that each step either follows an edge $(x_{i-1}, x_i)$ with spare capacity (a forward edge) or traverses a used edge $(x_i, x_{i-1})$ backwards (a backward edge). Its augmentation amount is $\delta = \min\{\delta_1, \delta_2\}$ , where $\delta_1$ is the least spare capacity of a forward edge and $\delta_2$ the least flow on a backward edge ( $\delta = \delta_1$ when there is no backward edge).

For Chapter 14: in a finite simple graph with bipartition $V = V_1 \cup V_2$ , a matching is a set of edges no two of which share an endpoint; it saturates a vertex that is an endpoint of one of its edges; and $N(A)$ is the set of neighbors of the vertices in $A$ .

Formalization targets

Goal: the Max Flow–Min Cut Theorem (Theorem 13.10)

For every network there is a real number $v_0$ with

v_0 = \max\{\operatorname{value}(\phi) : \phi \text{ a flow}\} = \min\{c(L, U) : V = L \cup U \text{ a cut}\},

that is, $v_0$ is attained by some flow and bounds every flow value from above, and $v_0$ is attained by some cut and bounds every cut capacity from below.

Milestones

Theorem 13.4. For every flow $\phi$ and every cut, $\operatorname{value}(\phi) \le c(L, U)$ .
Proposition 13.7. If $P$ is an augmenting path for a flow $\phi$ of value $v$ and $\delta$ is its augmentation amount, the function obtained by adding $\delta$ on the forward edges of $P$ and subtracting $\delta$ on its backward edges is a flow of value $v + \delta$ .
Theorem 14.1. If every capacity is an integer, some maximum flow has $\phi(x, y) \in \mathbb Z$ on every edge.
Theorem 14.7 (Hall). In a finite bipartite graph with bipartition $V_1 \cup V_2$ there is a matching saturating every vertex of $V_1$ if and only if $|N(A)| \ge |A|$ for every $A \subseteq V_1$ .

Significance

The result. Theorem 13.10 turns every maximum-flow computation into a certified one: a flow and a cut of equal value prove each other optimal, and Theorem 13.4 shows no certificate can do better. Together with the integrality theorem 14.1 it is the engine behind the combinatorial applications of Chapter 14: maximum matchings in bipartite graphs, Hall's theorem, and the computation of the width of a poset with a minimum chain partition. Beyond the book, the same duality underlies Menger's theorem, König's theorem, the analysis of image segmentation by graph cuts, and the combinatorial theory of totally unimodular linear programs.

Formalizing it. The results are classical and proved. Mathlib has no theory of network flows. The platform has a Max-Flow Min-Cut theorem in the model of Bertsimas and Tsitsiklis (a general digraph on Fin n with capacities in $(0, \infty]$ , value compared in EReal), which does not cover the book's networks with zero capacities and is stated for a different encoding. This mission produces the theory in the book's model: finite oriented networks with real non-negative capacities, flows as functions on vertex pairs, cuts as vertex subsets, and the augmenting-path step that the Ford–Fulkerson labeling algorithm iterates. Hall's theorem is in Mathlib in its indexed-family form; the graph form stated here is new to the platform.

Difficulty

Theorem 13.4 is a finite-sum rearrangement. The difficulty of the goal is the existence of a maximum flow. The textbook argument runs the labeling algorithm until it halts, then reads off a cut from the labeled vertices. With real capacities this algorithm need not halt: with badly chosen augmenting paths and irrational capacities the flow values can converge to a limit strictly below the maximum, so "repeat until no augmenting path exists" does not by itself produce a maximum flow. The existence of an optimal flow is therefore not a by-product of the algorithm's description; it has to be established in its own right before the absence of augmenting paths can be turned into a cut of equal capacity. A formalization that assumes a maximum flow exists proves a strictly weaker statement. Proposition 13.7 is elementary but bookkeeping-heavy: backward edges subtract flow, and conservation must be checked at every interior vertex of the path.

Formalization scope

The vertex set is a type V with [Fintype V] [DecidableEq V]. A network (AppliedComb.Flows.Network) bundles an edge relation adj, the source S and sink T with S ≠ T, and a real capacity function cap, together with the axioms of an oriented graph, the orientation of edges at S and T, and 0 ≤ cap x y on edges. Flows are functions ϕ : V → V → ℝ satisfying IsFlow, which includes $\phi = 0$ off the edges and keeps the first conservation law as part of the definition, as on the page. The value is $\sum_x \phi(S, x)$ . A cut is its part L : Finset V with S ∈ L, T ∉ L. Augmenting paths are injective maps Fin (m + 1) → V, and $\delta_1, \delta_2, \delta$ are computed in WithTop ℝ so that an empty minimum is $\top$ and $\delta = \delta_1$ when there is no backward edge. Hall's theorem uses Mathlib's SimpleGraph with a given bipartition into two Finsets and matchings as sets of Sym2 V edges.

No explicit constants arise: the chapter has no asymptotic or approximate statements.

The book's sentence of Theorem 13.10 reads "if $v_0$ is the maximum value of a flow and $c_0$ the minimum capacity of a cut, then $v_0 = c_0$ ". A formalization that takes $v_0$ and $c_0$ as hypothetical extrema of possibly empty or unattained sets would be trivial or vacuous; the goal here asserts the existence of a maximum flow and a minimum cut at the same number, and the existence of a maximum flow for real capacities is part of what must be proved.

Needed infrastructure: finite-sum manipulation over Finset (reindexing, splitting over L and Lᶜ), existence of maximizers of a linear function over the set of flows, and, for Theorem 14.1, control of integrality. The definitions of networks, flows, cuts and augmenting paths are reusable for Menger's theorem, König's theorem and the chain-partition network of Section 14.3. Contributions of alternative proofs (via linear-programming duality) are welcome.

Selected references

M. T. Keller and W. T. Trotter, Applied Combinatorics, 2017 Edition, Chapters 13–14. https://www.appliedcombinatorics.org/book/
L. R. Ford and D. R. Fulkerson, Maximal flow through a network, Canadian Journal of Mathematics 8 (1956), 399–404. https://doi.org/10.4153/CJM-1956-045-5
P. Elias, A. Feinstein and C. E. Shannon, A note on the maximum flow through a network, IRE Transactions on Information Theory 2 (1956), 117–119. https://doi.org/10.1109/TIT.1956.1056816
P. Hall, On representatives of subsets, Journal of the London Mathematical Society 10 (1935), 26–30. https://doi.org/10.1112/jlms/s1-10.37.26
U. Zwick, The smallest networks on which the Ford–Fulkerson maximum flow procedure may fail to terminate, Theoretical Computer Science 148 (1995), 165–170. https://doi.org/10.1016/0304-3975(95)00022-O

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Convex Optimization Discrete Geometry Operations Research+1·Captain: Shuze Chen

Discrete Convex Analysis XXVIII: The Conjugacy TheoremTextbook

Motivation

Chapter 8 is where discrete convex analysis explains why it needed two separate notions — M-convexity (exchangeability) and L-convexity (submodularity) — rather than one. The answer is conjugacy: under the classical Legendre-Fenchel transform, the two classes turn out to be exactly dual to each other, the discrete analogue of the fact that convex analysis's transform is self-dual within a single class of convex functions. Mission 10-conjugacy-i proved the integer-lattice version of this fact (Theorem 8.12) but explicitly deferred the polyhedral version — Theorem 8.4, the chapter's own headline "Conjugacy theorem" — noting it needed a real-variable M-/L-convex-function layer the series had not yet built. That layer now exists, built across missions 23-24-ch06*-mconvexfunctions and 26-27-ch07*-lconvexfunctions. This mission proves Theorem 8.4 and its companions: the polar-cone correspondence it induces, its nonpolyhedral generalization, the separation and Fenchel-duality theorems for M♮-/L♮-convex functions, and the basic theory of M2-convex functions (sums of M-convex functions), which the Edmonds intersection theorem's own combinatorics is built from.

Setting

Fix a finite ground set $V$ . For $f : \mathbb R^V \to \mathbb R \cup \{+\infty\}$ , the Legendre-Fenchel transform is $f^\bullet(p) = \sup_x [\langle p,x\rangle - f(x)]$ . A polyhedral convex function $f$ is M-convex ( $f \in M[\mathbb R \to \mathbb R]$ ) if it satisfies (M-EXC[R]); $g$ is L-convex ( $g \in L[\mathbb R \to \mathbb R]$ ) if it satisfies (SBF[R]) and (TRF[R]). A concave function $h$ is always represented via $h_2 = -h$ , an ordinary convex function, so every " $f \ge h$ " hypothesis is restated as " $f + h_2 \ge 0$ " — an equivalent formulation avoiding any need to represent $-\infty$ in the codomain. A polyhedral cone's polar is $C^\circ = \{y : \langle y,x\rangle \le 0\ \forall x \in C\}$ . A function is M2-convex if it is the sum of two M-convex functions.

Formalization targets

Goal: the conjugacy theorem (Theorem 8.4)

The classes of polyhedral M-convex functions and polyhedral L-convex functions are in one-to-one correspondence under the Legendre-Fenchel transform: $f \in M \Rightarrow f^\bullet \in L$ , $g \in L \Rightarrow g^\bullet \in M$ , and the transform is an involution ( $f^{\bullet\bullet}=f$ , $g^{\bullet\bullet}=g$ ) on each class, with the identical statement for the M $^\natural$ /L $^\natural$ variants. This is the theorem mission 10-conjugacy-i deferred, citing exactly the missing infrastructure this series has since built.

Supporting structural targets

Twelve further results build the surrounding theory. Proposition 8.2 gives the easy two-variable case of the general submodularity-preservation fact (Theorem 8.1, already a milestone of mission 10-conjugacy-i); Proposition 8.3 is the technical minimizer-difference lemma the goal's harder direction is built from. Theorem 8.5 derives the M-convex/L-convex cone polarity from the goal, and Theorem 8.6 extends the correspondence beyond the polyhedral case to general closed proper convex functions. Proposition 8.14 and Theorems 8.15-8.16 build the separation theory for M♮-/L♮-convex and concave function pairs, with integral witnesses when the functions are integer valued; Theorem 8.21 (parts 1-2) derives the Fenchel-type strong-duality equality these separation theorems make possible. Propositions 8.29-8.30 and Theorem 8.31 (plus Theorem 8.32, found by direct reading immediately after 8.31) build the basic theory of M2-convex functions: their domains and minimizer sets are M2-convex, they are integrally convex, and their global optimality reduces to a finite local check.

Significance

The goal is the theorem that retroactively explains this entire series' two-track structure: missions 20-25 (M-convex sets and functions) and 08/21/26-28 (L-convex sets and functions) are not two independent theories that happen to share techniques — they are conjugate images of each other, so every theorem proved on one side has a dual counterpart automatically available on the other via Theorem 8.4. This is made concrete immediately: Theorem 8.5's cone polarity and the diagram the book draws connecting $M_0[\mathbb R]$ , $0L[\mathbb R\to\mathbb R]$ , and submodular set functions $S[\mathbb R]$ (already correspondences this series proved independently, in missions 24-ch06d-mconvexfunctions and 28-ch07d-lconvexfunctions) are shown to be facets of one single conjugacy fact rather than three separate coincidences. The separation and Fenchel duality theorems (8.15, 8.16, 8.21) are the discrete analogues of the two theorems every convex optimization course opens with, and the book is explicit that they are not corollaries of the classical versions plus convex extensibility — they carry genuinely combinatorial content, specializing to Frank's discrete separation theorem and Edmonds's intersection theorem as examples the book itself gives.

None of these results are open — they are Murota's account of the duality at the heart of discrete convex analysis, the reason the theory needed two dual notions rather than one. What this mission contributes is a faithful, machine-checked formal statement of each, completing a theorem mission 10-conjugacy-i explicitly left for a future session once the necessary polyhedral apparatus existed, and including one result (Theorem 8.32) the platform's own automated extractor missed; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The naive approach to the goal's harder direction (L⇒M) would try to verify the exchange inequality for $g^\bullet$ directly from the definition of the transform; the book's actual proof instead identifies the exchange inequality with a statement about weighted minimizers of $g$ itself via Proposition 8.3 (the minimizer-difference bound), converting a claim about the conjugate function into a claim about $g$ 's own combinatorial structure — a genuine change of perspective, not a direct calculation. Proposition 8.3's own proof is the hardest single argument in this block: it derives the minimizer-difference bound by a contradiction argument that constructs an explicit pair of "worse" minimizers via a join/meet perturbation and derives a strict inequality from Theorem 7.29's translation inequality — a multi-step combinatorial argument with no direct shortcut.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; convex functions are WithTop ℝ valued throughout (never EReal, except for the Legendre-Fenchel transform itself, whose defining supremum/infimum can genuinely be infinite). All thirteen numbered results found in this chunk's page range — the twelve in BRIEF.md's own table plus Theorem 8.32 — are placed, with one documented scope reduction: Theorem 8.21 states only its real-attainment parts (1)-(2), not the integer-attainment refinement of parts (3)-(4), which needs a separate argument no other result in this chunk requires — see HARD.md. Concave functions $h$ are always represented via $h_2 = -h$ and every inequality $f \ge h$ restated as $f + h_2 \ge 0$ , avoiding WithTop ℝ negation entirely. This chunk's own BRIEF.md inherited the chapters-4-7 page-offset boilerplate (printed = PDF $-$ 19); chapter 8 uses offset 18, confirmed against the PDF's own footers — every citation here uses the corrected offset. This mission's base vocabulary is redeclared from missions 10-conjugacy-i, 20-ch04b-mconvexsets, 21-ch05b-lconvexsets, 23-24-ch06*-mconvexfunctions, and 26-27-ch07*-lconvexfunctions rather than imported, since sibling drafts in this series cannot yet reference one another. Contributions completing any of the thirteen sorrys are welcome; the goal and Proposition 8.3 carry the most independent proof content.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
K. Murota and A. Shioura, "M-convex function on generalized polymatroid," Mathematics of Operations Research, 24 (1999), pp. 95-105 [152] (the polyhedral M-/L-convex conjugacy theory this mission's real-variable results are drawn from).

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Combinatorics Graph Theory Probability·Captain: mikedeng1

Applied Combinatorics VI: Ramsey's Theorem and Erdős's Lower BoundTextbook

Motivation

Ramsey theory studies the principle that complete disorder is impossible: every sufficiently large structure contains a large, perfectly uniform substructure. Its most familiar instance concerns graphs. In any graph on six vertices there are three vertices that are pairwise adjacent or three that are pairwise non-adjacent, and the same phenomenon persists at every scale. The quantity that measures it, the Ramsey number $R(m, n)$ , is one of the most studied and least understood functions in combinatorics. Only a handful of values are known exactly ( $R(3,3) = 6$ , $R(4,4) = 18$ ), while $R(5,5)$ is only known to lie between 43 and 49 (Radziszowski, Small Ramsey Numbers).

The subject has a short, well-documented history. F. P. Ramsey proved the general theorem in 1930 as a lemma in decidability (Ramsey 1930). Erdős and Szekeres (1935) gave the upper bound $R(m, n) \le \binom{m+n-2}{m-1}$ (Erdős–Szekeres 1935). In 1947 Erdős proved an exponential lower bound for the diagonal numbers $R(n, n)$ by counting graphs (Erdős 1947); this argument is now regarded as the origin of the probabilistic method. In 1959 Erdős used the same method to show that graphs of large girth and large chromatic number exist (Erdős 1959). For decades the exponential bases $\sqrt 2$ and $4$ stood essentially unchanged; the upper base was lowered below $4$ only in 2023 (Campos–Griffiths–Morris–Sahasrabudhe).

This mission formalizes these results as presented in Chapter 11 of Keller and Trotter, Applied Combinatorics (2017 Edition).

Setting

A graph $G$ is a finite simple graph: a finite vertex set with a symmetric, irreflexive adjacency relation (no loops, no multiple edges). A complete subgraph on $m$ vertices is a set of $m$ pairwise adjacent vertices; an independent set of size $n$ is a set of $n$ pairwise non-adjacent vertices.

A non-negative integer $N$ is a Ramsey bound for $(m, n)$ if every graph with at least $N$ vertices contains a complete subgraph on $m$ vertices or an independent set of size $n$ . The Ramsey number $R(m, n)$ is the least positive Ramsey bound. In Lean these are AppliedComb.Ramsey.IsRamseyBound m n N and AppliedComb.Ramsey.ramseyNumber m n.

More generally, write $[n] = \{1, \dots, n\}$ and $C(X, s)$ for the family of $s$ -element subsets of $X$ . For a string $h = (h_1, \dots, h_r)$ , the number $R(s : h_1, \dots, h_r)$ is the least positive $N$ such that for every $n \ge N$ and every colouring $\phi : C([n], s) \to [r]$ some colour $\alpha$ has a set $H_\alpha \subseteq [n]$ of size $h_\alpha$ all of whose $s$ -subsets receive colour $\alpha$ (hypergraphRamseyNumber s r h).

The girth of a graph is the smallest number of vertices on a cycle, and infinite for a forest; the chromatic number $\chi(G)$ is the least number of colours in a proper vertex colouring.

Formalization targets

Goal: Erdős's lower bound (Theorem 11.4)

For every positive integer $n$ ,

R(n, n) \;\ge\; \frac{n}{e\sqrt 2}\, 2^{n/2}.

Equivalently, below this threshold there is a graph on each number of vertices with neither a complete subgraph on $n$ vertices nor an independent set of size $n$ . The statement is for every $n \ge 1$ , with no asymptotic slack.

Milestones

Lemma 11.1. Every graph with at least six vertices has a complete subgraph on 3 vertices or an independent set of size 3.
Theorem 11.2 (Ramsey's Theorem for Graphs). For positive integers $m, n$ the least positive integer $R(m, n)$ exists.
Theorem 11.6. For positive integers $r, s$ and $h_1, \dots, h_r \ge s$ , the least positive integer $R(s : h_1, \dots, h_r)$ exists.
Theorem 11.7 (Erdős). For all integers $g \ge 3$ and $t$ there is a graph with $\chi(G) > t$ and girth greater than $g$ .

A further item, not a milestone because the book does not number it, records the bound that the proof of Theorem 11.2 establishes: every graph with at least $\binom{m+n-2}{m-1}$ vertices has a complete subgraph on $m$ vertices or an independent set of size $n$ .

Significance

The goal is the diagonal lower bound that every later improvement is measured against. With the upper bound from the proof of Theorem 11.2 it shows that $R(n,n)$ grows exponentially, with base between $\sqrt 2$ and $4$ . No explicit construction is known to give $R(n, n) > c^n$ for any constant $c > 1$ . Theorem 11.7 is the standard example of a statement whose only known proofs for decades were probabilistic, and it shows that chromatic number is not a local property.

On the formal side, Mathlib has cliques, independent sets, girth and chromatic number, but no Ramsey numbers, no Erdős lower bound and no high-girth theorem. The platform already has weaker or differently shaped relatives, all checked for this mission. Erdos1947.ramsey_lower_bound gives a graph on $2^{\lfloor k/2 \rfloor}$ vertices without monochromatic $k$ -sets, a weaker bound than the goal's. BookSixth.high_girth_chromatic is a single-parameter form of Theorem 11.7 in another Lean environment. ramsey_theory_upper_bound is a diagonal $4^k$ bound. The goal statement carries the constant $1/(e\sqrt2)$ exactly, which requires an explicit, non-asymptotic lower bound for $n!$ where the book writes "Stirling's approximation".

Difficulty

The obvious route to the goal is to count graphs with a large clique or independent set and compare the result with the total number of graphs. Two steps of that route do not go through as written in the text. First, the book replaces $n!$ by its Stirling approximation, which is only asymptotic; a statement for every $n \ge 1$ needs an inequality valid for all $n$ , and the constant $1/(e\sqrt2)$ leaves no room for a cruder estimate such as $n! \ge (n/e)^n$ alone. Second, the counting argument yields a graph on each $t$ below the threshold, whereas $R(n,n)$ is defined as a least threshold over all graphs with at least that many vertices; the two have to be connected.

Theorem 11.7 needs random graphs with edge probability depending on $n$ , a first-moment bound on short cycles and on independent sets, and a deletion step. The book states Theorem 11.6 without proof.

Formalization scope

Graphs are Mathlib SimpleGraph V on a finite type V : Type (Theorems 11.1, 11.2, 11.4) or on Fin N (Theorem 11.7). Cliques and independent sets are SimpleGraph.IsNClique and SimpleGraph.IsNIndepSet on a Finset.
IsRamseyBound m n N quantifies over every graph with at least $N$ vertices, as the book does; ramseyNumber m n is the sInf of the positive Ramsey bounds. Theorem 11.2 is stated as IsLeast {N | 0 < N ∧ IsRamseyBound m n N} (ramseyNumber m n), so its content is the nonemptiness of that set. The same pattern is used for Theorem 11.6.
The goal compares real numbers: (n : ℝ) / (Real.exp 1 * Real.sqrt 2) * (2 : ℝ) ^ ((n : ℝ) / 2) ≤ (ramseyNumber n n : ℝ), with a real power. Explicit constant: the book says "use the Stirling approximation … after some algebra"; the statement keeps the book's constant $1/(e\sqrt 2)$ and holds for every $n \ge 1$ with no threshold.
The bound of the proof of Theorem 11.2 is stated as the Ramsey property at $\binom{m+n-2}{m-1}$ , not as an inequality on ramseyNumber, so it cannot hold through an empty defining set.
Girth is Mathlib's SimpleGraph.egirth (valued in $\mathbb N \cup \{\infty\}$ , $\infty$ for forests), not SimpleGraph.girth, which is $0$ on forests. Chromatic number is SimpleGraph.chromaticNumber in $\mathbb N \cup \{\infty\}$ . The parameter $t$ of Theorem 11.7 is a natural number; negative $t$ is trivial.
Theorem 11.6 is printed with typos: $h_i \ge s$ is read for all $i = 1, \dots, r$ , the undefined $n_0$ is read as $R(s : h_1, \dots, h_r)$ , and $C([n], s]$ as $C([n], s)$ . Colourings are functions on the subtype of $s$ -element subsets of Fin n, with colours in Fin r.
Trivializing formalizations ruled out. A Ramsey number defined as an arbitrary upper bound, or as a supremum with junk value $0$ , would make the goal vacuous or false. Here the goal's right-hand side is positive, so it forces the defining set to be nonempty, and every graph is simple on exactly its vertex type, with no loops or multiple edges.
Reusable infrastructure: IsRamseyBound/ramseyNumber, the hypergraph version, an all- $n$ lower bound for $n!$ , and counting over the $2^{\binom{t}{2}}$ labelled graphs on $t$ vertices. Proofs of any milestone are welcome contributions.

Selected references

M. T. Keller and W. T. Trotter, Applied Combinatorics, 2017 Edition, Chapter 11, pp. 229–238. https://www.appliedcombinatorics.org/
F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30 (1930), 264–286. https://doi.org/10.1112/plms/s2-30.1.264
P. Erdős and G. Szekeres, A combinatorial problem in geometry, Compositio Math. 2 (1935), 463–470. http://www.numdam.org/item/CM_1935__2__463_0/
P. Erdős, Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947), 292–294. https://doi.org/10.1090/S0002-9904-1947-08785-1
P. Erdős, Graph theory and probability, Canad. J. Math. 11 (1959), 34–38. https://doi.org/10.4153/CJM-1959-003-9
S. Radziszowski, Small Ramsey Numbers, Electron. J. Combin. Dynamic Survey DS1. https://doi.org/10.37236/21
M. Campos, S. Griffiths, R. Morris and J. Sahasrabudhe, An exponential improvement for diagonal Ramsey, 2023. https://arxiv.org/abs/2303.09521

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Combinatorics Linear algebra·Captain: mikedeng1

Applied Combinatorics V: Linear Recurrence Equations and the Advancement OperatorTextbook

Motivation

Linear recurrences with constant coefficients are among the first tools of enumerative combinatorics and the analysis of algorithms: the Fibonacci numbers, the number of binary strings avoiding a pattern, the running time of a divide-and-conquer loop and the number of tilings of a strip all satisfy relations of the form $c_0 a_{n+k} + c_1 a_{n+k-1} + \cdots + c_k a_n = 0$ . Chapter 9 of Keller and Trotter's Applied Combinatorics (appliedcombinatorics.org) treats such relations as linear equations in an operator, the advancement operator, in direct analogy with linear differential equations with constant coefficients. The chapter's Principal Theorem says that the solution set of such an equation is a vector space whose dimension equals the order of the recurrence; everything else in the chapter (general solutions for distinct and repeated roots, and the reduction of nonhomogeneous equations to homogeneous ones) is organised around it.

This mission formalizes that theorem, in the book's model of functions on all of $\mathbb{Z}$ , together with the numbered lemmas of Section 9.5 on which its outline rests.

Setting

Let $V$ be the real vector space of all functions $f : \mathbb{Z} \to \mathbb{R}$ , with pointwise addition and scalar multiplication. The advancement operator $A : V \to V$ is

A f(n) = f(n+1) \qquad (n \in \mathbb{Z}),

a linear operator with $A^p f(n) = f(n+p)$ for $p \ge 0$ . For a nonnegative integer $k$ and real constants $c_0, c_1, \dots, c_k$ , the advancement operator polynomial is

p(A) = c_0 A^k + c_1 A^{k-1} + c_2 A^{k-2} + \cdots + c_k,

so that $p(A) f(n) = c_0 f(n+k) + c_1 f(n+k-1) + \cdots + c_k f(n)$ . The solution space of the homogeneous equation $p(A) f = 0$ is

W = \{ f \in V : p(A) f = 0 \},

the kernel of $p(A)$ , a linear subspace of $V$ . In Lean, $A$ is AppliedComb.Recurrence.advance : Module.End ℝ (ℤ → ℝ), $p(A)$ is opPoly k c for a coefficient vector c : Fin (k + 1) → ℝ with c i multiplying $A^{k-i}$ , and $W$ is solutionSpace k c := LinearMap.ker (opPoly k c). A factor $A - r$ is advance - r • 1.

Formalization targets

Goal: Theorem 9.18 (the Principal Theorem)

For a positive integer $k$ and real constants $c_0, \dots, c_k$ with $c_0 \neq 0$ and $c_k \neq 0$ ,

\dim_{\mathbb{R}} \{ f : \mathbb{Z} \to \mathbb{R} \;:\; (c_0 A^k + c_1 A^{k-1} + \cdots + c_k) f = 0 \} = k.

Milestones

Lemma 9.19. If $r \neq 0$ and $(A - r) f = 0$ , then $f(n) = f(0)\, r^n$ for every $n \in \mathbb{Z}$ .
Lemma 9.20. If $c_0, c_k \neq 0$ , $g \in V$ is arbitrary and $p(A) f_0 = g$ , then every solution of $p(A) f = g$ is $f = f_0 + f_1$ with $f_1 \in W$ .
Theorem 9.21. If $r_1, \dots, r_k$ are distinct nonzero reals, every solution of $(A - r_1)(A - r_2)\cdots(A - r_k) f = 0$ has the form

f(n) = c_1 r_1^n + c_2 r_2^n + \cdots + c_k r_k^n \qquad (n \in \mathbb{Z}).

Lemma 9.22. If $k \ge 1$ and $r \neq 0$ , then $(A - r)^k f = 0$ holds exactly when

f(n) = c_1 r^n + c_2 n r^n + c_3 n^2 r^n + \cdots + c_k n^{k-1} r^n \qquad (n \in \mathbb{Z})

for some real constants $c_1, \dots, c_k$ .

Significance

The result itself. Theorem 9.18 is what justifies the standard recipe for solving a recurrence: once $k$ linearly independent solutions are found, every solution is a linear combination of them, and $k$ initial values pin down a unique solution. Theorems 9.21 and 9.22 name those $k$ solutions when the characteristic polynomial splits over $\mathbb{R}$ , and Lemma 9.20 reduces nonhomogeneous recurrences, the ones arising from counting problems with a forcing term, to one particular solution plus the homogeneous space.

Formalizing it. The book states Theorem 9.18 without a full proof ("we won't prove the full result") and leaves Lemma 9.22 as an exercise, so a formalization supplies arguments the source omits. Mathlib's LinearRecurrence develops the $\mathbb{N}$ -indexed, monic version (LinearRecurrence.solSpace_rank); the bi-infinite, two-sided setting of the book, in which the constant term $c_k$ must be nonzero, is not in Mathlib or on the platform as of this mission.

Difficulty

The subspace part of Theorem 9.18 is immediate; the content is the dimension count, in both directions. On $\mathbb{Z}$ a solution must extend to negative indices as well as positive ones, so an argument that works for sequences indexed by $\mathbb{N}$ does not transfer: there the constant term plays no role and the dimension is $k$ whatever the constant term, while on $\mathbb{Z}$ a vanishing $c_k$ changes the answer. The outline in the book proceeds through factorisations of $p(A)$ , which over $\mathbb{R}$ need not exist (complex roots), so the goal cannot be obtained by combining Theorems 9.21 and 9.22 alone. Lemma 9.22 requires showing that the functions $n^j r^n$ solve $(A - r)^k f = 0$ and that they exhaust its solutions, for $n$ ranging over all integers.

Formalization scope

Functions are $\mathbb{Z} \to \mathbb{R}$ (the real space $V$ of Section 9.5, p. 199), not $\mathbb{N} \to \mathbb{R}$ and not $\mathbb{Z} \to \mathbb{C}$ . Powers $r^n$ with $n$ negative are integer powers (zpow), always of a nonzero base.
Dimension in the goal is Module.rank ℝ (solutionSpace k c) = k, a cardinal equality, so it also asserts finite-dimensionality.
The coefficient vector is c : Fin (k + 1) → ℝ; c 0 is the leading coefficient $c_0$ and c (Fin.last k) the constant term $c_k$ .
Theorem 9.21 asserts one inclusion, as on the page ("every solution has the form"); Lemma 9.22 asserts both ("the general solution"), as an iff.
Lemma 9.22 carries the hypothesis $r \neq 0$ , the standing assumption of Section 9.5.2 (p. 200), which the lemma's own sentence does not repeat.
No O(·), "≈" or unspecified threshold occurs in these statements, so no explicit constant is instantiated.
Ruling out the trivial reading: $W$ is defined as the kernel of the operator polynomial $p(A)$ , never as the span of $k$ chosen functions, so the goal is not a statement about the rank of a spanning family.

Needed infrastructure: linear operators on the function space ℤ → ℝ, powers and products in Module.End, Module.rank/Module.finrank, and finite sums; Mathlib's LinearRecurrence may be adapted for the forward half. The definitions advance, opPoly and solutionSpace are reusable for any later mission on recurrences or on generating functions for bi-infinite sequences. Contributions of the milestone lemmas, of a proof of the goal that avoids factorisation over $\mathbb{C}$ , and of a complex-valued variant are welcome.

Selected references

Mitchel T. Keller and William T. Trotter, Applied Combinatorics, 2017 Edition, Chapter 9 (Recurrence Equations), CC BY-SA 4.0. appliedcombinatorics.org
Mathlib, Mathlib/Algebra/LinearRecurrence.lean (ℕ-indexed linear recurrences, LinearRecurrence.solSpace_rank). Mathlib docs
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, 1994, Section 7.3 (solving recurrences). ISBN 978-0-201-55802-9.

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Convex Optimization Discrete Geometry Operations Research+1·Captain: Shuze Chen

Discrete Convex Analysis X: The Lagrangian Saddle-Point TheoremTextbook

Motivation

Chunk 10 formalized the discrete conjugacy theorem — the Legendre-Fenchel transform's bijection between the classes of M-convex and L-convex functions — and, along the way, a function-level generalization of Edmonds's intersection theorem. Section 8.4 turns that machinery toward a different question: not "how are two convexity classes related," but "when does a discrete optimization problem have a dual that meets it with equality." The classical route to such strong-duality results in continuous convex programming — Lagrangian relaxation, an embedding of the problem in a family of perturbed problems, and a saddle-point characterization of when primal and dual values coincide — has a discrete analogue that needs no continuity, no differentiability, and no convexity in the classical sense at all: only the elementary fact that the Legendre-Fenchel transform, once discretized, is still an involution on the right class of functions. This mission formalizes that discrete Lagrangian duality framework and its central saddle-point theorem, in full generality — before the book specializes it, in the section that follows, to the specific M-convex perturbation that gives the chapter's headline strong-duality result for M-convex programs.

Setting

Let $V$ and $U$ be finite ground sets. A perturbation of an optimization problem $\min\{f(x) : x \in \mathbb Z^V\}$ is a function $F : \mathbb Z^V \times \mathbb Z^U \to \mathbb Z \cup \{+\infty\}$ such that $F(x,0) = f(x)$ for all $x$ (Eq. (8.54)) and, for each fixed $x$ , $F(x,\cdot)$ is self-biconjugate: $F(x,\cdot)^{\bullet\bullet} = F(x,\cdot)$ under the discrete Legendre-Fenchel transform of chunk 10 (Eq. (8.55)). The Lagrangian function is $K(x,y) = \inf\{F(x,u) + \langle u,y\rangle : u \in \mathbb Z^U\}$ (Eq. (8.58)), valued in $\mathbb Z \cup \{\pm\infty\}$ (formalized in EReal, since both the infimum and the supremum below can be genuinely unbounded). The dual objective is $g(y) = \inf\{K(x,y) : x \in \mathbb Z^V\}$ (Eq. (8.60)). Writing $\inf(P) = \inf_x f(x)$ , $\sup(D) = \sup_y g(y)$ , $\operatorname{opt}(P) = \{x : f(x) = \inf(P)\}$ , $\operatorname{opt}(D) = \{y : g(y) = \sup(D)\}$ , the primal problem $P$ is to minimize $f$ over $\mathbb Z^V$ and the dual problem $D$ is to maximize $g$ over $\mathbb Z^U$ .

Formalization targets

Goal: Theorem 8.54 (the saddle-point theorem)

Assuming $F$ is self-biconjugate (Eq. (8.55)): both $\inf(P)$ and $\sup(D)$ are finite and $\min(P) = \max(D)$ if and only if there exist $\bar x \in \mathbb Z^V$ , $\bar y \in \mathbb Z^U$ with $K(\bar x,\bar y)$ finite and $K(x,\bar y) \le K(\bar x,\bar y) \le K(\bar x,y)$ for all $x,y$ — a saddle point of the Lagrangian kernel. When this holds, $\bar x \in \operatorname{opt}(P)$ and $\bar y \in \operatorname{opt}(D)$ .

Milestones: Theorem 8.52, Proposition 8.51(1)-(2)

Theorem 8.52 (weak duality): $\inf(P) \ge \sup(D)$ always, with no biconjugacy hypothesis on $F$ at all — the baseline the saddle-point theorem sharpens to equality. Proposition 8.51(1)-(2): under self-biconjugacy, the perturbation $F$ (and hence the primal objective $f$ ) is itself recoverable from the Lagrangian kernel $K$ by a supremum, $F(x,u) = \sup_y\{K(x,y) - \langle u,y\rangle\}$ and $f(x) = \sup_y K(x,y)$ — the algebraic identity the saddle-point theorem's proof turns on directly.

Significance

The result itself. The saddle-point theorem is the general-purpose engine behind every strong-duality result the book proves for specific classes of discrete optimization problems: the book's own next section specializes it (via a particular choice of $F$ built from an M-convex regularizer $r$ ) to obtain strong duality for M-convex programs, but the theorem itself needs no M-convexity, no submodularity, and no exchange axiom — only the elementary self-biconjugacy of a perturbation under the discrete Legendre-Fenchel transform. It is, in that sense, the most general and most reusable strong-duality statement in the book: any future mission proving strong duality for a specific class of discrete programs (M-convex, M2-convex, network flow, or otherwise) by exhibiting a self-biconjugate perturbation can cite this theorem directly rather than reproving the saddle-point argument from scratch.

Formalizing it. No matching item exists on the platform for a discrete Lagrangian saddle- point theorem, discrete weak duality, or this perturbation-based duality framework. (A prior-art search turned up an unrelated continuous Lagrangian saddle-point theorem for convex cones, Luenberger's Chapter 8 §8.4, formalized as VectorSpaceOpt.lagrangian_saddle_sufficient_pointed — a genuinely different setting: no discreteness, no biconjugacy hypothesis, and a one-directional sufficiency statement rather than this mission's iff. Not reused.) This mission gives the first formal statement of discrete Lagrangian duality, and directly reuses chunk 10's ConvexConjugate apparatus (self-biconjugacy is stated using chunk 10's own conjugate-of-conjugate composition), demonstrating exactly the kind of shared-substrate payoff the discrete conjugacy theorem was built to provide.

Difficulty

The saddle-point theorem's "only if" direction is not a routine unwinding of definitions: given $\min(P) = \max(D)$ at finite common value, one must construct the saddle point $(\bar x,\bar y)$ — the book's proof takes $\bar x \in \operatorname{opt}(P)$ , $\bar y \in \operatorname{opt}(D)$ (which exist because the infimum/supremum are attained at a finite optimum) and verifies the sandwiching inequality using Proposition 8.51(2)'s identity $f(x) = \sup_y K(x,y)$ together with weak duality, rather than by any direct algebraic manipulation of $K$ alone. Skipping straight to a "trivial" biconditional that never invokes Proposition 8.51 would misrepresent the actual proof structure the book relies on for exactly this direction.

Formalization scope

$V, U$ are Fintype ground types; LagrangianKernel, DualObjective, InfP, SupD are EReal-valued to keep both the defining infima/suprema total (a complete lattice) without an artificial finiteness side-condition; OptP, OptD compare PrimalValue/DualObjective against InfP/SupD after casting through chunk 10's ToEReal, mirroring that chunk's own round-trip convention. "Finite" throughout is formalized as ≠ ⊤ ∧ ≠ ⊥ in EReal. The perturbation $F$ itself is left fully abstract (an arbitrary function satisfying the self-biconjugacy hypothesis where needed) — this mission does not draft the specific M-convex perturbation $F_r$ (Eq. (8.61)) that the book's next subsection (§8.4.3) uses to specialize this framework to M-convex programs, nor Theorem 8.59 (the resulting M-convex strong-duality theorem) itself, which needs that specific perturbation plus its own regularity conditions (REG)/(OBJ) and a chain of M-convex-specific propositions (8.55–8.58) beyond what the general framework built here provides. A trivializing formalization would state the saddle-point theorem's sandwiching inequality with a weaker order (e.g., only one of the two directions) or would omit the " $\bar x \in \operatorname{opt}(P), \bar y \in \operatorname{opt}(D)$ " consequence clause; neither is done — both inequalities and the full consequence clause are included exactly as the book states them.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.

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Combinatorics·Captain: mikedeng1

Applied Combinatorics IV: Newton's Binomial Theorem and the Central Binomial ConvolutionTextbook

Motivation

Generating functions are the standard device of enumerative combinatorics for turning a counting sequence into a single algebraic or analytic object: a sequence $\{a_n : n \ge 0\}$ is recorded as the power series $\sum_{n\ge0} a_n x^n$ , and operations on series (products, powers, derivatives) become operations on the counts. Chapter 8 of Keller and Trotter's Applied Combinatorics (appliedcombinatorics.org), an open textbook used in undergraduate combinatorics courses, develops the method up to one of its classical applications: extending the binomial theorem to real exponents, as Newton did, and reading off an identity about central binomial coefficients that is awkward to prove by direct counting.

The identity in question,

2^{2n} = \sum_{k=0}^{n} \binom{2k}{k}\binom{2n-2k}{n-k},

appears in standard collections of binomial identities such as Graham, Knuth and Patashnik's Concrete Mathematics (1994).

Setting

For a real number $p$ and a nonnegative integer $k$ , the book defines a number $P(p, k)$ by the recursion

P(p, 0) = 1, \qquad P(p, k) = p\,P(p-1, k-1) \quad (k > 0),

so that $P(p, k) = p(p-1)\cdots(p-k+1)$ with no requirement $p \ge k$ . The generalized binomial coefficient is

\binom{p}{k} = \frac{P(p,k)}{k!}.

For integers $p \ge k \ge 0$ this is the usual binomial coefficient; for integers $0 \le p < k$ it is $0$ ; for other real $p$ it is in general nonzero for every $k$ . In the Lean development $P(p,k)$ is AppliedComb.GenFun.fallingP p k and $\binom pk$ is AppliedComb.GenFun.binomReal p k.

The generating function of a real sequence $a = (a_n)_{n\ge0}$ is the formal power series $A(x) = \sum_{n\ge0} a_n x^n$ , in Lean PowerSeries.mk a : PowerSeries ℝ. When a closed-form function such as $(1+x)^p$ or $(1-4x)^{-1/2}$ is called the generating function of a sequence, the statements below read this as convergence of $\sum_n a_n x^n$ to the function's value on an explicit real interval around $0$ .

The central binomial coefficients are $\binom{2n}{n}$ : $1, 2, 6, 20, 70, \dots$

Formalization targets

Goal: Corollary 8.14

For every integer $n \ge 0$ ,

2^{2n} = \sum_{k=0}^{n} \binom{2k}{k}\binom{2n-2k}{n-k},

stated as an identity of natural numbers.

Milestones

Lemma 8.11. For every real $p$ and integer $k \ge 0$ , $P(p, k+1) = P(p, k)\,(p - k)$ .
Lemma 8.12. For every integer $k \ge 0$ , $\binom{-1/2}{k} = (-1)^k \binom{2k}{k} / 2^{2k}$ .
Theorem 8.10 (Newton's Binomial Theorem). For real $p \ne 0$ and real $|x| < 1$ ,

(1+x)^p = \sum_{n=0}^{\infty} \binom{p}{n} x^n .

Theorem 8.13. For real $|x| < 1/4$ ,

(1-4x)^{-1/2} = \sum_{n=0}^{\infty} \binom{2n}{n} x^n .

Proposition 8.3. For real sequences $a$ , $b$ , the product of their generating functions is the generating function of $\bigl(\sum_{k=0}^n a_k b_{n-k}\bigr)_{n\ge0}$ .
Theorem 8.16 (already on the platform). For each $n \ge 1$ , the number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts.

The first five follow the chapter's own chain toward the goal; Theorem 8.16 is the chapter's other main result and is included as a reference.

Significance

Corollary 8.14 says that the sequence of central binomial coefficients convolved with itself is the sequence $4^n$ ; equivalently, the generating function of $\binom{2n}{n}$ is a square root of $1/(1-4x)$ . Central binomial coefficients count lattice paths with $n$ up-steps and $n$ down-steps, and the identity says that the pairs consisting of a balanced path of length $2k$ and one of length $2n-2k$ , summed over $k$ , are equinumerous with all $4^n$ strings over a four-letter alphabet. The same generating function reappears in the book's Section 9.7, and Newton's theorem with exponent $-1/2$ or $1/2$ is the standard route to closed forms for Catalan-type sequences.

On the formalization side, Mathlib has the central binomial coefficient (Nat.centralBinom), formal power series, and Newton's series in the complex-analytic form Complex.one_add_cpow_hasFPowerSeriesOnBall_zero, which is also published on the platform as FamousTheorems.newton_binomial_series_6b and included in this mission as a reference. It does not contain the convolution identity of Corollary 8.14, and it does not contain the book's recursive $P(p,k)$ or the closed form of $\binom{-1/2}{k}$ . The mission produces a machine-checked version of the chapter's chain from the book's own definitions to the identity.

Difficulty

The identity is not a special case of the Vandermonde convolution $\sum_k \binom{a}{k}\binom{b}{n-k} = \binom{a+b}{n}$ : both factors depend on the summation index in their upper argument as well as their lower one. Induction on $n$ does not close directly, since the sum for $n+1$ is not a simple combination of the sum for $n$ . A counting proof is possible but not obvious, which is why the chapter's route goes through a generating function with a non-integer exponent. That route passes from formal power series to real analysis: $(1-4x)^{-1/2}$ is a real function, and the passage from an identity of functions on an interval to an identity of coefficients requires the uniqueness of power series coefficients on an open interval and the product of two convergent series. The book asserts Newton's theorem without proof.

Formalization scope

Numbers. $P(p,k)$ and $\binom pk$ are real-valued, defined by the book's recursion (Definition 8.8) and quotient (Definition 8.9) in the definition item AppliedComb.GenFun.binomReal. Mathlib's descPochhammer and Ring.choose compute the same values; they are not used in the statements so that Lemma 8.11 is a statement about the book's recursion rather than a definitional unfolding.
Pinned readings. The book treats generating functions as formal power series and states Theorems 8.10 and 8.13 without a domain for $x$ . Here both are stated analytically: Theorem 8.10 for real $p \ne 0$ and real $x$ with $|x| < 1$ , Theorem 8.13 for real $x$ with $|x| < 1/4$ , with the real power Real.rpow of a positive base on the left and HasSum (unconditional convergence of the series) on the right. The book's hypothesis $p \ne 0$ is kept. No other explicit constants replace informal ones: the chapter's statements contain no $O(\cdot)$ , " $\approx$ " or "sufficiently large".
Proposition 8.3 is stated for formal power series PowerSeries ℝ with the sum written as $\sum_{k=0}^{n} a_k b_{n-k}$ over Finset.range (n + 1).
Goal. Corollary 8.14 is an identity in ℕ with Nat.choose; the subtractions $2n-2k$ and $n-k$ occur only for $k \le n$ and are exact. A statement asserting only that the square of the formal power series $\sum_n \binom{2n}{n}X^n$ equals $\sum_n 4^n X^n$ , or a purely formal version of Theorem 8.13, would hide the identity in a coefficient comparison and is not the book's statement; the goal is the explicit sum identity.
Theorem 8.16 is referenced as FamousTheorems.card_odds_eq_card_distincts, stated for all $n$ with Mathlib's Nat.Partition, whose partitions are multisets of positive integers, as in the book (p. 168); the case $n = 0$ it adds is immediate.

A complete development needs real power series on an interval (Cauchy products, identity theorem for coefficients), the real power function, and elementary manipulation of binomial coefficients; the identification of binomReal with Ring.choose is reusable for any later statement using the generalized binomial coefficient. Proofs of any milestone are welcome independently, and so is a direct proof of Corollary 8.14 that bypasses the analytic chain.

Selected references

M. T. Keller and W. T. Trotter, Applied Combinatorics, 2017 Edition, CC BY-SA 4.0. Chapter 8. https://www.appliedcombinatorics.org/
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, 1994. ISBN 978-0-201-55802-9.
Mathlib, Complex.one_add_cpow_hasFPowerSeriesOnBall_zero (Newton's binomial series). https://github.com/leanprover-community/mathlib4

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Convex Optimization Discrete Geometry Operations Research+1·Captain: Shuze Chen

Discrete Convex Analysis IX: The Discrete Conjugacy TheoremTextbook

Motivation

The Legendre-Fenchel transform is the single most structurally important operation in convex analysis: for a proper closed convex function $f$ , its conjugate $f^\bullet(p) = \sup_x \{\langle p,x\rangle - f(x)\}$ is again proper closed convex, and the transform is an involution — $f^{\bullet\bullet} = f$ . This one fact underlies duality theory across optimization: every strong-duality theorem is, at bottom, a statement about conjugate pairs. Chapters 6 and 7 of this book developed M-convex and L-convex functions as if they were two separate theories, each with its own exchange axiom, optimality criterion, and proximity theorem. Chapter 8 reveals they were never separate: the Legendre-Fenchel transform, suitably discretized, is a bijection between the two classes. This mission formalizes that discrete conjugacy theorem together with its classical real-valued precursor and a genuine function-level generalization of Edmonds's intersection theorem, completing the picture that chunks 06 through 09 built the two halves of.

Setting

Let $V$ be a finite ground set. For $f : \mathbb R^V \to \mathbb R \cup \{+\infty\}$ , the Legendre-Fenchel transform is $f^\bullet(p) = \sup\{\langle p,x\rangle - f(x) : x \in \mathbb R^V\}$ ; $f$ is submodular if $f(x)+f(y) \ge f(x\vee y)+f(x\wedge y)$ and supermodular under the reverse inequality. For $f : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ , the discrete Legendre-Fenchel transform restricts the same supremum formula to $p \in \mathbb Z^V$ : $f^\bullet(p) = \sup\{\langle p,x\rangle - f(x) : x \in \mathbb Z^V\}$ for $p \in \mathbb Z^V$ — a genuinely different object from the real-valued transform, since the supremum is now over integer $x$ only, and the codomain is checked back against the discrete M-/L-convexity axioms of chunks 06–09. The integer biconjugate $f^{\bullet\bullet}$ is the transform applied twice. $f$ is integer valued if every finite value it takes is an integer (the classes $M[\mathbb Z\to\mathbb Z]$ , $L[\mathbb Z\to\mathbb Z]$ of the goal theorem are exactly the M-/L-convex functions with this property).

Formalization targets

Goal: Theorem 8.12 (the discrete conjugacy theorem)

(1) The classes $M[\mathbb Z\to\mathbb Z]$ and $L[\mathbb Z\to\mathbb Z]$ are in one-to-one correspondence under the discrete Legendre-Fenchel transform: for $f \in M[\mathbb Z\to\mathbb Z]$ and $g \in L[\mathbb Z\to\mathbb Z]$ , $f^\bullet \in L[\mathbb Z\to\mathbb Z]$ , $g^\bullet \in M[\mathbb Z\to\mathbb Z]$ , $f^{\bullet\bullet}=f$ , and $g^{\bullet\bullet}=g$ . (2) The same correspondence holds between $M^\natural[\mathbb Z\to\mathbb Z]$ and $L^\natural[\mathbb Z\to\mathbb Z]$ .

Milestones: Theorem 8.1, Proposition 8.11, Theorem 8.17

Theorem 8.1: the conjugate of a real-valued submodular function is always supermodular — the classical warm-up, and evidence that submodularity/supermodularity is not symmetric under conjugation on its own (the converse fails). Proposition 8.11: the integer biconjugate recovers $f$ at any point with a nonempty integer subdifferential — the fact that makes discrete biconjugation meaningful at all. Theorem 8.17 (the M-convex intersection theorem): a point jointly minimizes a sum of two M $^\natural$ -convex functions if and only if a single linear functional separately certifies it as a minimizer of each perturbed function — the function-level generalization of chunk 04's Edmonds's intersection theorem for M-convex sets.

Significance

The result itself. The discrete conjugacy theorem is, in the book's own words, "the unifying result of the entire book": every theorem proved separately for M-convex functions (chunks 06–07) has an exact mirror for L-convex functions (chunks 08–09) precisely because the Legendre-Fenchel transform carries one class to the other. Theorem 8.17's function-level Edmonds generalization shows the payoff directly — the classical matroid-intersection-style min-max duality of chunk 04 was never really about sets; it is a special case (indicator functions) of a duality that holds for the whole class of M-convex functions.

Formalizing it. No matching item exists on the platform for conjugate functions, discrete conjugacy, or this generality of intersection theorem. This mission gives the first formal statement of the discrete conjugacy theorem, distinguishing it carefully from its real-valued (polyhedral) precursor, Theorem 8.4 — a genuinely different, harder theorem this mission does not draft (see Formalization scope), since the integer bijection needs the M-/L-proximity theorems of chunks 06–09 to control integrality under convex extension, while the real-valued case does not.

Difficulty

The obvious approach — try to prove the discrete conjugacy theorem directly by mimicking the real-valued proof (Theorem 8.4) with ℤ in place of ℝ everywhere — fails, because the real-valued proof's key step (Proposition 8.3, an infimal-convolution argument comparing arg min sets of perturbed polyhedral functions) has no immediate discrete analogue: a discrete arg min need not vary continuously with the perturbation the way a polyhedral one does. The book's actual strategy instead routes through the convex extension of the discrete function (chunk 06/08's bridge to chapter 3's integral convexity), applies the already-proved real-valued conjugacy theorem to the extension, and then must separately argue that the resulting conjugate, restricted back to integer points, is again integer-valued and satisfies the discrete exchange axiom — an argument that needs different treatment depending on whether the original function's domain is bounded or unbounded (an exhaustion argument via restriction to a growing integer interval, invoking chunk 06's proximity theorem to control convergence). Skipping this discreteness argument and treating the real-valued theorem as if it settled the integer case would silently discard exactly the chapter's own point.

Formalization scope

The ground set $V$ is a Fintype with DecidableEq. ConvexConjugate (the discrete transform) has domain and codomain both (V → ℤ) → WithTop ℝ, obtained by taking the defining supremum in EReal (a complete lattice, so it is always total) and projecting back via a new FromEReal map — this is what lets the biconjugate f•• typecheck as an equality of functions of the same type as f. ConvexConjugateR (the real-valued transform, used only by the milestone Theorem 8.1) is a separate object with no shared code, per the explicit warning against conflating the two transforms; the two never appear in the same item.

A trivializing formalization of the goal would draft only the real-valued case (Theorem 8.4) as if it were the discrete theorem, or would silently allow WithTop ℝ's subtraction-avoidance convention to change which values are compared; neither is done. Theorem 8.4 itself (the polyhedral conjugacy theorem) is not drafted in this mission at all — it would require a fresh, otherwise-unused polyhedral M-/L-convex-function layer on Rⱽ that no other item here needs (see MODERATION_NOTES.md). The M-/L-separation theorems (8.15, 8.16) and the Fenchel-type duality theorem (8.21) are likewise left for a follow-on mission; contributions building the polyhedral bridge or the separation theorems, which depend on machinery this mission establishes, are welcome.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.

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Combinatorics·Captain: mikedeng1

Applied Combinatorics II: Dilworth's Chain Covering TheoremTextbook

Motivation

Partially ordered sets model precedence: tasks that must wait for other tasks, versions that supersede others, files nested in directories, alternatives ranked by several criteria at once. Two questions about such a structure recur in scheduling and in the design of algorithms. How many sequential "threads" are needed so that every item lies on a thread in which all items are mutually comparable? How many "rounds" are needed so that no round contains two comparable items? Dilworth's theorem answers the first and its dual, due to Mirsky, answers the second: in both cases the obvious lower bound, given by one large set of mutually incomparable (resp. comparable) items, is attained.

This mission formalizes Chapter 6 of Keller and Trotter's open textbook Applied Combinatorics (2017 Edition, appliedcombinatorics.org), whose capstone is Dilworth's theorem, together with the chapter's other three proved theorems on the same objects.

Timeline. Sperner (1928, doi:10.1007/BF01171114) showed that the largest family of subsets of a $t$ -set in which no member contains another has $\binom{t}{\lfloor t/2\rfloor}$ members. Dilworth (1950, doi:10.2307/1969503) proved that a finite poset whose largest antichain has $w$ elements can be partitioned into $w$ chains. Mirsky (1971, doi:10.2307/2316481) recorded the dual statement for antichain partitions. Fishburn (1970, doi:10.1016/0022-2496(70)90062-3) characterized the posets that arise from comparing real intervals as those with no subposet $\mathbf 2 + \mathbf 2$ .

Setting

A poset $\mathbf P = (X, P)$ is a set $X$ with a reflexive, antisymmetric and transitive relation $\le$ . Here $X$ is finite and is represented by a finite type $\alpha$ with a partial order. A subset $C \subseteq X$ is a chain if every two distinct points of $C$ are comparable; a subset $A \subseteq X$ is an antichain if every two distinct points of $A$ are incomparable. The empty set is both.

The width of $\mathbf P$ is the largest size of an antichain, and the height the largest size of a chain:

\operatorname{width}(\mathbf P) = \max_{A \text{ antichain}} |A|, \qquad \operatorname{height}(\mathbf P) = \max_{C \text{ chain}} |C|.

A chain partition of $X$ into $k$ parts is a family $C_1, \dots, C_k$ of pairwise disjoint chains with union $X$ ; an antichain partition is defined the same way with antichains.

The subset lattice $\mathbf 2^t$ is the family of all subsets of a $t$ -element set, ordered by inclusion.

A poset is an interval order if each point $x$ can be assigned a closed real interval $[a_x, b_x]$ , degenerate intervals allowed, such that $x < y$ exactly when $b_x < a_y$ . The poset $\mathbf 2 + \mathbf 2$ consists of two disjoint two-element chains with no comparabilities between them, and $\mathbf P$ excludes $\mathbf 2 + \mathbf 2$ if no four points of $X$ induce a copy of it.

Formalization targets

Goal: Dilworth's theorem (Theorem 6.17)

For every finite poset $\mathbf P$ with $w = \operatorname{width}(\mathbf P)$ ,

\exists\, C_1, \dots, C_w \text{ chains partitioning } X, \quad\text{and}\quad X = C_1 \cup \dots \cup C_k \text{ a chain partition} \implies w \le k.

Milestones

Theorem 6.18 (dual of Dilworth). With $h = \operatorname{height}(\mathbf P)$ , $X$ has a partition into $h$ antichains, and every antichain partition has at least $h$ parts.
Theorem 6.27 (Sperner). For $t \ge 1$ ,

\operatorname{width}(\mathbf 2^t) = \binom{t}{\lfloor t/2 \rfloor}.

Theorem 6.29 (Fishburn). A finite poset is an interval order if and only if it excludes $\mathbf 2 + \mathbf 2$ .

The milestones stand on the same definitions as the goal (width, height, chain and antichain partitions, subposets) and are ordered from the one closest to the goal's statement to the one furthest from it.

Significance

The result itself. Dilworth's theorem is a min–max theorem: a minimum over covers equals a maximum over obstructions. It is equivalent to König's theorem on bipartite matchings and to Hall's marriage theorem, and it underlies the computation of minimum chain covers by bipartite matching and network flows (Chapter 14 of the same book). The dual theorem gives the layer decomposition used in scheduling with precedence constraints. Sperner's theorem is the first result of extremal set theory, and Fishburn's theorem is the basic structure theorem for interval orders, which model preferences with thresholds of indifference.

Formalizing it. All four results are classical and proved. In Mathlib at the environment revision of this mission there is no Dilworth or Mirsky theorem; Sperner's inequality (the upper bound on antichains in a Boolean lattice) is available through the LYM inequality, and there is no theory of interval orders. The Prove2Me library holds the easy half of Dilworth (an antichain is no larger than any chain partition) in a different Mathlib environment and over a different chain-partition structure. The mission asks for complete machine-checked statements of the book's four theorems over one common set of definitions.

Difficulty

For Dilworth's theorem the lower bound is a pigeonhole argument; the content is the existence of a partition into exactly $w$ chains. Greedy constructions fail: removing a longest chain, or a maximal chain through a chosen point, can leave a poset whose width has not decreased, so a partition built by repeatedly peeling chains may use more than $w$ of them. No local rule visible from a single chain decides whether its removal keeps the count at $w$ .

For Sperner's theorem the lower bound (the middle rank is an antichain) is immediate and the upper bound needs a counting argument over maximal chains; both halves are required, since the target is an equality. For Fishburn's theorem one direction is a four-line contradiction; the other requires constructing real intervals from nothing but the absence of $\mathbf 2 + \mathbf 2$ .

Formalization scope

The ground set is a type α with [PartialOrder α] [Fintype α]. Chains and antichains are Mathlib's IsChain (· ≤ ·) and IsAntichain (· ≤ ·). width α and height α are maxima of Finset.card over all antichains (resp. chains) of α; the family is never empty, since the empty set qualifies, so both are well defined and equal $0$ for the empty poset. Width is defined from antichains, not as the least number of chains in a partition; a definition of the latter kind would make the goal true by definition and is ruled out. Chain and antichain partitions are families indexed by Fin k of pairwise disjoint Finsets whose union is all of α; empty parts are not excluded, which does not change either theorem. The "no fewer" clauses quantify over every k and every such family. The subset lattice $\mathbf 2^t$ is Finset (Fin t) ordered by inclusion, $\lfloor t/2 \rfloor$ is t / 2 in ℕ, and Sperner's theorem keeps the book's hypothesis $t \ge 1$ . An interval representation is a pair of maps a b : α → ℝ with a x ≤ b x; $\mathbf 2 + \mathbf 2$ is Fin 2 ⊕ Fin 2 with the disjoint-sum order, and "excludes" means there is no order embedding of it into α. Fishburn's theorem is stated for finite posets: the book's construction of a representation is finite, and the equivalence fails for some uncountable posets.

No statement contains an asymptotic bound, so no explicit constant is instantiated.

A complete development needs finite-poset combinatorics (induction on the ground set, subposets as subtypes, maximal and minimal elements), which is reusable across order theory; results proving the definitions agree with Mathlib's notions of chains in Finset lattices are welcome, as is an alternative proof of Dilworth via Hall's theorem, which Mathlib has.

Selected references

M. T. Keller and W. T. Trotter, Applied Combinatorics, 2017 Edition, Chapter 6, pp. 113–140. https://www.appliedcombinatorics.org
R. P. Dilworth, A decomposition theorem for partially ordered sets, Annals of Mathematics 51 (1950), 161–166. https://doi.org/10.2307/1969503
L. Mirsky, A dual of Dilworth's decomposition theorem, American Mathematical Monthly 78 (1971), 876–877. https://doi.org/10.2307/2316481
E. Sperner, Ein Satz über Untermengen einer endlichen Menge, Mathematische Zeitschrift 27 (1928), 544–548. https://doi.org/10.1007/BF01171114
P. C. Fishburn, Intransitive indifference with unequal indifference intervals, Journal of Mathematical Psychology 7 (1970), 144–149. https://doi.org/10.1016/0022-2496(70)90062-3

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Convex Optimization Discrete Geometry Operations Research+1·Captain: Shuze Chen

Discrete Convex Analysis XXVI: Polyhedral L-Convex FunctionsTextbook

Motivation

L-convex functions were defined purely combinatorially, on the integer lattice. Chapter 6's M-convex theory showed that combinatorial definition always extends to a genuine convex function on real space (missions 23-ch06c-mconvexfunctions/24-ch06d-mconvexfunctions); this mission carries out the identical program for the L side. It first shows L $^\natural$ -convexity is exactly integral convexity plus ordinary submodularity — a clean synonym that also explains why submodular set functions are a natural special case — then builds the entire polyhedral (real-variable) theory of L-convex functions: the axioms (SBF[R])/(TRF[R]), two practical local criteria for verifying submodularity without checking every pair of points, the fact that the classical Lovász extension of a submodular set function is itself a polyhedral L-convex function, the six-operation closure toolkit, and — this mission's goal — the L-optimality criterion in its full polyhedral generality, characterizing global optimality by finitely many directional derivatives.

Setting

Fix a finite ground set $V$ . A polyhedral convex function $g : \mathbb R^V \to \mathbb R \cup \{+\infty\}$ with nonempty effective domain is polyhedral L-convex, $g \in L[\mathbb R \to \mathbb R]$ , if it satisfies (SBF[R]): $g(p)+g(q) \ge g(p\vee q)+g(p\wedge q)$ , and (TRF[R]): $\exists r \in \mathbb R$ , $g(p+\alpha\mathbf 1) = g(p)+\alpha r$ for all $p \in \mathbb R^V$ , $\alpha \in \mathbb R$ ; it is polyhedral L $^\natural$ -convex if its lift to one extra real coordinate is polyhedral L-convex. A function $g : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ is integrally convex if its convex closure agrees, at every real point, with the closure taken using only that point's integral neighborhood. The Lovász extension $\hat\rho$ of a submodular set function $\rho$ is the piecewise-linear interpolation built from the sorted distinct components of $p \in \mathbb R^V$ . The directional derivative $g'(p;d)$ is $\inf_{t>0}(g(p+td)-g(p))/t$ .

Formalization targets

Goal: the polyhedral L-optimality criterion (Theorem 7.33)

For a polyhedral L-convex function $g$ and $p \in \operatorname{dom}_{\mathbb R} g$ : $g(p) \le g(q)$ for all $q$ if and only if $g'(p;\chi_Y) \ge 0$ for every $Y \subseteq V$ and $g'(p;\mathbf 1)=0$ ; for polyhedral L $^\natural$ -convex $g$ , the criterion simplifies to $g'(p;\pm\chi_Y) \ge 0$ for every $Y$ . This is the direct L-side mirror of mission 24-ch06d-mconvexfunctions's M-optimality criterion (Theorem 6.52) and the polyhedral generalization of mission 08-lconvex-functions-i's integer-domain L-optimality criterion (Theorem 7.14): checking global optimality against exponentially many points reduces to $|V|+1$ (or $2|V|$ ) directional-derivative inequalities.

Supporting structural targets

Twelve further results build the polyhedral theory from the ground up. Theorems 7.20-7.21 identify L $^\natural$ -convexity with the conjunction of ordinary submodularity and integral convexity — a genuinely different, function-analytic characterization from the exchange-axiom- style definitions used so far. Propositions 7.23-7.24 give two practical sufficient conditions for verifying (SBF[R]) locally, at a single scale, rather than globally. Proposition 7.25 shows the Lovász extension of any submodular set function is automatically polyhedral L-convex — not in this chunk's own extraction table (its label is preceded by an unlabeled restatement of the same fact, which evidently confused the extractor), found and placed by direct reading. Theorem 7.26 shows an L-convex function's convex extension, when polyhedral, inherits polyhedral L-convexity, continuing mission 26-ch07b-lconvexfunctions's Theorem 7.19. Theorems 7.28-7.32 restate the discrete theory's core equivalences (translation submodularity, the L/L $^\natural$ correspondence, the six basic operations, restrictions) in the polyhedral setting, and Proposition 7.34 shows minimizer sets of linearly-perturbed polyhedral L-convex functions are themselves L-convex polyhedra — flagged by the book itself as a partial result whose full converse characterization (Theorem 7.45) lies beyond this chunk's range.

Significance

Theorems 7.20-7.21's synonym is structurally important: it means every algorithm and theorem already known for submodular-function minimization over $\{0,1\}^V$ -type domains applies, after a midpoint-convexity check, to the vastly larger class of integer-lattice L $^\natural$ -convex functions, with no new proof technique required. Proposition 7.25 is the bridge that lets the combinatorial Lovász extension — the workhorse of submodular optimization for forty years — be recognized as a special case of the polyhedral L-convex function theory this mission builds, explaining why algorithms for one transfer so readily to the other. The goal, Theorem 7.33, is the precise tool chapter 10's continuous-relaxation algorithms for L-convex-function minimization actually verify against: a scaling algorithm's claimed optimum is confirmed correct exactly by checking the criterion's finitely many directional-derivative inequalities.

None of these results are open — they are Murota's account of how the integer-lattice theory of L-convexity survives, result by result, the passage to polyhedral convex functions on $\mathbb R^V$ , mirroring chapter 6's identical program for M-convexity. What this mission contributes is a faithful, machine-checked formal statement of each, including one result (Proposition 7.25) the platform's own automated extractor missed, extending the shared Lean vocabulary (SBFR, TRFR, LovaszExtension, DirDeriv) this series builds on; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The naive approach to the goal would try to verify $g(p) \le g(q)$ directly against every $q \in \mathbb R^V$ ; the book's actual proof instead reduces this to the finite family of directional derivatives via Theorem 7.20's integral-convexity fact (an L $^\natural$ -convex function's local behavior determines its global behavior) applied to the polyhedral setting through Theorem 3.21's general optimality criterion for integrally convex functions — a two-layer reduction (polyhedral $\to$ integral-convexity $\to$ finite local check) with no direct one-step argument. The genuine combinatorial content in this block is in Proposition 7.25's proof: showing the Lovász extension is submodular requires the finite-valued case (a direct calculation split on whether the two perturbed coordinates land in the same or different threshold sets) and then a limiting argument over a sequence of finite-valued truncations $\rho_k \to \rho$ for the general, possibly-infinite case — a genuine two-step argument, not a single inequality chase.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; polyhedral L-(natural-)convex functions are (V→ℝ)→WithTop ℝ. All thirteen numbered results found in this chunk's page range are placed, with one documented scope reduction: Proposition 7.24 states only part (1) (the unconditional-on-magnitude sufficient condition), not part (2)'s sharper, sorted-index-restricted version, which needs the same SortedValues apparatus a second time for no other result's benefit — see HARD.md. " $g_U > -\infty$ " and its variants are replaced by the equivalent (DomR ...).Nonempty hypothesis throughout, matching mission 24-ch06d-mconvexfunctions's identical substitution. "Domain is closed"/"domain is an interval" (Propositions 7.23-7.24) are stated via Mathlib's IsClosed and Set.OrdConnected respectively, the latter being the precise order-theoretic notion of "interval" in a pointwise-ordered space. This mission's base vocabulary is redeclared from missions 08-lconvex-functions-i, 20-ch04b-mconvexsets (for the Lovász extension machinery), 21-ch05b-lconvexsets, 23-ch06c-mconvexfunctions/ 24-ch06d-mconvexfunctions, and 26-ch07b-lconvexfunctions rather than imported, since sibling drafts in this series cannot yet reference one another. Contributions completing any of the thirteen sorrys are welcome; the goal and Proposition 7.25 carry the most independent proof content.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
K. Murota and A. Shioura, "Extreme points of a generalized polymatroid," Discrete Applied Mathematics, 152 (2005), pp. 268-278 [152] (the polyhedral L-convex function theory this mission's real-variable results are drawn from).

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Convex Optimization Discrete Geometry Operations Research+1·Captain: Shuze Chen

Discrete Convex Analysis XXV: L-Convex Functions via Minimizer PolyhedraTextbook

Motivation

Chapter 5 characterized L-convex sets — sublattice-closed, translation-periodic subsets of $\mathbb Z^V$ — and showed they interact cleanly with integral convexity. Chapter 7 asks the functional analogue: which functions on the integer lattice deserve to be called convex in the "L" sense, and how do they relate back to L-convex sets? This mission (continuing mission 08-lconvex-functions-i, which built the axioms (SBF[Z])/(TRF[Z])/(SBF $^\natural$ [Z]) and proved the L-optimality and L-proximity theorems) answers the second question at its sharpest: an L-convex function is exactly a function whose every weighted-minimizer set is an L-convex polyhedron — the discrete analogue of the fact that a convex function is determined by the convex geometry of its sublevel sets. Along the way it settles the chapter's basic toolkit: operations that preserve L-convexity, the local nature of submodularity, the correspondence with ordinary submodular set functions, and the first two structural facts about the convex extension every L-convex function admits.

Setting

Fix a finite ground set $V$ . A function $g : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ with nonempty effective domain is L-convex, $g \in L[\mathbb Z \to \mathbb R]$ , if it satisfies submodularity (SBF[Z]): $g(p)+g(q) \ge g(p\vee q) + g(p\wedge q)$ , and translation invariance (TRF[Z]): $\exists r \in \mathbb R$ , $g(p+\mathbf 1) = g(p) + r$ for all $p$ . It is L $^\natural$ -convex if its lift to one extra coordinate (Eq. (7.2)) is L-convex. A set function $\rho : 2^V \to \mathbb R \cup \{+\infty\}$ is submodular if $\rho(X)+\rho(Y) \ge \rho(X\cup Y) + \rho(X\cap Y)$ ; it corresponds to an L $^\natural$ -convex function supported on $\{0,1\}^V$ via $g(\chi_X) = \rho(X)$ (Eq. (7.5)). The convex closure $\bar g$ of $g$ is its extension to $\mathbb R^V$ by finite convex combinations.

Formalization targets

Goal: L-convexity via minimizer polyhedra (Theorem 7.17)

For $g : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ with bounded nonempty effective domain: $g$ is L-convex if and only if $\arg\min g[-x]$ is an L-convex set for every $x \in \mathbb R^V$ ; the L $^\natural$ analogue holds with L $^\natural$ -convex sets. This is the direct mirror of mission 23-ch06c-mconvexfunctions's own goal (Theorem 6.43, characterizing M-convex functions via M-convex weighted-minimizer polyhedra) — the book's text calls it exactly "how the concept of L-convex functions can be defined from that of L-convex sets."

Supporting structural targets

Eleven further results build the chapter's basic vocabulary. Theorem 7.2 strengthens translation submodularity to allow negative shifts; Theorem 7.3 places L-convexity inside L $^\natural$ - convexity; Proposition 7.4 identifies submodular set functions with a subclass of L $^\natural$ -convex functions via the indicator embedding, and Theorem 7.15 derives the classical submodular-minimizer local-optimality criterion as its corollary; Proposition 7.5 shows submodularity is a local property, needing only unit-distance pairs; Proposition 7.8 transfers L-(natural-)convexity from functions to their effective domains; Proposition 7.9 and Theorem 7.10–7.11 give the chapter's basic examples (univariate and pairwise-difference functions) and its six-operation closure toolkit (scaling, affine reparametrization, linear perturbation, projection, infimal convolution with a separable function, and sums), both for L-convex and L $^\natural$ - convex functions, the latter also admitting interval and coordinate restrictions; Proposition 7.16 shows minimizer sets of L-convex functions are themselves L-convex, the special case ( $x=0$ ) the goal generalizes to every linear perturbation; Theorem 7.19 begins the convex-extension program this chapter's next chunk completes, establishing that the convex closure agrees with $g$ on $\mathbb Z^V$ and inherits its translation constant.

Significance

The goal is significant for the same structural reason as its M-side counterpart: it says L-convexity is not merely a combinatorial condition on lattice differences but is equivalent to a purely polyhedral-geometric one, closing the loop between chapters 5 and 7 the way Theorem 6.43 closes the loop between chapters 4 and 6. Theorem 7.15's corollary status is itself instructive: the well-known fact that a submodular set function's global minimizer needs only local verification against comparable sets — the theoretical basis of every submodular-minimization algorithm in chapter 10 — falls out of the L-optimality criterion (mission 08-lconvex-functions-i's Theorem 7.14) applied to the indicator embedding, rather than needing an independent proof. Theorem 7.10–7.11's six operations are the toolkit every later construction in this chapter and chapter 9's network transformations builds new L-convex functions from old.

None of these results are open — they are Murota's account of the basic function-level theory of L-convexity, mirroring chapter 6's M-convex function theory chunk-by-chunk. What this mission contributes is a faithful, machine-checked formal statement of each, extending the shared Lean vocabulary (SBF, TRF, LNaturalConvex, LConvexSet) that missions 08-lconvex-functions-i and 21-ch05b-lconvexsets began; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The naive approach to the goal would try to verify L-convexity's submodularity inequality directly against the definition of an L-convex set applied to each minimizer family; the book's actual proof instead routes through Theorem 7.10 (3)'s closure of L-convexity under linear perturbation and Proposition 7.16's minimizer-is-L-convex-set fact for the forward direction, and defers the converse entirely to a later note (Note 7.47, outside this chunk and mission 08's combined range) proved via the integral-convexity machinery of section 7.7 onward. The genuine combinatorial difficulty in this block is upstream, in Theorem 7.10 (5)'s infimal-convolution operation: proving L-convexity of the perturbed function requires a four-term submodularity inequality assembled from the separable function's own convexity and $g$ 's submodularity applied at the optimal $q_1,q_2$ simultaneously — a genuine two-hypothesis combination with no single-inequality shortcut.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; L-(natural-)convex functions are (V→ℤ)→WithTop ℝ. All twelve numbered results found in this chunk's page range are placed, with one documented scope reduction: Theorem 7.19 states only parts (3)-(4) (that the convex closure agrees with $g$ on $\mathbb Z^V$ and inherits its translation constant), not the explicit Lovász-extension-formula construction of parts (1)-(2) and (5), which needs a sorted-distinct- component apparatus no other result in this chunk requires — see HARD.md. " $g_U > -\infty$ " and its variants are replaced by the equivalent (DomZ ...).Nonempty hypothesis throughout, matching mission 24-ch06d-mconvexfunctions's identical substitution (WithTop ℝ has no $-\infty$ element). This mission's base vocabulary (SBF, TRF, LNaturalConvex, etc.) is redeclared verbatim from mission 08-lconvex-functions-i rather than imported, since sibling drafts in this series cannot yet reference one another; LConvexSet is likewise redeclared from mission 21-ch05b-lconvexsets. Contributions completing any of the twelve sorrys are welcome; the goal and Theorem 7.10 carry the most independent proof content.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
K. Murota, "Discrete convex analysis," Mathematical Programming, 83 (1998), pp. 313–371 (the original account of L-convex functions this chapter's basic theory is drawn from).

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Analysis Functional Analysis Partial Differential Equations·Captain: mikedeng1

Notes on Partial Differential Equations X: Galerkin Existence of Weak Solutions of Parabolic PDEsTextbook

Motivation

Linear second-order parabolic equations such as $u_t + Lu = f$ model diffusion with advection, growth or decay, and external sources. The heat equation is the prototype. For coefficients that are only bounded and measurable, classical solutions need not exist, and the natural well-posedness theory is the $L^2$ energy theory of weak solutions. In that theory the solution is a curve $t \mapsto u(t)$ in a Sobolev space, and the equation holds in the dual space. This framework, developed by J.-L. Lions and others in the 1950s–60s (Lions–Magenes 1972), underlies the numerical analysis of parabolic problems through finite elements and spectral Galerkin methods. It also underlies the existence theory for nonlinear evolution equations such as the Navier–Stokes equations (Temam 1977).

This mission formalizes Chapter 6 of J. K. Hunter's Notes on Partial Differential Equations (UC Davis, revised 6/18/2014): the existence, uniqueness and energy estimate for weak solutions of the parabolic initial–boundary value problem, proved by the Galerkin method. It also formalizes the vector-valued calculus on Hilbert triples that the proof rests on, and the Ehrling and Aubin–Lions compactness results of §6.6.

Setting

Let $\Omega \subset \mathbb{R}^n$ be a bounded open set and $T > 0$ . $H^1_0(\Omega)$ is the closure of $C_c^\infty(\Omega)$ in the norm $\|u\|_{H^1_0}^2 = \int_\Omega (u^2 + |Du|^2)\,dx$ . $H^{-1}(\Omega)$ is its dual, with pairing $\langle f, v\rangle$ . Every $u \in L^2(\Omega)$ acts on $H^1_0(\Omega)$ by $v \mapsto \int_\Omega uv\,dx$ .

The operator is $Lu = -\sum_{i,j}\partial_i(a^{ij}\partial_j u) + \sum_j b^j\partial_j u + cu$ . Its coefficients satisfy Assumption 6.1: $a^{ij}, b^j, c \in L^\infty(\Omega\times(0,T))$ , $a^{ij} = a^{ji}$ , and uniform ellipticity $\sum a^{ij}\xi_i\xi_j \ge \theta|\xi|^2$ for some $\theta > 0$ . The associated form is

a(u, v; t) = \sum_{i,j}\int_\Omega a^{ij}\partial_i u\,\partial_j v\,dx + \sum_j\int_\Omega b^j\partial_j u\,v\,dx + \int_\Omega c\,uv\,dx.

A function $u : [0,T] \to H^1_0(\Omega)$ has weak time derivative $u_t = w$ if $\int_0^T \phi'(t)u(t)\,dt = -\int_0^T\phi(t)w(t)\,dt$ for all $\phi \in C_c^\infty(0,T)$ , with Bochner integrals in $H^{-1}(\Omega)$ . Given $f \in L^2(0,T;H^{-1}(\Omega))$ and $g \in L^2(\Omega)$ , $u$ is a weak solution of

u_t + Lu = f \ \text{in } \Omega\times(0,T), \qquad u = 0 \ \text{on } \partial\Omega, \qquad u(0) = g

when three conditions hold. First, $u \in L^2(0,T;H^1_0)$ and $u_t \in L^2(0,T;H^{-1})$ . Second, $\langle u_t(t), v\rangle + a(u(t), v; t) = \langle f(t), v\rangle$ for every $v \in H^1_0(\Omega)$ and a.e. $t$ . Third, $u(0) = g$ , where the value at $0$ is that of the continuous representative in $C([0,T];L^2(\Omega))$ .

The Galerkin approximations use an $L^2$ -orthonormal basis $\{w_k\}$ of Dirichlet eigenfunctions and the spaces $E_N = \langle w_1, \dots, w_N\rangle$ . The approximation $u_N : [0,T] \to E_N$ solves the projected equation $(u_{Nt}, v)_{L^2} + a(u_N, v; t) = \langle f, v\rangle$ for $v \in E_N$ , with $u_N(0) = P_N g$ .

A Hilbert triple $\mathcal{V} \hookrightarrow \mathcal{H} \hookrightarrow \mathcal{V}'$ consists of two separable Hilbert spaces $\mathcal{V}$ , $\mathcal{H}$ and the dual $\mathcal{V}'$ , with dense continuous injective embeddings and $\langle f, v\rangle = (f, v)_{\mathcal{H}}$ for $f \in \mathcal{H}$ . The model is $H^1_0 \hookrightarrow L^2 \hookrightarrow H^{-1}$ .

Formalization targets

Goal: Theorem 6.3

For every $f \in L^2(0,T;H^{-1}(\Omega))$ and $g \in L^2(\Omega)$ there is a unique weak solution $u \in C([0,T];L^2) \cap L^2(0,T;H^1_0)$ with $u_t \in L^2(0,T;H^{-1})$ . There is also a constant $C = C(\Omega, T, L)$ such that

\|u\|_{L^\infty(0,T;L^2)} + \|u\|_{L^2(0,T;H^1_0)} + \|u_t\|_{L^2(0,T;H^{-1})} \le C\big(\|f\|_{L^2(0,T;H^{-1})} + \|g\|_{L^2}\big).

Milestones

Theorem 6.41. If $u \in L^2(0,T;\mathcal{V})$ and $u_t \in L^2(0,T;\mathcal{V}')$ , then $u \in C([0,T];\mathcal{H})$ . Moreover $\frac{d}{dt}(u,v)_{\mathcal{H}} = \langle u_t, v\rangle$ , $\frac{d}{dt}\|u\|^2_{\mathcal{H}} = 2\langle u_t, u\rangle$ , and $\|u\|_{L^\infty(\mathcal{H})} \le C(\|u\|_{L^2(\mathcal{V})} + \|u_t\|_{L^2(\mathcal{V}')})$ .
Theorem 6.42. $\int_0^T\langle u_t, v\rangle\,dt = (u(T),v(T))_{\mathcal{H}} - (u(0),v(0))_{\mathcal{H}} - \int_0^T\langle u, v_t\rangle\,dt$ .
Proposition 6.5. Each Galerkin problem has a unique solution $u_N$ .
Proposition 6.6. The energy estimate above holds for $u_N$ with $C$ independent of $N$ .
Proposition 6.7. A subsequence of $u_N$ converges weakly to a weak solution satisfying the estimate.
Lemma 6.10 (Ehrling) and Theorem 6.9 (Aubin–Lions). These are further results of the chapter, stated for Banach spaces $X \hookrightarrow Y \hookrightarrow Z$ with $X$ compactly embedded in $Y$ .

Significance

Theorem 6.3 is the basic well-posedness theorem for linear parabolic equations with rough coefficients. It gives existence and uniqueness, and continuous dependence on the data through the linear estimate. The same argument gives the Galerkin convergence that justifies spectral and finite-element semi-discretizations. The Hilbert-triple calculus of Theorems 6.41–6.42 is the standard setting for linear and nonlinear evolution equations. The Aubin–Lions lemma is the compactness tool for nonlinear problems, such as the semilinear heat equation of §6.6 and the Navier–Stokes equations of §6.7.

All of these results are classical and proved in the notes, except that Theorem 6.41's proof leans on vector-valued mollification. As far as a search of the platform shows, none of them is formalized there. Mathlib has the Bochner integral, $L^p$ spaces of Banach-valued functions and Gronwall's inequality. It does not have weak derivatives of vector-valued functions, Hilbert triples, or evolution-equation existence theorems. The work here is therefore to formalize known proofs, and in doing so to build reusable infrastructure for time-dependent PDEs.

Difficulty

The main obstacle is that the natural objects are equivalence classes of vector-valued functions, not functions. The initial condition $u(0) = g$ is meaningless for an element of $L^2(0,T;H^1_0)$ until Theorem 6.41 provides a continuous representative. Its proof needs extension across the endpoints and mollification of $\mathcal{V}$ -valued functions. Passing to the limit in the Galerkin scheme needs weak compactness in $L^2(0,T;H^1_0)$ and $L^2(0,T;H^{-1})$ , identified with a dual Bochner space. Proposition 6.5 is a linear ODE system whose coefficients are only $L^\infty$ in time, so the Picard–Lindelöf theorem for continuous right-hand sides does not apply directly. The $H^{-1}$ bound on $u_{Nt}$ needs the projection $P_N$ to be bounded on $H^1_0$ as well as on $L^2$ , which is why the basis is taken to be eigenfunctions. Uniqueness needs the energy identity (6.46) and a Gronwall argument in weak form.

Formalization scope

$\mathbb{R}^n$ is EuclideanSpace ℝ (Fin n), with 0-based coordinates. H10 n Ω is the closure of the jets $(\phi, \nabla\phi)$ , $\phi \in C_c^\infty(\Omega)$ , in $L^2(\Omega;\mathbb{R}\times\mathbb{R}^n)$ , so its norm is the full $H^1$ norm. Hm1 n Ω is StrongDual ℝ (H10 n Ω). l2Embed is the embedding $H^1_0 \hookrightarrow L^2 \hookrightarrow H^{-1}$ .
$L^p(0,T;X)$ is MemLp for Lebesgue measure on $(0,T)$ , and norms are eLpNorm with values in $[0,\infty]$ . Constants are finite nonnegative reals. Every constant is quantified before the data it must not depend on: before $f$ , $g$ in Theorem 6.3, and before the basis, $f$ , $g$ , $N$ in Propositions 6.6–6.7.
The weak time derivative is (6.14) in its standard form $\int\phi' u = -\int\phi w$ . The page prints $\phi$ and $\phi'$ interchanged, which is a typo.
Weak solutions take $g \in L^2(\Omega)$ as in Assumption 6.1(3). The printed Theorem 6.3 says $g \in H^1_0(\Omega)$ , a special case. $u(0) = g$ is read through a continuous representative $\tilde u \in C([0,T];L^2)$ , and uniqueness means equality for a.e. $t$ .
Coefficient conditions and ellipticity hold almost everywhere on $\Omega\times(0,T)$ . The Galerkin basis is a hypothesis: an orthonormal basis of $L^2(\Omega)$ of weak Dirichlet eigenfunctions.
Proposition 6.7 states weak convergence of $u_{N_k}$ in $L^2(0,T;H^1_0)$ and of $u_{N_k t}$ in $L^2(0,T;H^{-1})$ , in the explicit integral form of p. 186. This is at least as strong as the printed claim.
Hilbert triples are structures with two continuous injective dense-range embeddings, separability, and the compatibility identity. Reflexivity in Theorem 6.9 is surjectivity of the canonical map into the bidual.
Ruling out trivializations. Every Bochner integral in a definition has an integrable integrand under the stated $L^2$ hypotheses, so no weak-derivative or weak-solution condition holds through the integral's junk value $0$ . The initial condition is imposed on a continuous representative, so it is not vacuous on a null set.
Reusable infrastructure welcome: vector-valued mollification and extension, the Hilbert-triple calculus, and dual identification of $L^2(0,T;X)'$ .

Selected references

J. K. Hunter, Notes on Partial Differential Equations, UC Davis lecture notes, revised 6/18/2014, Chapter 6. https://www.math.ucdavis.edu/~hunter/pdes/pde_notes.pdf
L. C. Evans, Partial Differential Equations, 2nd ed., Graduate Studies in Mathematics 19, AMS, 2010, §7.1. https://doi.org/10.1090/gsm/019
J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications I, Springer, 1972. https://doi.org/10.1007/978-3-642-65161-8
R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis, North-Holland, 1977; AMS Chelsea reprint 2001. https://doi.org/10.1090/chel/343
J. Simon, Compact sets in the space $L^p(0,T;B)$ , Ann. Mat. Pura Appl. 146 (1987), 65–96. https://doi.org/10.1007/BF01762360

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Machine Learning Probability Statistics·Captain: mikedeng1

On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities II: The Uniform Deviation BoundResearch Paper

Motivation

Bernoulli's law of large numbers says that the relative frequency of a single event $A$ in $l$ independent trials converges in probability to $P(A)$ . Statistics and learning theory need more: the probabilities of a whole class $S$ of events are judged from one and the same sample, so the frequencies must converge uniformly over the class. Uniform convergence can fail even for simple classes (all open subsets of $[0,1]$ ), so one needs a criterion that says when it holds and how fast.

Vapnik and Chervonenkis gave the first distribution-free answer in On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities (Theory Probab. Appl. 16 (1971) 264–280, doi:10.1137/1116025). Its Theorem 2, now called the VC inequality, bounds the probability of a uniform deviation larger than $\varepsilon$ by a combinatorial quantity of the class times an exponentially small factor.

Timeline:

1933: Glivenko and Cantelli prove uniform almost-sure convergence of the empirical distribution function on the line (the class of rays $\{x \le a\}$ ).
1971: Vapnik and Chervonenkis publish the growth function, the VC inequality (Theorem 2), almost-sure convergence under polynomial growth (Theorem 3) and the entropy criterion (Theorem 4).
1972: Sauer and Shelah independently prove the polynomial bound on the growth function (the paper's Lemma 1).
From the late 1970s: the inequality is sharpened in its constants and extended to empirical processes (Dudley, Pollard, Talagrand).

Setting

Let $(X, P)$ be a probability space and $S$ a collection of measurable events $A \subseteq X$ , with probabilities $P_A$ . A sample of size $l$ is a sequence $x_1, \dots, x_l$ of points of $X$ drawn independently with law $P$ , so the sample has the product law $P^l$ on $X^l$ . The relative frequency of $A$ in the sample is $\nu_A^{(l)} = n_A / l$ , where $n_A$ is the number of sample terms in $A$ . The uniform deviation is

\pi^{(l)} = \sup_{A \in S} \bigl|\nu_A^{(l)} - P_A\bigr|.

Each $A \in S$ induces in a sample $x_1, \dots, x_r$ the subsample of terms lying in $A$ . The index $\Delta^S(x_1, \dots, x_r)$ is the number of different subsamples so induced (at most $2^r$ ), and the growth function is $m^S(r) = \max \Delta^S(x_1, \dots, x_r)$ over all samples of size $r$ .

For a double sample $x_1, \dots, x_{2l}$ let $\nu'_A$ and $\nu''_A$ be the frequencies of $A$ in the two semi-samples $x_1, \dots, x_l$ and $x_{l+1}, \dots, x_{2l}$ , and let

\rho^{(l)} = \sup_{A \in S} \bigl|\nu'_A - \nu''_A\bigr|.

Following the paper, $\pi^{(l)}$ and $\rho^{(l)}$ are assumed to be measurable functions of the sample for every $l$ .

Formalization targets

Goal: Theorem 2 (p. 269)

For every $\varepsilon > 0$ and every $l \ge 2/\varepsilon^2$ ,

P\bigl(\pi^{(l)} > \varepsilon\bigr) \le 4\, m^S(2l)\, e^{-\varepsilon^2 l/8}.

The constants $4$ and $1/8$ and the growth function at $2l$ are the paper's.

Milestones, in the order of the proof

Lemma 2 (p. 268): for $l \ge 2/\varepsilon^2$ , $P\{\rho^{(l)} \ge \varepsilon/2\} \ge \tfrac12 P\{\pi^{(l)} > \varepsilon\}$ .
Eq. (11) (p. 270): $P\{\rho^{(l)} \ge \varepsilon/2\} = \int \frac{1}{(2l)!} \sum_{T} \theta\bigl(\rho^{(l)}(T X_{2l}) - \varepsilon/2\bigr)\, dP$ , the sum over all permutations $T$ of the $2l$ positions ( $\theta$ the indicator of $[0, \infty)$ ).
The $\Gamma$ estimate (p. 271): for $0 \le m \le 2l$ ,

\Gamma = \sum_{k : |2k/l - m/l| \ge \varepsilon/2} \frac{\binom{m}{k}\binom{2l-m}{l-k}}{\binom{2l}{l}} \le 2e^{-\varepsilon^2 l/8}.

The per-sample permutation bound (p. 271): for every fixed double sample, $\frac{1}{(2l)!}\sum_T \theta\bigl(\rho^{(l)}(T X_{2l}) - \varepsilon/2\bigr) \le 2\Delta^S(x_1, \dots, x_{2l})\, e^{-\varepsilon^2 l/8}$ .
The semi-sample bound (p. 271): $P\{\rho^{(l)} \ge \varepsilon/2\} \le 2\, m^S(2l)\, e^{-\varepsilon^2 l/8}$ for every $l \ge 1$ .

Further items (consequences, not milestones)

Corollary (p. 269): if $m^S(l) \le l^n + 1$ for all $l$ and some finite $n$ , then $P(\pi^{(l)} > \varepsilon) \to 0$ for every $\varepsilon > 0$ .
Theorem 3 (p. 271): under the same condition, $P(\pi^{(l)} \to 0) = 1$ for an infinite i.i.d. sequence.

Significance

The bound holds for every distribution $P$ and depends on the class only through $m^S(2l)$ . Together with the paper's Theorem 1 (the growth function is either $2^r$ for every $r$ or bounded by $r^n + 1$ ), it shows that every class whose growth function is not identically $2^r$ enjoys uniform convergence at an exponential rate in probability and almost surely. The Glivenko–Cantelli theorem is the special case of rays on the line. The inequality underlies sample-complexity bounds for empirical risk minimization, the "finite VC dimension implies learnability" direction of the fundamental theorem of statistical learning, and the theory of empirical processes indexed by sets.

The result has been proved for more than fifty years; what is missing is a machine-checked proof of it in its original form. Formal libraries contain Hoeffding-type inequalities for independent variables and textbook uniform-convergence statements with other constants, stated for hypothesis classes and loss functions. This mission produces the 1971 statement itself, with its constants and its sequence-based index, together with the combinatorial tail bound for sampling without replacement that the paper states without proof.

Difficulty

The obvious argument applies Hoeffding's inequality to each $A \in S$ and takes a union bound. This fails as soon as $S$ is infinite, and the classes of interest are uncountable. The growth function can only enter after the probabilities $P_A$ have been removed from the event, because only then does the event depend on the finitely many subsamples that $S$ induces on a finite sample. Lemma 2 does this at the price of the condition $l \ge 2/\varepsilon^2$ and a factor $2$ .

The second difficulty is combinatorial. Under a random rearrangement of a fixed double sample, the number of points of an event that fall into the first half is hypergeometric, not binomial. The paper states the required tail bound $\Gamma \le 2e^{-\varepsilon^2 l/8}$ and omits the "simple but long computation". Mathlib has no tail bound for sampling without replacement.

Formalization scope

Samples are functions Fin l → X with 0-based positions; repetitions are allowed. The subsample induced by $A$ is the set of positions {i | x i ∈ A}, so the index counts distinct sets of positions and the growth function maximizes over sequences, not finite point sets (the paper's model; the two differ when $X$ has fewer than $r$ points). The sample law is Measure.pi (fun _ : Fin l => P). The double sample is Fin (l + l) → X, read through Fin.castAdd and Fin.natAdd, and $m^S(2l)$ is growth S (2 * l). Suprema are real suprema over the events of $S$ (values in $[0, 1]$ ; $0$ for $S = \emptyset$ ). Probabilities are values in $[0, \infty]$ and the bounds enter through ENNReal.ofReal. Theorem 3 uses the infinite product Measure.infinitePi and evaluates $\pi^{(l)}$ on the first $l$ coordinates.

The measurability of $\pi^{(l)}$ (p. 265) and of $\rho^{(l)}$ (p. 268) are the paper's own assumptions and are carried as hypotheses. Without them Theorem 2 can fail for uncountable classes; replacing them by a stronger condition such as countability of $S$ would weaken the theorem.

A trivializing formalization is ruled out as follows: $\varepsilon > 0$ is stated, which forces $l \ge 1$ and so avoids the value $0/0 = 0$ of the frequency. The supremum runs over the events of $S$ , not over all subsets. The index counts distinct subsamples, not sets.

Corrections of the printed text:

Lemma 2 is printed for $l > 2/\varepsilon^2$ , but its proof concludes for $l \ge 2/\varepsilon^2$ , and Theorem 2 uses $l = 2/\varepsilon^2$ . The $\ge$ form is stated, which is the stronger statement.
The text before the permutation bound describes the averaged quantity as counting arrangements with $|\nu'_A - \nu''_A| \le \tfrac12\varepsilon$ ; the indicator and the index set of $\Gamma$ count those with $\ge \tfrac12\varepsilon$ , which is what is stated.
Slips in the proof of Lemma 2 that do not affect any statement: $\varepsilon/3$ printed for $\varepsilon/2$ on p. 269, and $<$ , $>$ where Chebyshev's inequality gives $\le$ , $\ge$ .
Theorem 2 prints "more then" for "more than".

Needed infrastructure:

the invariance of Measure.pi under permutations of coordinates;
the splitting of Fin (l + l) into two halves under the product measure;
Chebyshev's inequality for binomial frequencies;
a hypergeometric (sampling without replacement) tail bound.

The last two are reusable beyond this mission. Proofs of any milestone are welcome.

Selected references

V. N. Vapnik and A. Ya. Chervonenkis, On the uniform convergence of relative frequencies of events to their probabilities, Theory Probab. Appl. 16(2) (1971) 264–280. https://doi.org/10.1137/1116025
W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963) 13–30. https://doi.org/10.1080/01621459.1963.10500830
N. Sauer, On the density of families of sets, J. Combin. Theory Ser. A 13 (1972) 145–147. https://doi.org/10.1016/0097-3165(72)90019-2
S. Shalev-Shwartz and S. Ben-David, Understanding Machine Learning: From Theory to Algorithms, Cambridge University Press, 2014, Ch. 6 and 28. https://doi.org/10.1017/CBO9781107298019

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Convex Optimization Discrete Geometry Operations Research+1·Captain: Shuze Chen

Discrete Convex Analysis VIII: Quasi L-Convex Functions and the Quasi-Proximity TheoremTextbook

Motivation

Milgrom and Shannon's theory of quasi-supermodularity, developed for monotone comparative statics in economics, showed that many of the consequences of lattice submodularity survive under a much weaker, purely ordinal relaxation of the defining inequality. Chapter 7's final section imports this idea into discrete convex analysis: does L-convexity's optimality and proximity theory survive when the additive submodularity inequality is relaxed to an ordinal condition on the sign pattern of the two relevant differences, rather than their sum? This mission formalizes the chapter's answer for the strongest of the relevant relaxations, (SSQSB) (semistrict quasi submodularity): yes, and the class is large enough to include every strictly increasing rescaling of an L-convex function — exactly mirroring chunk 07's result for the M-convex side, and completing the "quasi" theory on both halves of the exchange-axiom framework before chapter 8 unifies them under conjugacy.

Setting

Let $V$ be a finite ground set and $g : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ . Building on chunk 08's submodularity axiom (SBF[Z]), this section introduces four ordinal relaxations. $g$ is quasi submodular, satisfying (QSB), if for every $p, q \in \mathbb Z^V$ , $g(p \wedge q) \le g(p)$ or $g(p \vee q) \le g(q)$ . $g$ is semistrictly quasi submodular, satisfying (SSQSB), if additionally $g(p \vee q) \ge g(q) \implies g(p \wedge q) \le g(p)$ and symmetrically. The weak variants (QSBw) and (SSQSBw) restrict attention to points of the effective domain and compare $\max(g(p), g(q))$ against $\min(g(p \wedge q), g(p \vee q))$ directly, with (SSQSBw) additionally allowing the four-way tie $g(p) = g(q) = g(p\wedge q) = g(p \vee q)$ . The linear perturbation of $g$ by $x : V \to \mathbb R$ is $g[x](p) = g(p) + \langle p, x \rangle$ .

Formalization targets

Goal: Theorem 7.54 (the quasi L-proximity theorem)

Let $g$ satisfy (SSQSB) and $g(p) = g(p + \mathbf 1)$ for all $p$ , $n = |V|$ , $\alpha$ a positive integer. If $p_\alpha \in \operatorname{dom} g$ satisfies $g(p_\alpha) \le g(p_\alpha + \alpha \chi_Y)$ for all $Y \subseteq V$ , then $\arg\min g \ne \emptyset$ and there is $p^* \in \arg\min g$ with the componentwise bound $p_\alpha \le p^* \le p_\alpha + (n-1)(\alpha-1) \mathbf 1$ — verbatim the same conclusion, and the same exact bound, as chunk 08's Theorem 7.18(1), now established for the strictly larger class satisfying (SSQSB) rather than (SBF[Z]).

Milestones: Theorems 7.49, 7.53

Theorem 7.49: the full nesting chain (SBF[Z]) $\Rightarrow$ (SSQSB) $\Rightarrow$ (QSB), (SSQSB) $\Rightarrow$ (SSQSBw) $\Rightarrow$ (QSBw), together with the collapse theorem that (SBF[Z]) holds if and only if every linear perturbation of $g$ satisfies (QSBw) — precisely quantifying how weak (QSBw) is pointwise and how the classes reunite under universal perturbation. Theorem 7.53 (the quasi L-optimality criterion): the direct analogue of chunk 08's Theorem 7.14, showing that global (or, for the weaker (QSBw) case, unique-up-to-translation) optimality still reduces to a purely local check against the $2^n - 2$ nontrivial sign-pattern neighbors $p + \chi_X$ .

Significance

The result itself. As with the M-convex case (chunk 07), the proximity theorem is what algorithms actually need: an L-convex-flavored objective transformed by any strictly increasing scalar rescaling (a common device — expressing a network-flow cost in a different currency, or applying a monotone risk adjustment) retains a scaling algorithm's correctness guarantee with exactly the same distance bound, even though the rescaled function is generally no longer L-convex itself.

Formalizing it. No matching item exists on the platform for quasi submodularity or quasi L-convexity in any form. Together with chunk 07 (the M-side quasi-convexity theory), this mission completes the "quasi" relaxation on both halves of the exchange-axiom framework the book develops, immediately before chapter 8 unifies M-convexity and L-convexity under a single conjugacy relationship.

Difficulty

As with chunk 07's quasi M-proximity theorem, the temptation is to imitate chunk 08's L-proximity proof line by line. The overall architecture does survive — translate so $p_\alpha = 0$ , find a lattice-minimal sufficiently-good point, and bound the gap using submodularity — but chunk 08's proof uses (SBF[Z])'s additive inequality directly to compare four function values at once, while this proof must instead route every such comparison through (SSQSB)'s two one-directional implications (Proposition 7.50's quasi-version of the same two-sided inequality), which only ever license moving in one direction at a time depending on which side of a comparison is tight. The book's proof handles this by working with the specific implications (7.43)–(7.44) in place of the L♮-approach property used in chunk 08's proof — an ordinal substitute for the same additive step, at the cost of a case analysis chunk 08's proof did not need.

Formalization scope

This mission builds directly on chunk 08's published items (SBF, DomZ, ArgMin, IndicatorVec), per the platform's textbook convention that a later chapter section of the same book imports an earlier one's definitions; its own namespace DiscreteConvex.LConvexFunctions.Quasi nests under chunk 08's DiscreteConvex.LConvexFunctions accordingly. Note the sign convention of the linear perturbation here, $g[x](p) = g(p) + \langle p,x\rangle$ , is the opposite of the M-side's $f[p](x) = f(x) - \langle p,x\rangle$ (chunks 06–07) — verified against the book's own formula rather than assumed by analogy.

A trivializing formalization of the goal would silently strengthen (SSQSB) back to plain (SBF[Z]) (making this mission redundant with chunk 08's Theorem 7.18) or loosen the exact bound $(n-1)(\alpha-1)$ ; neither is done. Only (QSB), (SSQSB), (QSBw), (SSQSBw) are drafted, matching exactly what the chosen three items need; the polyhedral L-convex-function bridge (§7.8–7.9, Theorems 7.40–7.46) and the level-set characterizations (Theorems 7.51–7.52) are left for a follow-on mission. Contributions building the 0L ↔ S correspondence (Theorem 7.40, a bridge back to chunk 04's submodular-set-function vocabulary) or the scaled quasi L-minimizer-cut analogue are welcome.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
P. Milgrom, C. Shannon, "Monotone comparative statics," Econometrica, 62(1), 1994, pp. 157–180.

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Convex Optimization Functional Analysis Operations Research·Captain: mikedeng1

On the Maximal Monotonicity of Subdifferential Mappings II: Subdifferentials Are Exactly the Maximal Cyclically Monotone Operators, Unique up to an Additive ConstantResearch Paper

Motivation

A differentiable convex function on $\mathbb{R}^n$ is determined, up to an additive constant, by its gradient, and a vector field is a gradient of a convex function exactly when it satisfies a monotonicity condition along closed cycles. Convex analysis and optimization need the same statement for nonsmooth and extended-valued functions on infinite-dimensional spaces: the subdifferential replaces the gradient, and the question becomes which multivalued maps from a Banach space to its dual arise as subdifferentials, and how much of the function they determine. The answer underlies the treatment of optimality conditions, variational inequalities and evolution equations governed by subdifferentials, where one works with the operator $\partial f$ and needs to recover $f$ from it.

Timeline.

1966: R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math. 17 (DOI 10.2140/pjm.1966.17.497), studied cyclically monotone operators and stated the characterization of subdifferentials as the maximal cyclically monotone operators (its Theorem 3), together with the maximal monotonicity of subdifferentials (its Theorem 4).
1969: H. Brézis pointed out a gap in the 1966 proofs of maximality and uniqueness: a family of dual vectors $x_\varepsilon^*$ used in the argument might increase unboundedly in norm as $\varepsilon \to 0$ .
1970: Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (DOI 10.2140/pjm.1970.33.209), repaired the argument for arbitrary real Banach spaces, proving Theorem A (maximal monotonicity of $\partial f$ ) and Theorem B (the characterization treated here).

Setting

Let $E$ be a real Banach space with dual $E^*$ , and write $\langle x, x^* \rangle$ for the value of $x^* \in E^*$ at $x \in E$ . A proper convex function on $E$ is a function $f : E \to (-\infty, +\infty]$ , not identically $+\infty$ , with $f((1-\lambda)x + \lambda y) \le (1-\lambda)f(x) + \lambda f(y)$ for all $x, y \in E$ and $0 < \lambda < 1$ . It is lower semicontinuous (lsc) in the norm topology. Its subdifferential is the multivalued map

\partial f(x) = \{\, x^* \in E^* \mid f(y) \ge f(x) + \langle y - x, x^* \rangle \ \ \forall y \in E \,\}.

A multivalued map $T : E \to E^*$ is a cyclically monotone operator if

\langle x_0 - x_1, x_0^* \rangle + \cdots + \langle x_{n-1} - x_n, x_{n-1}^* \rangle + \langle x_n - x_0, x_n^* \rangle \ge 0 \qquad\text{whenever } x_i^* \in T(x_i),\ i = 0, \dots, n,

and maximal cyclically monotone if, in addition, its graph $\{(x, x^*) \mid x^* \in T(x)\}$ is not properly contained in the graph of any other cyclically monotone operator. The conjugate of $f$ is $f^*(x^*) = \sup_{x \in E} (\langle x, x^* \rangle - f(x))$ on $E^*$ , and $j(x) = \tfrac12\|x\|^2$ . In the Lean development these are ProperConvex, subdiff, IsCyclicallyMonotone, IsMaximalCyclicallyMonotone, conj and halfSqNorm, in the namespace RockafellarMaxMono.Cyclic.

Formalization targets

Goal: Theorem B (p. 210)

For every multivalued map $T : E \to E^*$ on a real Banach space $E$ ,

\bigl(\exists f \text{ lsc proper convex with } T = \partial f\bigr) \iff T \text{ is maximal cyclically monotone},

and if $f$ and $g$ are lsc proper convex with $\partial f = T = \partial g$ , then $g = f + c$ for a real constant $c$ . Both halves are one statement.

Milestones, in attack order

(3.7) For a finite continuous convex function $f$ on a real Banach space, $\partial f(x)$ is nonempty and weak* compact and $f'(x;u) = \max\{\langle u, x^* \rangle \mid x^* \in \partial f(x)\}$ .
Finite continuous case (pp. 214–215). For finite continuous convex $f, g$ on a real Banach space, $\partial g(x) \supset \partial f(x)$ for all $x$ implies $g = f + \mathrm{const}$ .
(3.1) $\partial(f + j)(x) = \partial f(x) + \partial j(x)$ for lsc proper convex $f$ .
Proposition 1 $x^{**} \in \partial f^*(x^*)$ if and only if $x^{**}$ is a weak** limit of a bounded net $x_i$ with $x_i^* \in \partial f(x_i)$ , $x_i^* \to x^*$ in norm.
(p. 213) $(f + j)^*$ is finite and continuous on $E^*$ .
(p. 211) $f^{**}$ restricted to $E$ is $f$ .
(3.6) For lsc proper convex $f, g$ : $\partial g(x) \supset \partial f(x)$ for all $x$ implies $g = f + \mathrm{const}$ .

Significance

The result. Theorem B gives an intrinsic description of subdifferential maps: an operator is the subdifferential of a closed proper convex function if and only if it satisfies the cycle inequality and cannot be enlarged without violating it. The uniqueness clause says that a closed convex function is recovered from its subdifferential up to a constant, the nonsmooth counterpart of recovering a function from its gradient. Milestone 7 is stronger than uniqueness: a one-sided inclusion $\partial f \subseteq \partial g$ already forces $g = f + c$ , and this is what gives maximality.

Formalizing it. The theorem is proved in the paper, and nothing of it is formalized on the platform. Mathlib has convex functions, continuous duals, biduals and weak-* topologies, but not extended-valued subdifferentials on Banach spaces, conjugate duality in the nonreflexive setting, or monotone operator theory. This mission produces formal statements of the paper's steps, the standard max formula for directional derivatives on a Banach space, and the Fenchel conjugate facts the argument uses, each as a separate target.

Difficulty

In a reflexive space the argument is short, because $\partial f^*$ is the inverse of $\partial f$ . In a nonreflexive space it is not: $\partial f^*$ maps $E^*$ into $E^{**}$ , and points of $E^{**} \setminus E$ appear. The naive route, transferring the inclusion $\partial f \subseteq \partial g$ to the conjugates by inverting the maps, breaks down there, and the 1966 argument failed at a related step, where the dual vectors in an approximation could be unbounded. Relating $\partial f^*$ to $\partial f$ without reflexivity is where the difficulty sits; the boundedness of the approximating nets in Proposition 1 is essential and cannot be dropped. The finite continuous case and the max formula (3.7) are needed on an arbitrary Banach space, including the dual $E^*$ , not only on $E$ .

Formalization scope

$E$ is a real Banach space (NormedAddCommGroup, NormedSpace ℝ, CompleteSpace); $E^*$ is StrongDual ℝ E with the operator norm, $E^{**}$ is StrongDual ℝ (StrongDual ℝ E), and $E \hookrightarrow E^{**}$ is NormedSpace.inclusionInDoubleDual. No reflexivity, inner product or finite dimension is assumed. Explicit readings:

Values in $(-\infty, +\infty]$ are EReal with the requirement $f(x) \ne -\infty$ ; convexity is the paper's inequality for $0 < \lambda < 1$ in EReal arithmetic. Lower semicontinuity is in the norm topology.
A multivalued map is E → Set (StrongDual ℝ E); $T = \partial f$ means $T(x) = \partial f(x)$ for every $x$ .
The cycle inequality quantifies over all $n \in \mathbb{N}$ and points indexed by Fin (n + 1) with wrap-around addition, so the last term is $\langle x_n - x_0, x_n^* \rangle$ . Maximality is graph inclusion among cyclically monotone operators, not among monotone operators.
The uniqueness constant is a real number, never $\pm\infty$ .
" $\supset$ " in (3.6) is non-strict inclusion, and the hypothesis is one-sided.
A net is a nonempty directed partially ordered index type with convergence along atTop; weak** convergence is pointwise convergence on $E^*$ ; "bounded" is a uniform norm bound.
"Finite and continuous" for $(f+j)^*$ means equal everywhere to a continuous real-valued function. The max in (3.7) is IsGreatest, so it is attained; the directional derivative is the limit along $\lambda \to 0^+$ .
The print's " $\partial(f + j) = \partial f(x) + \partial j(x)$ " in (3.1) is read as $\partial(f+j)(x)$ .

A formalization that drops lower semicontinuity, allows an extended-real constant, or replaces "maximal cyclically monotone" by "maximal monotone" states a different, and in the first two cases false or trivial, theorem; the statements here keep all three.

The proof reduces Theorem B to milestone 7 through Theorem 1 of Rockafellar (1966) and its Corollary 2, which are not stated in this paper and are not milestones; formal statements of them are welcome as supporting theorems. Contributions of reusable infrastructure are welcome: extended-valued subdifferentials and conjugates on normed spaces, the Fenchel–Moreau identity on $E$ , the sum rule with a continuous function, and the max formula for directional derivatives.

Selected references

R. T. Rockafellar, On the maximal monotonicity of subdifferential mappings, Pacific J. Math. 33 (1970), 209–216. https://doi.org/10.2140/pjm.1970.33.209
R. T. Rockafellar, Characterization of the subdifferentials of convex functions, Pacific J. Math. 17 (1966), 497–510. https://doi.org/10.2140/pjm.1966.17.497
G. J. Minty, On the monotonicity of the gradient of a convex function, Pacific J. Math. 14 (1964), 243–247. https://doi.org/10.2140/pjm.1964.14.243
J.-J. Moreau, Fonctionnelles convexes, mimeographed lecture notes, Collège de France, 1967.
R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970. https://doi.org/10.1515/9781400873173

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Analysis Functional Analysis Partial Differential Equations·Captain: mikedeng1

Notes on Partial Differential Equations VIII: Strongly Continuous Semigroups and the Lumer–Phillips TheoremTextbook

Motivation

Many linear partial differential equations of evolution type — the heat equation, the Schrödinger equation, linear wave and Klein–Gordon equations, linearized reaction–diffusion and Kuramoto–Sivashinsky equations — can be written as an ordinary differential equation

u_t = Au, \qquad u(0) = f,

in a Banach space $X$ of functions, where $A$ is a spatial differential operator. When $A$ is bounded, the solution is the operator exponential $e^{tA}f$ . Differential operators are unbounded, however, and then the solution operators $f \mapsto u(t)$ are only continuous in time in the strong sense and may exist only forward in time. Semigroup theory makes this picture rigorous: it characterizes exactly which operators $A$ generate a well-posed evolution, and it is the standard route from energy estimates to existence and uniqueness for linear and semilinear evolution equations (Duhamel's formula, §5.4.5 of the notes, used for the semilinear heat equation in §5.5).

This mission formalizes §5.4 of J. K. Hunter's Notes on Partial Differential Equations (UC Davis, revised 6/18/2014): uniformly continuous groups, strongly continuous semigroups and their generators, the Hille–Yosida theorem, the Lumer–Phillips theorem and Stone's theorem.

Timeline. The generation theorem for contraction semigroups is due independently to E. Hille (1948) and K. Yosida (1948); the general version with constants $M$ , $a$ to Feller, Miyadera and Phillips (1952–53). G. Lumer and R. S. Phillips (1961) recast the contraction case in terms of dissipative operators. Stone's theorem on one-parameter unitary groups dates from 1930–32. Standard modern accounts are Pazy (1983) and Engel–Nagel (2000).

Setting

Let $X$ be a Banach space over $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$ , and $\mathcal{L}(X)$ the Banach space of bounded linear operators on $X$ with the operator norm.

A uniformly continuous group is a family $\{T(t) : t \in \mathbb{R}\} \subset \mathcal{L}(X)$ with $T(0) = I$ , $T(s)T(t) = T(s+t)$ , and $T(h) \to I$ in operator norm as $h \to 0$ . A strongly continuous (C₀) semigroup is a family $\{T(t) : t \ge 0\} \subset \mathcal{L}(X)$ with $T(0) = I$ , $T(s)T(t) = T(s+t)$ for $s, t \ge 0$ , and $T(h)f \to f$ as $h \to 0^+$ for every $f \in X$ ; it is a contraction semigroup if $\|T(t)\| \le 1$ for all $t \ge 0$ . A C₀ group has the same properties for all $s, t \in \mathbb{R}$ ; on a complex Hilbert space it is unitary if every $T(t)$ is unitary.

An operator $A : \mathcal{D}(A) \subset X \to X$ is a linear map defined on a subspace $\mathcal{D}(A)$ . The generator of a C₀ semigroup is the operator $A$ whose domain is the set of $f$ for which

Af = \lim_{h \to 0^+} \frac{T(h)f - f}{h}

exists in norm, with $Af$ this limit. $A$ is closed if its graph $\{(f, Af) : f \in \mathcal{D}(A)\}$ is closed in $X \times X$ , and densely defined if $\mathcal{D}(A)$ is dense. A scalar $\lambda$ is in the resolvent set of $A$ if $\lambda I - A : \mathcal{D}(A) \to X$ is a bijection, and then $R(\lambda, A) = (\lambda I - A)^{-1}$ is the resolvent. $A$ is dissipative if $\lambda\|f\| \le \|(\lambda I - A)f\|$ for all real $\lambda > 0$ and $f \in \mathcal{D}(A)$ , and m-dissipative if in addition $\lambda I - A$ is onto $X$ for some $\lambda > 0$ . On a complex Hilbert space $\mathcal{H}$ , $A$ is self-adjoint if $\mathcal{D}(A)$ is dense, $\mathcal{D}(A) = \mathcal{D}(A^*)$ , and $(x, Ay) = (Ax, y)$ for $x, y \in \mathcal{D}(A)$ .

Formalization targets

Goal: the Lumer–Phillips theorem (Theorem 5.38)

For an operator $A : \mathcal{D}(A) \subset X \to X$ ,

A \text{ generates a C₀ contraction semigroup on } X \iff A \text{ is closed, densely defined and m-dissipative.}

Milestones

Theorem 5.24. A uniformly continuous group is smooth in operator norm, $A = T'(0) \in \mathcal{L}(X)$ exists, and $T(t) = e^{tA}$ for all $t \in \mathbb{R}$ .
Theorem 5.32. The generator of a C₀ semigroup is closed and densely defined.
Theorem 5.35 (Hille–Yosida). $A$ generates a C₀ semigroup if and only if, for some $M \ge 1$ and $a \in \mathbb{R}$ , $A$ is closed and densely defined, $(a, \infty)$ lies in the resolvent set, and

\|R(\lambda, A)^n\| \le \frac{M}{(\lambda - a)^n} \qquad (\lambda > a,\ n \in \mathbb{N});

in that case $\|T(t)\| \le M e^{at}$ for $t \ge 0$ . 4. Corollary 5.36. The contraction case: $M = 1$ , $a = 0$ , with $\|R(\lambda, A)\| \le 1/\lambda$ . 5. Theorem 5.42 (Stone). On a complex Hilbert space, $iA$ generates a C₀ unitary group if and only if $A$ is self-adjoint.

Significance

The result itself. The Lumer–Phillips theorem is the working form of generation theory for PDEs: to show that $u_t = Au$ is well posed with $\|u(t)\| \le \|u(0)\|$ , it suffices to prove an energy inequality (dissipativity) and to solve a single resolvent equation $\lambda f - Af = g$ . The notes apply it to the Laplacian on $H^2(\mathbb{R}^n) \subset L^2(\mathbb{R}^n)$ and, after a shift, to the linearized Kuramoto–Sivashinsky operator; later chapters use the resulting semigroups in Duhamel's formula to construct solutions of semilinear heat and Schrödinger equations. Hille–Yosida gives the general growth bound $\|T(t)\| \le Me^{at}$ , and Stone's theorem identifies unitary quantum dynamics with self-adjoint Hamiltonians.

Formalizing it. All results are classical and proved in the literature; the notes themselves state 5.32, 5.35, 5.36, 5.38 and 5.42 without proof, referring to standard texts. Mathlib currently has the operator exponential on Banach algebras, partially defined linear maps (LinearPMap) with closedness, closures and adjoints, but no theory of C₀ semigroups or their generators. A complete development would give Lean a reusable generation theory for unbounded operators, on which parabolic and dispersive PDE existence results can rest.

Difficulty

The central obstacle is the passage from resolvent information to a semigroup. Formally $T(t) = e^{tA}$ , but for unbounded $A$ the exponential series is meaningless, and the necessity directions require showing that the resolvent is a Laplace transform of the semigroup, which needs integration of strongly (not norm) continuous operator-valued functions. The sufficiency directions must construct a semigroup from $A$ by an approximation of the unbounded generator and a limit that converges only strongly, and then identify the generator of the limit with $A$ — including its domain, which is where most careless arguments fail. In the Lumer–Phillips theorem, surjectivity of $\lambda I - A$ is assumed for a single $\lambda > 0$ and must be propagated to every $\lambda > 0$ . In Stone's theorem both directions need domain arguments for unbounded operators on Hilbert space; symmetry of $A$ alone is not enough.

Formalization scope

Scalars: a field 𝕜 with [RCLike 𝕜] (ℝ or ℂ), and X a Banach space over it ([NormedAddCommGroup X] [NormedSpace 𝕜 X] [CompleteSpace X]). Stone's theorem is over ℂ on an InnerProductSpace ℂ H with [CompleteSpace H]. Theorem 5.24 additionally equips X with a compatible real structure ([NormedSpace ℝ X] [IsScalarTower ℝ 𝕜 X]) so that $t \mapsto T(t)$ can be differentiated; $e^{tA}$ is Mathlib's NormedSpace.exp, and $C^\infty$ is ContDiff ℝ ∞.
Operators $A : \mathcal{D}(A) \subset X \to X$ are X →ₗ.[𝕜] X (LinearPMap); closedness is LinearPMap.IsClosed (closed graph).
Semigroups are functions T : ℝ → X →L[𝕜] X; for semigroups only the values at $t \ge 0$ enter. Composition is multiplication in X →L[𝕜] X.
The generator relation IsGenerator T A states, for all $f, g$ : the right difference quotient $h^{-1}(T(h)f - f)$ tends to $g$ as $h \to 0^+$ if and only if $f \in \mathcal{D}(A)$ and $Af = g$ . So " $A$ generates a semigroup" means that $A$ equals the generator including its domain; a formalization in which $A$ merely agrees with the limit on some subspace, or in which the generator is a restriction, would make the theorems different (and false) statements, and is ruled out.
Resolvent norm bounds $\|R(\lambda, A)^n\| \le c$ are stated pointwise, $\|R(\lambda,A)^n g\| \le c\|g\|$ for all $g$ , which is equivalent. In (5.24) $n = 0$ is included, which adds nothing because $M \ge 1$ . The "in that case" clause of Theorem 5.35 is a second conjunct: for every semigroup generated by $A$ and every $M, a$ satisfying the three conditions, $\|T(t)\| \le Me^{at}$ .
Closedness and density, which the notes build into the definitions of the resolvent set and of dissipativity, are stated explicitly wherever they are used (they appear as condition (1) in every theorem).
The generator of a C₀ group (Stone) is the generator of its forward part, as in Definition 5.30; the operator $iA$ is Complex.I • A, with the same domain as $A$ . Self-adjointness is Definition 5.41 as printed, not Mathlib's IsSelfAdjoint for LinearPMap, although the two agree.
Useful infrastructure, reusable beyond this mission: Riemann/Bochner integration of strongly continuous operator families, the Laplace-transform representation of the resolvent, Yosida approximations, and dissipativity–resolvent equivalences. Contributions of any of these as separate lemmas are welcome.

Selected references

J. K. Hunter, Notes on Partial Differential Equations, UC Davis lecture notes, revised 6/18/2014, §5.4. https://www.math.ucdavis.edu/~hunter/pdes/pde_notes.pdf
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. https://doi.org/10.1007/978-1-4612-5561-1
K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer GTM 194, 2000. https://doi.org/10.1007/b97696
G. Lumer and R. S. Phillips, Dissipative operators in a Banach space, Pacific J. Math. 11 (1961), 679–698. https://doi.org/10.2140/pjm.1961.11.679
M. H. Stone, On one-parameter unitary groups in Hilbert space, Ann. of Math. 33 (1932), 643–648. https://doi.org/10.2307/1968538
K. Yosida, On the differentiability and the representation of one-parameter semigroup of linear operators, J. Math. Soc. Japan 1 (1948), 15–21. https://doi.org/10.2969/jmsj/00110015

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Dynamic Programming Operations Research Optimization+1·Captain: Shuze Chen

Markov Decision Processes XIX: Theory of Optimal Stopping ProblemsTextbook

Motivation

A gambler watching a sequence unfold has to decide, at each moment and knowing only the past, whether to take what is on the table or wait for something better. That is the whole of optimal stopping, and it is one of the few problems in stochastic control with a clean and completely general answer: the value of the problem is the smallest superharmonic function dominating the immediate payoff. Snell (1952) proved the martingale form; the dynamic-programming form is due to Chow, Robbins and Siegmund. It is the structure behind the pricing of American options, the secretary problem, sequential hypothesis testing, and the bandit problems of Chapter 5.

Bäuerle and Rieder's Chapter 10 (Markov Decision Processes with Applications to Finance, Springer, 2011) derives this from their own Markov-decision machinery rather than from martingale theory, which makes the whole development elementary and self-contained: a stopping problem is a Markov Decision Problem whose action space is {continue, stop}, so Chapter 2's finite-horizon theory and Chapter 7's unbounded-horizon theory apply to it verbatim. The chapter then runs the resulting theory on three classical problems and solves each one in closed form.

Setting

The problem. A Markov process (X_n) on a Borel space E is observed. A stopping time is a random time τ with {τ ≤ n} ∈ F_n — "upon observing the process until time n we can decide whether or not τ has already occurred". Stopping at τ collects

R_\tau := \sum_{k=0}^{\tau-1} c_k(X_k) + g_\tau(X_\tau),

a running reward c_k while continuing and a stopping reward g_τ at the end, and the problem is to find V_N^*(x) := sup_{τ ≤ N} E_x[R_τ] (10.1). Assumption (B_N) — finiteness of the supremum of the positive parts — is what makes this well posed.

The reduction (Theorem 10.1.2). Take A = {0,1}, let a = 0 mean continue and a = 1 mean stop, and make the transition law uncontrollable on continuation and absorbing on stopping. A policy π = (f_0,…,f_{N-1}) induces the stopping time τ_π = inf{n | f_n(X_n) = 1} ∧ N, and conversely every stopping time is a history-dependent policy. The theorem says the two suprema agree: the extra history buys nothing.

The recursion (Theorems 10.1.3, 10.1.5). The Bellman operator becomes a two-branch maximum,

\mathcal{T}v(x) = \max\Big\{g(x),\ c(x) + \beta\int v(x')Q^X(dx'|x)\Big\},

with no action variable left in it. In the stationary case J_0 = g, J_n = \mathcal{T}J_{n-1}; the J_n increase, the sets S_n^* = {J_n = g} shrink — "the tendency to stop is non-decreasing as time goes by" — and the optimal rule is "stop on first entry into S_{N-n}^*".

The unbounded horizon (§10.2). Now the reward is discounted, R_τ = Σ β^k c(X_k) + β^τ g(X_τ) for τ < ∞, the value is V_∞^*(x) = sup_{τ<∞} E_x[R_τ], and there is no terminal condition to induct from. Three candidate values present themselves: V_∞^*; G = sup_π liminf_n J_{nπ}, a supremum over policies of limits of finite-horizon values; and J = lim_n J_n, which exists by monotonicity. Theorem 10.2.2, the goal, says all three coincide, that the common value solves J = \mathcal{T}J and satisfies 0-free bounds, and — the characterization — that it is the smallest c-superharmonic function majorizing g.

Turning the value into a rule (Theorems 10.2.3, 10.2.7, Corollaries 10.2.6, 10.2.8). Knowing the value is not knowing when to stop. Theorem 10.2.3 produces the stopping region as S^* = {J = g} = {d ≥ 0} where d = lim_n d_n, under two conditions that Corollary 10.2.6 then gives three checkable sufficient conditions for. Theorem 10.2.7 is the practical one, the One-Step-Look-Ahead Rule: if the set where stopping now beats stopping one step later is closed under the transition law, then the myopic rule is globally optimal. Corollary 10.2.8 adds monotonicity and gets a threshold.

Three applications (§10.3). The house seller who receives i.i.d. offers and pays maintenance on each rejection should accept the first offer above an explicit threshold, obtained as the maximiser of a one-dimensional function (Theorem 10.3.1). The secretary problem's value function is computed exactly (Proposition 10.3.2), giving the classical rule — reject the first k^*, then take the first leader — with success probability (k^*/N)h(k^*) and k^*(N)/N → 1/e (Theorem 10.3.3). And when the offers' distribution has an unknown parameter, MTP_2 of the likelihood propagates into monotonicity of the value in the information state (Theorem 10.3.4), with a fully explicit solution for the exponential/Inverse-Gamma conjugate pair (Theorem 10.3.6).

What is being asked

Formalize Theorem 10.2.2 in full: the three-way equality of V_∞^*, G and J, the fixed point equation, and — the part that carries the theorem — minimality among all functions that are both c-superharmonic and above g. Asserting only that J is such a function, or only one of the two conditions, is a strictly weaker and different claim.

The twelve milestones are the rest of the chapter, in attack order: the reduction and the two recursions, then the unbounded-horizon apparatus, then the three worked problems.

The stopping-time apparatus is built rather than assumed — the chain's law pinned by its finite-dimensional distributions, stopping times valued in ℕ ∪ {∞}, rewards vanishing at ∞ — because every theorem here is the identification of a supremum over stopping times with something computable, and carrying the value as an abstract function would make them vacuous. Every supremum is taken as a least upper bound against an explicit set of achievable values rather than by sSup, so that a set unbounded above is not silently given the value 0.

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Analysis Harmonic Analysis Partial Differential Equations·Captain: mikedeng1

Notes on Partial Differential Equations II: The Newtonian Potential and Hölder Estimates for Poisson's EquationTextbook

Motivation

Poisson's equation $-\Delta u = f$ is the model elliptic partial differential equation. It describes the electrostatic potential of a charge density, the gravitational potential of a mass density, and the steady temperature of a body with internal heat sources. Its solution on the whole space by convolution with a point-source potential, and the fact that this solution has exactly two more derivatives than the data when smoothness is measured in Hölder norms, are the starting point of the regularity theory of second-order elliptic equations.

This mission formalizes §2.6–2.7 of J. K. Hunter's Notes on Partial Differential Equations (UC Davis, revised 6/18/2014), which follow the classical line: the fundamental solution, the Newtonian potential, an integral representation of its second derivatives, and the Hölder estimate. The estimate goes back to J. Schauder (1934), who used it to build an existence theory for elliptic equations with Hölder continuous coefficients, now called Schauder theory (Gilbarg–Trudinger, Ch. 4 and 6). The singular integral operators that appear in the second derivatives were later placed in a general framework by A. P. Calderón and A. Zygmund (1952), whose theory gives the $L^p$ analogue.

Setting

Throughout, $n \ge 2$ and $\mathbb{R}^n$ carries the Euclidean norm $|x|$ and Lebesgue measure. Let $\alpha_n$ be the volume of the open unit ball. The fundamental solution of Laplace's equation is

\Gamma(x) = \frac{1}{n(n-2)\alpha_n}\,\frac{1}{|x|^{n-2}} \quad (n \ge 3), \qquad \Gamma(x) = -\frac{1}{2\pi}\log|x| \quad (n = 2),

defined for $x \neq 0$ . With this sign convention (as in Evans, opposite to Gilbarg–Trudinger) $\Gamma$ is harmonic away from the origin and $-\Delta\Gamma = \delta$ in the sense of distributions. $\Gamma$ is locally integrable, and so is its gradient, but its second derivatives behave like $|x|^{-n}$ and are not.

For a smooth function $f$ with compact support, $f \in C_c^\infty(\mathbb{R}^n)$ , the Newtonian potential of $f$ is the convolution

u(x) = (\Gamma * f)(x) = \int_{\mathbb{R}^n} \Gamma(x - y)\, f(y)\, dy.

Partial derivatives are written $\partial_i$ , with $\partial_{ij} = \partial_i \partial_j$ , and $\delta_{ij}$ is the Kronecker delta. For $0 < \alpha \le 1$ the Hölder seminorm of $v : \mathbb{R}^n \to \mathbb{R}$ is

[v]_{0,\alpha} = \sup_{x \neq y} \frac{|v(x) - v(y)|}{|x - y|^{\alpha}} \in [0, \infty].

In the Lean development these objects are fundamentalSolution n, unitBallVolume n, newtonianPotential n f, partialDeriv u i, secondPartial u i j and holderSeminorm α Set.univ v, all in the namespace HunterPDE.Newtonian.

Formalization targets

Goal: the Hölder estimate (Theorem 2.28)

For $0 < \alpha < 1$ there is a constant $C$ , depending only on $\alpha$ and $n$ , such that for every $f \in C_c^\infty(\mathbb{R}^n)$ and all indices $i, j$ ,

[\partial_{ij}(\Gamma * f)]_{0,\alpha} \le C\,[f]_{0,\alpha}.

The constant is not specified: the goal asserts the shape of the estimate, not a numerical value.

Milestones

Eq. (2.14). $\Gamma$ is $C^\infty$ on $\mathbb{R}^n \setminus \{0\}$ and $\partial_i\Gamma(x) = -\dfrac{1}{n\alpha_n}\dfrac{1}{|x|^{n-1}}\dfrac{x_i}{|x|}$ for $x \neq 0$ .
Eq. (2.23). $\Gamma * (\Delta f) = -f$ for $f \in C_c^\infty(\mathbb{R}^n)$ .
Theorem 2.25. $u = \Gamma * f$ is well defined, $u \in C^\infty(\mathbb{R}^n)$ , and $-\Delta u = f$ .
Corollary 2.27. For every $x$ and every open ball $B_R(x)$ containing the support of $f$ ,

\partial_{ij} u(x) = \int_{B_R(x)} \partial_{ij}\Gamma(x - y)\,[f(y) - f(x)]\,dy - \frac{1}{n} f(x)\,\delta_{ij},

with the integral absolutely convergent.

Significance

Theorem 2.25 is the existence theorem for Poisson's equation on the whole space. Corollary 2.27 expresses $D^2 u$ as a singular integral operator applied to $f$ plus a multiple of $f$ ; it is the prototype of the representations behind both the Hölder (Schauder) and the $L^p$ (Calderón–Zygmund) theories. Theorem 2.28 is the first Schauder estimate. With it, a priori estimates for equations with variable Hölder coefficients follow by freezing the coefficients, and existence follows by the method of continuity. The notes point out that the Hölder scale is essential: there are continuous compactly supported $f$ for which $\Gamma * f$ is not $C^2$ , so no estimate of $D^2 u$ by the maximum of $|f|$ holds. The endpoint $\alpha = 1$ is excluded from the theorem.

All four statements are classical and proved in the notes; none is new mathematics. As far as the Prove2Me corpus and Mathlib show, none has been machine-checked. Mathlib has the Laplacian on inner product spaces, convolution, and the volume of Euclidean balls, but no fundamental solution of the Laplacian in general dimension, no Newtonian potential, and no Hölder estimates for it. A formalization would supply reusable infrastructure: the fundamental solution with its derivative formulas, differentiation under the integral sign with a locally integrable kernel, and estimates for kernels homogeneous of degree $-n$ with mean zero on spheres.

Difficulty

The obvious route to the second derivatives, differentiating twice under the integral sign, fails: $\partial_{ij}\Gamma$ is of order $|x|^{-n}$ near the origin and is not locally integrable, so $\int \partial_{ij}\Gamma(x-y) f(y)\,dy$ need not exist. Corollary 2.27 is precisely the statement that the correct formula carries a subtracted value $f(x)$ and a boundary contribution $-\frac1n f(x)\delta_{ij}$ , and computing that contribution needs integrals over spheres, which Mathlib supports only indirectly. For the goal, a pointwise bound on the kernel is not enough. Estimating $|\partial_{ij}\Gamma(x-y)|\,|f(y)-f(x)|$ in absolute value gives a bound that grows with the radius of the ball containing the support of $f$ , while the theorem's constant must be uniform over all $f$ , all indices and all pairs of points. The estimate depends on cancellation in the kernel, and the maximum-norm version of the same statement is false.

Formalization scope

$\mathbb{R}^n$ is EuclideanSpace ℝ (Fin n) with volume. Every theorem assumes 2 ≤ n; the definition of $\Gamma$ returns junk values for $n = 0, 1$ , which the book excludes.
$\Gamma(0)$ is undefined in the book; in Lean it equals $0$ (from $1/0 = 0$ and $\log 0 = 0$ ), one point of measure zero.
Coordinates are 0-based (i : Fin n). $\partial_i u(x)$ is fderiv ℝ u x (EuclideanSpace.single i 1), and $\partial_{ij} u = \partial_i(\partial_j u)$ . fderiv is $0$ where $u$ is not differentiable, so $\partial_{ij}\Gamma(x-y)$ in Corollary 2.27 has an arbitrary value at the single point $y = x$ .
"Smooth" is ContDiff ℝ ∞ ( $C^\infty$ ). In this Mathlib ContDiff ℝ ⊤ means analytic, and it is not used. Compact support is HasCompactSupport, and "the support of $f$ " is the closed support tsupport f.
$\Delta$ is Mathlib's InnerProductSpace.laplacian.
The convolution $\Gamma * f$ is the Lebesgue integral $\int \Gamma(x-y) f(y)\,dy$ , with no integrability assumed of $\Gamma$ . Lean's Bochner integral is $0$ for a non-integrable integrand, so Theorem 2.25 asserts integrability of $y \mapsto \Gamma(x-y)f(y)$ for every $x$ , and Corollary 2.27 asserts integrability of its integrand on the ball. Both are true and are implicit in the book's formulas; they rule out reading an identity between junk values as the theorem.
The Hölder seminorm is a supremum in $[0,\infty]$ over all pairs of distinct points of $\mathbb{R}^n$ , not Mathlib's HolderWith, which fixes a constant. In the goal $C$ is a finite nonnegative real quantified before $f$ , $i$ and $j$ . The inequality therefore also asserts that $\partial_{ij} u$ is uniformly $\alpha$ -Hölder, because $[f]_{0,\alpha} < \infty$ for $f \in C_c^\infty$ . A formalization that takes $C$ after $f$ , lets it depend on the support of $f$ , or includes the excluded endpoint $\alpha = 1$ is not the theorem.
Theorem 2.26 of the notes (the same representation over an arbitrary smooth domain, with a surface integral over $\partial\Omega$ ) is not stated, because a faithful statement needs a surface measure on general smooth boundaries. The singular-integral discussion of §2.8 is commentary and is not stated.

Useful infrastructure, all reusable beyond this mission: differentiation of $\Gamma$ away from the origin; integration in polar coordinates and integrals of $|x|^{-a}$ over balls and annuli; surface integrals over spheres, or the divergence theorem on balls; differentiation under the integral sign for convolutions with a locally integrable kernel. Contributions of any of these as separate lemmas are welcome.

Selected references

J. K. Hunter, Notes on Partial Differential Equations, UC Davis lecture notes, revised 6/18/2014, §2.6–2.8 and Definition 1.1. https://www.math.ucdavis.edu/~hunter/pdes/pde_notes.pdf
L. C. Evans, Partial Differential Equations, 2nd ed., Graduate Studies in Mathematics 19, AMS, 2010, §2.2. https://doi.org/10.1090/gsm/019
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001 (reprint of the 1998 ed.), Ch. 4. https://doi.org/10.1007/978-3-642-61798-0
J. Schauder, Über lineare elliptische Differentialgleichungen zweiter Ordnung, Mathematische Zeitschrift 38 (1934), 257–282. https://doi.org/10.1007/BF01170635
A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Mathematica 88 (1952), 85–139. https://doi.org/10.1007/BF02392130

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Dynamic Programming Operations Research Optimization+1·Captain: Shuze Chen

Markov Decision Processes XVIII: Terminal Wealth in Jump Markets and Trade ExecutionTextbook

Motivation

Two problems in this mission, both about markets that do not behave the way the textbook Black–Scholes market does, and both solved by the same technique.

The first is portfolio choice in a pure jump market. Prices move by jumps at the epochs of a Poisson process, not by continuous Brownian fluctuation. This is not a technical variation: the market is incomplete, there is no replicating portfolio, and the machinery of stochastic analysis that makes the diffusion case tractable is unavailable. What is available instead is that the wealth process is piecewise deterministic — between jumps it follows an ODE, and all the randomness is in when the jumps happen and how big they are. Chapter 8's technique embeds such a process in its jump chain and turns the continuous-time control problem into a discrete-time Markov Decision Model with infinite horizon; Chapter 7's contracting theory then solves that.

The second is trade execution in an illiquid market. An agent must sell a large block of shares by a deadline. Placing the whole order at once moves the price against them, and in a traditional order book other participants can see the intention and trade against it — so the order goes to a dark pool, where there is no order book and matches arrive at random. The agent can only sell when a counterparty happens to appear, and whatever is unsold at the deadline must be dumped on the traditional market at once. The question is how much to offer at each opportunity.

The two problems have opposite curvature — the first is a concave maximisation of utility, the second a convex minimisation of cost — and the section is a good demonstration that the same embedding technique handles both, with each problem's structure entering only through which set of functions the value function is sought in.

Setting

The jump market (§9.3). The bond is S⁰_t = e^{ρt}; the risky assets follow dS^k_t = S^k_{t-}(μ_k dt + dC^k_t) where C_t = Σ_{n≤N_t} Y_n is a compound Poisson process of intensity λ whose jumps Y_n are supported in (-1,∞)^d, which keeps prices positive. Short-sellings are prohibited, so the admissible fractions of wealth form the compact set 𝒰 = {u ≥ 0, u·e ≤ 1}, and the wealth follows

dX_t = X_{t-}\big((\rho + \pi_t\cdot(\mu-\rho e))dt + \pi_t dC_t\big). \tag{9.10}

The investor maximises E^π_{tx}[U(X_T)] for a strictly increasing, strictly concave U.

The embedded model's state is (t,x) — a jump time and the wealth just after it — and its action is a whole control path α : [0,T] → 𝒰, followed until the next jump. Between jumps the wealth is φ^α_t(x) = x exp(∫₀^t (ρ + α_s·(μ-ρe))ds) (9.13), and the transition kernel is substochastic: with probability e^{-λ(T-t)} no further jump arrives before the horizon, and the reward r(t,x,α) = e^{-λ(T-t)}U(φ^α_{T-t}(x)) is collected instead.

The trade execution model (§9.4). A Poisson process of intensity λ delivers the trading epochs; selling a shares costs C(a) with C strictly increasing and strictly convex (the discrete form (9.19)), C(0) = 0; the inventory X_t = x₀ - ∫₀^t π_s dN_s is what remains, and C(X_T) is the terminal dump. Here the flow is uncontrolled — the inventory does not move between epochs — which makes the embedded model simpler.

What is being asked

The goal is Theorem 9.3.4, the main result for the terminal wealth problem, in all six of its parts: the value function is the limit of the value iteration and lies in IM_cv; it is the unique fixed point of the dynamic programming operator there; value iteration converges at the explicit geometric rate α_b^n/(1-α_b); there exists an optimal Markov portfolio strategy given by a single decision rule; policy iteration holds; and Howard's policy improvement algorithm holds. Parts a)–c) describe the value; parts d)–f) produce the strategy, and a formalization of the first three alone would omit the entire control half of the theorem.

The seven milestones are the rest of §9.3–9.4: the reduction from continuous to discrete time, the bounding function and its explicit contraction modulus, the invocation of Chapter 7's Structure Theorem, the iff-characterization of when holding only the bond is optimal, the stability of the value and of the optimal policies under perturbation of the utility, and then the trade execution problem's own bounding function and its monotone, unit-Lipschitz optimal execution rate.

Two formalization conventions run through everything here. The operator of §9.3 is a supremum over a space of control paths, and since Mathlib's sSup of a set unbounded above is 0 — with an unbounded reward that is a live risk, not a formality — it is carried as a relation defined by least upper bounds against explicit sets of achievable values, with its iterates a chain of such relations. And the continuous-time side is built, not assumed: Theorem 9.3.1 is the identification of the continuous-time value with the discrete-time one, so the law of the embedded jump chain is pinned by the one-step conditional law the book displays, and the terminal wealth is read off that chain.

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Algorithmic Game Theory Machine Learning Operations Research·Captain: mikedeng1

Calibrated Learning and Correlated Equilibrium II: For Almost Every Game, Limits of Calibrated Learning Are Exactly the Correlated EquilibriaResearch Paper

Motivation

A correlated equilibrium (Aumann, 1974) is a joint distribution over the players' strategy profiles from which no player gains by deviating from a recommended strategy. A central question of learning in games is which equilibria repeated play of simple, myopic rules can reach. Foster and Vohra (1997) answered it for calibrated forecasting: if each player forecasts the opponent with a calibrated rule and best-responds to the forecast, the empirical distribution of play approaches the set of correlated equilibria (their Theorem 1). That result has become a basic reference point for no-regret and calibration-based learning in games (Hart and Mas-Colell, 2000).

This mission formalizes the paper's converse. Theorem 1 says calibrated learning ends up in the correlated equilibria; the converse says it can end up at any of them, for almost every game. Together the two results characterize exactly which long-run outcomes calibrated learning with best responses can produce.

Setting

Two players choose strategies from finite sets $S(1)$ with $m$ elements and $S(2)$ with $n$ elements; player $i$ receives payoff $u_i(x,y)\in\mathbb{R}$ and maximizes it. A game $G=(u_1,u_2)$ is a pair of real $m\times n$ matrices, that is, a point of $\mathbb{R}^{2mn}$ ; a set of games has measure zero if it is Lebesgue-null in $\mathbb{R}^{2mn}$ .

A joint distribution $D$ on $S(1)\times S(2)$ is a correlated equilibrium if for every map $\Phi:S(1)\to S(1)$ , $\sum_{x,y}D(x,y)u_1(\Phi(x),y)\le\sum_{x,y}D(x,y)u_1(x,y)$ , and symmetrically for player 2. $\pi(G)$ is the set of correlated equilibria.

Play is repeated in rounds $t=0,1,2,\dots$ . Before each round, player 1 issues a forecast $f_1(t)$ , a probability vector over $S(2)$ , produced by a deterministic forecasting rule from the history of play so far; player 2 likewise forecasts player 1. For a forecast sequence $f$ and the opponent's plays $z$ , let $N(p,t)$ be the number of the first $t$ rounds with forecast $p$ , and $\rho(p,j,t)$ the fraction of those rounds in which the opponent played $j$ (zero if $N(p,t)=0$ ). The forecasts are calibrated if for every $j$

\sum_p |\rho(p,j,t)-p_j|\,\frac{N(p,t)}{t}\ \longrightarrow\ 0 \qquad (t\to\infty).

Each player then plays $R_i$ of its forecast, where the best-reply function $R_i$ picks a best response to every forecast and does not depend on the round. $D_t(x,y)$ is the fraction of the first $t$ rounds in which $(x,y)$ was played.

$\lambda(G)$ , the set of limit points of calibrated forecasts, consists of the $D$ for which some best-reply functions $R_1,R_2$ and some calibrated forecasting rules make $D_t(x,y)\to D(x,y)$ for all $(x,y)$ .

Formalization targets

Goal: Theorem 2 (p. 47)

\text{for Lebesgue-almost every } G\in\mathbb{R}^{2mn}:\qquad \lambda(G)=\pi(G).

Milestones

$\lambda(G)\subseteq\pi(G)$ for every game (Theorem 1 restated, p. 46).
Every joint distribution $D$ is the limiting empirical distribution of a deterministic play sequence supported on $\{D>0\}$ (p. 47).
Along such a sequence the conditional forecasts $p_{1,t}=D(x_t,\cdot)/\sum_yD(x_t,y)$ and $p_{2,t}=D(\cdot,y_t)/\sum_xD(x,y_t)$ are calibrated (p. 47).
In a correlated equilibrium each recommended strategy is a best response to its conditional forecast (p. 47).
For almost every payoff matrix, each set $M_b(x)$ of forecasts to which $x$ is a best response is either empty or contains a forecast in the open simplex at which $x$ is the unique best response (p. 47).
The perturbed forecasts $p_i=(1-1/i)p^*+(1/i)q$ converge to $p^*$ and keep a unique best reply (p. 48).
For the $3\times3$ game of p. 48, a correlated equilibrium lies outside $\lambda(G)$ , and $\lambda(G)$ is the single point mass on $(C,2)$ .

Significance

Theorem 1 by itself leaves open whether calibration selects among correlated equilibria, for instance toward Nash equilibria or toward particular payoffs. Theorem 2 closes that question negatively for generic games: every correlated equilibrium is the genuine limit, not merely an accumulation point, of calibrated play with stationary best replies. As the paper notes, adding the assumption that the limit exists therefore does not refine the equilibrium reached, in contrast with Fudenberg and Kreps's result for asymptotically myopic Bayesian play. The $3\times3$ example shows that the genericity hypothesis cannot be dropped.

The results are proved in the 1997 paper; none is formalized. A complete development produces a reusable layer for repeated two-player games (calibration, empirical distributions, best-reply maps, correlated equilibria), a genericity lemma for best-response regions that is useful beyond this paper, and a machine-checked version of an argument that the paper gives only in outline.

Difficulty

The natural construction takes a correlated equilibrium $D$ , a play sequence realizing $D$ , and forecasts equal to the conditional distributions of $D$ . The forecasts are then calibrated and each played strategy is a best response. What fails is the best-reply function: two strategies $x'\ne x''$ can have the same conditional forecast $p^*$ , while a stationary $R_1$ maps $p^*$ to only one strategy. Separating them needs forecasts near $p^*$ at which each is the unique best response, and that exists only when the best-response regions have nonempty relative interior, a property that fails on a null set of games (the $3\times3$ example) and whose genericity must be proved. The perturbed forecasts are no longer exactly equal to the conditional frequencies, so calibration has to be re-established with errors that vanish along the sequence.

Formalization scope

Strategies are Fin m and Fin n; payoffs are real matrices; players maximize. A game is a point of (Fin m → Fin n → ℝ) × (Fin m → Fin n → ℝ) with Mathlib's volume (Lebesgue measure on $\mathbb{R}^{2mn}$ ), and "almost every" is ∀ᵐ.
A correlated equilibrium is the joint-distribution form of p. 44 (a joint distribution with the two deviation inequalities).
Forecasting rules map finite histories to probability vectors; the play is generated recursively from the rules and the best-reply functions. Best-reply functions are arbitrary stationary selections of a best response at every probability vector; they may not depend on the round, since round-dependent tie-breaking enlarges $\lambda(G)$ (the matching pennies example of p. 46).
Rounds are $0,\dots,t-1$ ; $D_0=0$ . The calibration sum runs over the forecasts issued so far, which is the paper's sum over all $p$ since the other terms vanish.
$\lambda(G)$ requires convergence of $D_t$ , not a subsequence. Replacing the limit by a limit point, or stating only $\pi(G)\subseteq\lambda(G)$ , does not formalize the theorem.
The page's sentence "Almost every game has the property that all the sets $M_b(x)$ have non-empty interior" is false for dominated strategies ( $M_b(x)=\emptyset$ on an open set of games). Milestone 5 states the dichotomy the page's argument proves: empty or with relative interior.
The page prints the denominator of $p_{2,t}$ as $\sum_{x}D(x_t,y)$ ; the mission uses $\sum_xD(x,y_t)$ .

The theorems are stated from the mission's own definitions of the game, calibration and $\lambda(G)$ ; no statement is vacuous: at $m=0$ or $n=0$ both sides of the goal are empty, and the $3\times3$ example exercises every definition. Contributions welcome: proofs of the milestones in any order, the genericity lemma (Lebesgue-null sets of linear degeneracies), and the calibration estimates for the perturbed forecasts.

Selected references

D. P. Foster and R. V. Vohra, Calibrated learning and correlated equilibrium, Games and Economic Behavior 21 (1997), 40–55. https://doi.org/10.1006/game.1997.0595
R. J. Aumann, Subjectivity and correlation in randomized strategies, Journal of Mathematical Economics 1 (1974), 67–96. https://doi.org/10.1016/0304-4068(74)90037-8
A. P. Dawid, The well-calibrated Bayesian, Journal of the American Statistical Association 77 (1982), 605–610. https://doi.org/10.1080/01621459.1982.10477856
D. Fudenberg and D. M. Kreps, Learning mixed equilibria, Games and Economic Behavior 5 (1993), 320–367. https://doi.org/10.1006/game.1993.1021
S. Hart and A. Mas-Colell, A simple adaptive procedure leading to correlated equilibrium, Econometrica 68 (2000), 1127–1150. https://doi.org/10.1111/1468-0262.00153

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Combinatorics Convex Optimization Discrete Geometry+2·Captain: Shuze Chen

Discrete Convex Analysis XXI: The Exchange Axiom as Local OptimalityTextbook

Motivation

Convexity on the integer lattice cannot be defined by the classical secant-line inequality alone: a function can be midpoint-convex along every line and still admit no useful global optimality theory, because integer points off a chosen line are invisible to it. M-convex functions, introduced by Murota, resolve this by replacing the secant condition with an exchange axiom directly generalizing the basis-exchange property of matroids and the convex-hull structure of network flows: a function on the integer lattice is M-convex if, whenever two points can be improved by moving one coordinate up and a compensating coordinate down, at least one such move weakly improves the sum of the two function values. This single axiom turns out to be equivalent to several strikingly different-looking properties — invariance under a wide family of domain operations, supermodularity in the M♮ (translation-invariant) case, and, most importantly, a local-to-global optimality principle: a point is a global minimizer of an M-convex function if and only if no single coordinate exchange improves it. This mission develops the algebraic core of that theory — the exchange axiom's basic consequences, its equivalent local and dynamic reformulations, and the operations that preserve it — building toward the theorem that recasts M-convexity itself as an algorithmically meaningful local-search guarantee.

Companion mission 06-mconvex-functions-i (Discrete Convex Analysis V) covers this chapter's own primary line of development: the equivalence of M-convexity and M♮-convexity with their respective exchange axioms (Theorem 6.2), the M-optimality criterion (Theorem 6.26), a minimizer-cut lemma (Theorem 6.28), and the M-proximity theorem (Theorem 6.37, its goal). This mission builds the vocabulary those results also need (redeclared here, since sibling drafts cannot yet import one another) and proves the results that chapter leaves for a second pass: the domain structure of M- and M♮-convex functions, worked examples (quadratic forms, quasi-separable functions), the operations that preserve M-convexity, supermodularity of the M♮-convex case, the descent-direction property, and — this mission's goal — the equivalence of the exchange axiom with a dynamic sequential-improvement property.

Setting

Fix a finite ground set $V$ . A function $f : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ with nonempty effective domain $\operatorname{dom} f$ is M-convex if it satisfies the exchange axiom (M-EXC[Z]): for $x, y \in \operatorname{dom} f$ and $u \in \operatorname{supp}^+(x-y)$ (coordinates where $x$ exceeds $y$ ), there is $v \in \operatorname{supp}^-(x-y)$ with

f(x) + f(y) \ge f(x - \chi_u + \chi_v) + f(y + \chi_u - \chi_v).

Writing $\tilde f(x_0, x) = f(x)$ when $x_0 = -x(V)$ and $+\infty$ otherwise (a lift to one extra coordinate), $f$ is M $^\natural$ -convex if $\tilde f$ is M-convex; M♮-convexity is a genuine generalization of M-convexity (every M-convex function is M♮-convex, but not conversely) and coincides with it exactly when $\operatorname{dom} f$ lies on a single hyperplane. The linear-weighted function $f[p](x) = f(x) - \langle p, x \rangle$ (for $p \in \mathbb R^V$ ) is the standard device for testing local optimality under an arbitrary reweighting.

Formalization targets

Goal: the exchange axiom as sequential improvement

f \text{ is M-convex} \iff \forall p \in \mathbb R^V,\ \forall x, y \in \operatorname{dom} f,\ f[p](x) > f[p](y) \implies f[p](x) > \min_{u \in \operatorname{supp}^+(x-y)}\ \min_{v \in \operatorname{supp}^-(x-y)} f[p](x - \chi_u + \chi_v),

with the analogous statement for M♮-convexity (Theorem 6.24). This is the weakest stable form: it makes no reference to a specific algorithm, only to the existence of an improving single exchange whenever the current point is suboptimal under any linear reweighting — a property a faster algorithm could exploit without invalidating the characterization itself.

Supporting structural targets

Eleven further results build the vocabulary and toolkit this goal draws on: the domain structure of M-convex and M♮-convex functions (Propositions 6.1, 6.7), the equivalence of the exchange axiom with a local, bounded-distance version (Theorem 6.4), worked examples establishing M-convexity for quadratic forms, univariate, conservation-law, and quasi-separable functions (Propositions 6.8-6.9), the domain and range operations preserving M-convexity (Theorem 6.13, Proposition 6.14), supermodularity of the M♮-convex case (Theorem 6.19), the descent-direction property (Proposition 6.23) that Theorem 6.24 generalizes, and a discrete subgradient inequality (Proposition 6.25).

Significance

Theorem 6.24 is the bridge between the static exchange axiom (a property of function values at pairs of points) and the dynamic behavior of local-search algorithms: it says a greedy single-coordinate-exchange step, applied to any linearly reweighted version of an M-convex function, always finds a strict improvement when one exists. This is exactly the guarantee that makes steepest-descent-type algorithms for M-convex function minimization correct, and it is the theorem chapter 10's algorithmic analysis (Schrijver-type methods) relies on implicitly whenever it argues that local exchange steps make global progress. The descent-direction property (Proposition 6.23) is the special case $p=0$ , isolating the core combinatorial fact before the reweighting machinery is added. The operations catalog (Theorem 6.13) is the practical toolkit that lets later chapters build complex M-convex functions (network flow costs, matroid rank functions composed with linear maps) from simple pieces without re-verifying the exchange axiom from scratch each time.

None of these results are open — they are Murota's own systematic development of the exchange- axiom theory, with worked examples drawn from classical quadratic and separable function theory. What this mission contributes is a faithful, machine-checked formal statement of each, sharing the Lean vocabulary (MExchangeAxiom, MNaturalConvex, LinearWeight) the rest of the Discrete Convex Analysis series builds on; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The forward direction of Theorem 6.24 (M-convex $\implies$ sequential improvement) follows in one step from Proposition 6.23 applied to $f[p]$ , itself M-convex by Theorem 6.13(3) — routine once those two pieces are in hand. The converse is the substantial direction: it must derive the full static exchange axiom from a property that only ever exhibits some improving exchange at some linear weighting, for every pair of suboptimal points — the proof constructs an explicit adversarial weighting $p$ designed so that failure of the local exchange step at that specific $p$ forces the domain itself to be M-convex (via Theorem 4.3) and then forces the local exchange axiom (M-EXCloc[Z]) via a bipartite-matching argument on the coordinates that differ, finally invoking Theorem 6.4 to lift locality to the full exchange axiom. No shortcut bypasses this two-stage reduction (domain structure, then local exchange) — attempting to verify (M-EXC[Z]) directly from (M-SI[Z]) without first pinning down that $\operatorname{dom} f$ is M-convex fails because the exchange axiom's own statement presupposes a well-structured domain.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; functions are (V → ℤ) → WithTop ℝ. SuppPos/SuppNeg are Finset V (not Set V), matching how the (M-SI[Z])/(M♮-SI[Z]) axioms and the descent-direction property use Finset.inf, whose value on an empty index set is ⊤ — exactly the book's own stated convention for an empty minimum. No Module ℝ or ConvexOn machinery is used for WithTop ℝ-valued arithmetic; scalar actions by positive reals (PosScalarMul, Theorem 6.13(1)) and by naturals (FCheck's flow coefficients, Proposition 6.25) are built directly from the native order and AddMonoid structure. No numeric constants are hard-coded anywhere in this mission (rule 7 is vacuous); Proposition 6.8's quadratic-form conditions are stated with the book's own literal coefficients ( $0$ , and the min/≥ structure of Eq. (6.25)-(6.28)), not a special case. Theorem 6.13's parts (7) (aggregation) and (8) (integer infimal convolution) are not restated here since the book itself proves them only later via Chapter 9's network-transformation machinery — see the Difficulty note and MODERATION_NOTES.md; this is not a trivializing omission, since the six operations that are included already exercise every domain- and range-transformation technique this mission's goal needs. This mission's definitions (MExchangeAxiom, MNaturalConvex, CharVec, DomZ, SuppPos, SuppNeg, CharVecOpt) are redeclared from chunk 06-mconvex-functions-i rather than imported, since sibling drafts in this series cannot yet reference one another. Contributions completing any of the twelve sorrys are welcome; the goal's converse direction and Theorem 6.13's operations are the two with the most independent proof content.

Selected references

K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
K. Murota, "Discrete convex analysis," Mathematical Programming, 83 (1998), pp. 313-371 (the exchange axiom and its equivalent local/dynamic reformulations).

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Tasty Bits of Several Complex Variables I: Holomorphic Functions and Power Series on PolydiscsTextbook

Why holomorphic functions of several variables

Setting

Formalization targets

Goal: Theorem 1.2.1 (power series on a polydisc)

Milestones

Significance

Difficulty

Formalization scope

Selected references

Discrete Convex Analysis XXX: Lagrangian Duality for M-Convex ProgramsTextbook

Motivation

Setting

Formalization targets

Goal: Strong duality for M-convex programs (Theorem 8.59)

Supporting structural targets

Significance

Difficulty

Formalization scope

Selected references

Applied Combinatorics X: The Lovász Local Lemma and the Gale–Ryser TheoremTextbook

Motivation

Setting

Formalization targets

Goal: the asymmetric local lemma (Lemma 16.14)

Milestone: the symmetric local lemma (Lemma 16.15)

Milestones: covers in P(t)\mathcal P(t)P(t) and Gale–Ryser (Proposition 16.11, Theorem 16.12)

Significance

Difficulty

Formalization scope

Selected references

Discrete Convex Analysis XXIX: M2-Convex and L2-Convex FunctionsTextbook

Motivation

Setting

Formalization targets

Goal: L2♮^\natural_22♮​-convex functions are integrally convex (Theorem 8.42)

Supporting structural targets

Significance

Difficulty

Formalization scope

Selected references

Applied Combinatorics IX: Pólya's Enumeration TheoremTextbook

Motivation

Setting

Formalization targets

Milestone: Proposition 15.8

Milestone: Lemma 15.9 (Burnside's Lemma)

Goal: Theorem 15.11 (Pólya's Enumeration Theorem)

Significance

Difficulty

Formalization scope

Selected references

Applied Combinatorics VIII: The Max Flow–Min Cut TheoremTextbook

Motivation

Setting

Formalization targets

Goal: the Max Flow–Min Cut Theorem (Theorem 13.10)

Milestones

Significance

Difficulty

Formalization scope

Selected references

Discrete Convex Analysis XXVIII: The Conjugacy TheoremTextbook

Motivation

Setting

Formalization targets

Goal: the conjugacy theorem (Theorem 8.4)

Supporting structural targets

Significance

Difficulty

Formalization scope

Selected references

Applied Combinatorics VI: Ramsey's Theorem and Erdős's Lower BoundTextbook

Motivation

Milestones: covers in $\mathcal P(t)$ and Gale–Ryser (Proposition 16.11, Theorem 16.12)

Goal: L $^\natural_2$ -convex functions are integrally convex (Theorem 8.42)