Discrete Convex Analysis

Motivation

Matroids abstract the combinatorial content of linear independence: which sets of columns of a matrix are independent, which are maximal (bases), and how bases relate to each other. This abstraction, isolated independently by Whitney (1935) and van der Waerden's school, turned out to be exactly the right level of generality for a large family of greedy and augmenting-path algorithms — a base of a matroid can always be reached from another by a sequence of single-element swaps, and this exchange property is what makes local search on bases correct and efficient.

A natural question, raised in the 1980s once matroid-based combinatorial optimization was mature, is what happens when bases are not merely present or absent but carry real-valued weights that must interact well with the exchange structure. Dress and Wenzel answered this with the notion of a valuated matroid: a real-valued function on the bases of a matroid satisfying a weighted strengthening of the exchange axiom. Their motivation was explicitly algorithmic — valuated matroids are exactly the structures for which a greedy algorithm computes an optimal basis under linear objectives, and more generally under the family of "tilted" objectives obtained by adding an arbitrary linear functional. Independently, valuated matroids arise from the classical Grassmann–Plücker relation applied to matrices over a field with a valuation (hence the name), connecting them to tropical geometry.

This mission formalizes the two theorems of Murota's Discrete Convex Analysis (2003, §2.4) that make this story precise: the classical correspondence between a matroid's base family and its rank function (Theorem 2.29), and the characterization of valuations by a perturbation-robustness property (Theorem 2.32). Theorem 2.32 is also historically the entry point of the book's central theme — it is the special case, for the two-valued lattice $\{0,1\}^V$, of the general local-exchange criterion for M-convex functions that occupies chapters 6 and 7.

Setting

Let $V$ be a finite set (the ground set). A matroid on $V$ is a pair $(V, \mathcal B)$ where $\mathcal B$, the base family, is a nonempty family of subsets of $V$ satisfying the simultaneous exchange axiom (B): for every $J, J' \in \mathcal B$ and every $i \in J \setminus J'$, there exists $j \in J' \setminus J$ such that both

$$J - i + j := (J \setminus \{i\}) \cup \{j\} \in \mathcal B \quad\text{and}\quad J' + i - j := (J' \setminus \{j\}) \cup \{i\} \in \mathcal B.$$

Equivalently (Theorem 2.29 below), a matroid can be described by its rank function $\rho : 2^V \to \mathbb Z$, a set function satisfying:

• (R1) $0 \le \rho(X) \le |X|$ for every $X \subseteq V$;
• (R2) monotonicity: $X \subseteq Y \implies \rho(X) \le \rho(Y)$;
• (R3) submodularity: $\rho(X) + \rho(Y) \ge \rho(X \cup Y) + \rho(X \cap Y)$.

A valuation of a base family $\mathcal B$ is a function $\omega : \mathcal B \to \mathbb R$ satisfying the axiom (VM): for every $J, J' \in \mathcal B$ and $i \in J \setminus J'$, there is $j \in J' \setminus J$ with $J - i + j, J' + i - j \in \mathcal B$ and

$$\omega(J) + \omega(J') \le \omega(J - i + j) + \omega(J' + i - j).$$

The pair $(V, \omega)$ is then a valuated matroid. For $p : V \to \mathbb R$, the perturbation of $\omega$ by $p$ is

$$\omega[-p](J) = \omega(J) - \sum_{j \in J} p(j).$$

Formalization targets

Goal: Theorem 2.32 (the valuated matroid characterization)

$$\omega \text{ is a valuation of } \mathcal B \iff \forall\, p : V \to \mathbb R,\ \{J \in \mathcal B : \omega[-p](J') \le \omega[-p](J)\ \forall J' \in \mathcal B\} \text{ is a nonempty family satisfying (B)}.$$

The right-hand side says: for every linear perturbation $p$, the set of $\omega[-p]$-maximal bases is again the base family of a matroid. The universal quantifier over $p$ is not optional — a version of this statement quantified over a single fixed $p$ is either vacuous or false, and does not capture what makes valuated matroids useful.

Milestone: Theorem 2.29 (the base-family / rank-function correspondence)

The maps

$$\rho(X) = \max\{|X \cap J| : J \in \mathcal B\}, \qquad \mathcal B = \{J \subseteq V : \rho(J) = |J| = \rho(V)\}$$

are mutually inverse bijections between nonempty families satisfying (B) and set functions satisfying (R1)-(R3). This is weaker groundwork than the goal, stated first because it fixes the exact axiomatic vocabulary — (B) and (R) — that Theorem 2.32 is built on.

Significance

The result itself. Theorem 2.32 is the reason valuated matroids are the right object for weighted combinatorial optimization on matroids: it says a function on bases behaves correctly under every linear re-weighting of the ground set exactly when it satisfies the local exchange inequality (VM). This is what guarantees, for instance, that a greedy algorithm which is correct for the unweighted matroid extends correctly to families of tilted objectives, and it is the germ of the general local-optimality criterion for M-convex functions (chapters 6–7), which underlies most of the algorithmic content of the rest of the book. Theorem 2.29 is the classical result — due jointly to the development of matroid theory from the 1930s onward — that the base-exchange and rank-submodularity axiomatizations of a matroid carry the same information; it is the finite, unweighted precursor of Theorem 2.32.

Formalizing it. Neither theorem has a machine-checked proof on the platform prior to this mission (see Formalization scope for the prior-art check). Theorem 2.29's own proof is elementary but has two independent halves (each map preserves its target axiom class, and the two maps compose to the identity in both directions) that must all be established; Theorem 2.32's proof, as given in the source, defers entirely to a later, more general chapter-6 theorem, so a solver working only from this mission must either reconstruct a direct combinatorial argument for this special case or await chunk 06 (DiscreteConvex.MConvexFunctions, a separate mission) and specialize its main theorem.

Difficulty

The obvious approach to Theorem 2.32 — fix an optimal basis $J$ for $\omega[-p]$ and try to show the exchange condition on maximizers directly from (VM) — proves one direction (VM implies the maximizer property) in a few lines, since perturbing does not change which exchange moves are available. The converse is the substantial direction: from "the maximizer set is always a matroid, for every $p$," one must recover the single global inequality (VM) that must hold for all pairs $J, J' \in \mathcal B$, not just optimal ones. The standard argument constructs, for a given non-optimal pair, a perturbation $p$ under which that specific pair becomes simultaneously optimal, and this construction is exactly the step the book skips by citing chapter 6's general theorem. A formalization attempting to bypass this by only checking the maximizer property for a finite or generic sample of perturbations would trivialize the statement to something false or vacuous — a pitfall the goal's explicit ∀ p is designed to prevent.

Formalization scope

The ground set $V$ is a Fintype with DecidableEq; $2^V$ is represented as Finset (Finset V), and $V \to \mathbb R$ as a plain function type. The rank function is $\mathbb Z$-valued (matching the book's own convention for matroid rank, as opposed to the $\mathbb R$-valued conventions used from chapter 6 onward for general M-convex functions); RankOfFamily is implemented with Finset.sup over $\mathbb N$ rather than a partial max', so that it is a total function — its junk value at the empty family is never invoked, since every hypothesis in this mission supplies nonemptiness explicitly, matching the book's own phrasing.

A trivializing formalization of the goal is one that quantifies over a single fixed $p$, or allows $\mathcal B$ to be empty; both are explicitly excluded by keeping $\mathcal B.\text{Nonempty}$ a hypothesis and $p$ universally quantified inside the theorem statement itself.

Checked against Mathlib (commit 0df444a360eaa60ab8c11dca51a86af692955474): Mathlib's Matroid structure is axiomatized via the single-element (asymmetric) exchange property, classically but not definitionally equivalent to Murota's simultaneous axiom (B) used throughout this book, and Mathlib provides no constructor recovering a base family or a Matroid from a bare rank function satisfying (R1)-(R3). Theorem 2.29 is therefore genuine, reusable infrastructure, not a restatement of existing Mathlib API. No reference item was found on the platform for either theorem (GET /theorems?q=matroid, q=valuated matroid return only unrelated tropical-geometry and k-server results). Contributions to a shared DiscreteConvex.Combinatorial definitions layer (the exchange and rank axioms) are welcome from later chunks of this series that build on matroid or base-polyhedron structure.

Selected references

• K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
• H. Whitney, "On the abstract properties of linear dependence," American Journal of Mathematics, 57(3), 1935, pp. 509–533.
• A. W. M. Dress, W. Wenzel, "Valuated matroids," Advances in Mathematics, 93(2), 1992, pp. 214–250.
• R. A. Brualdi, "Comments on bases in dependence structures," Bulletin of the Australian Mathematical Society, 1(2), 1969, pp. 161–167.

Motivation

Quadratic minimization problems with a combinatorial sign pattern in their Hessian arise throughout applied mathematics: discretizations of elliptic boundary-value problems such as the Poisson equation, resistor-network energy functionals, and the Dirichlet forms of Markov-process potential theory all produce a symmetric matrix whose off-diagonal entries are nonpositive and whose rows are diagonally dominant (Fukushima, Oshima, and Takeda, Dirichlet Forms and Symmetric Markov Processes, De Gruyter, 1994). Such matrices are exactly the diagonally dominant symmetric M-matrices of classical numerical linear algebra (Berman and Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, 1994). Murota's Discrete Convex Analysis (SIAM, 2003) identifies the combinatorial content of this sign pattern with a discrete convexity property — submodularity, and its strengthening translation submodularity — of the associated quadratic form, and shows that passing to the Legendre-Fenchel conjugate of such a quadratic form (i.e., inverting the matrix) transports this property to a dual combinatorial property, an exchange axiom, on the conjugate side. This mission formalizes that correspondence for the special, matrix-algebraic case of quadratic forms — the case in which Murota's book gives a self-contained proof using only the classical Farkas lemma, before generalizing the same conjugacy to a much broader class of functions in Chapter 8.

Setting

Let $V$ be a finite ground set (identified with $\{1,\dots,n\}$ in the book) and let $L = (\ell_{ij})_{i,j\in V}$ be a symmetric real matrix. $L$ has off-diagonal nonpositivity if $\ell_{ij}\le 0$ for all $i\ne j$, and diagonal dominance if $\sum_{j} \ell_{ij}\ge 0$ for every row $i$. The associated quadratic form is $g(p) = \tfrac12 p^\top L p$ for $p \in \mathbb R^V$. For $p,q\in\mathbb R^V$ write $p\vee q$, $p\wedge q$ for the componentwise maximum and minimum. A function $g:\mathbb R^V\to\mathbb R$ is submodular if $g(p)+g(q)\ge g(p\vee q)+g(p\wedge q)$ for all $p,q$, and has translation submodularity if the stronger inequality $g(p)+g(q)\ge g((p-\alpha\mathbf 1)\vee q)+g(p\wedge(q+\alpha\mathbf 1))$ holds for every $\alpha \ge 0$, where $\mathbf 1$ is the all-ones vector (ordinary submodularity is the case $\alpha=0$).

On the conjugate side, for $x\in\mathbb R^V$ write $\operatorname{supp}^+(x)=\{i : x_i>0\}$, $\operatorname{supp}^-(x)=\{i:x_i<0\}$, and let $\chi_i$ denote the $i$-th unit vector ($\chi_0$ denotes the zero vector). A function $f:\mathbb R^V\to\mathbb R$ has the exchange property if for all $x,y\in\mathbb R^V$ and $i\in\operatorname{supp}^+(x-y)$ there exist $j \in \operatorname{supp}^-(x-y)\cup\{0\}$ and $\alpha_0>0$ such that $f(x)+f(y)\ge f(x-\alpha(\chi_i-\chi_j))+f(y+\alpha(\chi_i-\chi_j))$ for every $\alpha\in[0,\alpha_0]$. The Legendre-Fenchel conjugate of $f$ is $f^\bullet(p) = \sup_x\{\langle p,x\rangle - f(x)\}$; two functions $g,f$ are conjugate to each other when $g=f^\bullet$ and $f=g^\bullet$. For positive-definite symmetric $M,L$, the quadratic forms $f(x)=\tfrac12x^\top Mx$ and $g(p)=\tfrac12p^\top Lp$ are conjugate to each other exactly when $M$ and $L$ are matrix inverses of one another.

Formalization targets

Goal (Theorem 2.11). For conjugate strictly convex quadratic forms $g$ and $f$ as above,

$$g \text{ has translation submodularity} \iff f \text{ has the exchange property.}$$

This is the mission's capstone: the statement leaves the correspondence at the level of the two named combinatorial properties, without hard-coding which of the two properties is verified in a given application, so it survives exactly as strongly as the underlying conjugacy fact does.

Supporting milestones, in the order the book develops them: Proposition 2.4 (off-diagonal nonpositivity plus diagonal dominance implies positive semidefiniteness); Proposition 2.6 (off-diagonal nonpositivity is equivalent to plain submodularity of $g$); Theorem 2.7 (the full sign pattern is equivalent to translation submodularity of $g$); Proposition 2.9 (conjugate quadratic forms correspond exactly to inverse matrix pairs); Theorem 2.12 (a nine-way equivalence, for a nonsingular symmetric $M$, among membership in the matrix class $\mathcal L^{-1}$, two sign-consistency inequalities on the columns of $M$ together with their strict forms, two directional-derivative reformulations of the exchange property together with their strict forms, and the exchange property itself together with its strict form); Proposition 2.13 (the Farkas lemma in equality form, together with the strict variant valid for a nonsingular coefficient matrix); and Proposition 2.14 (the class $\mathcal L^{-1}$ is closed under taking principal submatrices).

Significance

The M-natural exchange property is the function-level analogue of the base-exchange axiom for matroids, and translation submodularity is the analogue, on the "primal" side, of ordinary submodularity for set functions; Chapter 2's quadratic-form case is the historical and pedagogical entry point for the general conjugacy Chapter 8 proves for the full M-convex/ L-convex function classes. Establishing it here, in the self-contained matrix-algebraic setting, isolates exactly which properties of a quadratic form are combinatorial (tied to the coordinate axes) rather than purely convex-analytic (rotation-invariant): submodularity and the exchange property are not preserved by an orthogonal change of variables, in contrast to ordinary convexity, which Proposition 2.4 shows the same sign pattern also implies.

Formalizing this mission produces the first Lean statement, in this project's namespace, of a genuine conjugacy theorem between a primal-side and a dual-side combinatorial convexity property for a concrete function class; nothing of this kind is yet proved (or, so far as the platform's own search shows, formalized at all) elsewhere on the platform. The nine-way equivalence of Theorem 2.12 is a substantial independent contribution beyond the goal itself, since it is what makes the goal's proof possible via elementary linear algebra rather than the general convex-analytic machinery Chapter 8 needs.

Difficulty

The naive approach to Theorem 2.11 tries to derive the exchange property for $f$ directly from the defining supremum in the conjugate relation $f = g^\bullet$, differentiating under the sup; this fails because the exchange property compares $f$ along a specific combinatorial direction $\chi_i - \chi_j$ tied to two coordinates, not along an arbitrary direction, and no naive first-order argument isolates the right pair $(i,j)$ without already knowing the sign pattern of $M = L^{-1}$. The book's actual route is Theorem 2.12: it reduces the exchange property to a column-wise sign-consistency statement on $M$ itself (conditions (b)/(c)) via the identity $f'(x;d) = x^\top Md$, and closes the loop back to membership in $\mathcal L^{-1}$ using the Farkas lemma applied to the linear system $ML = I$ — a genuinely matrix-algebraic argument that does not generalize verbatim to non-quadratic M-/L-convex functions, which is exactly why Chapter 8 needs a different (convex-analytic) proof for the general case.

Formalization scope

Vectors and matrices are indexed by a general finite type V ([Fintype V] [DecidableEq V]) rather than a fixed Fin n, matching this project's convention elsewhere and letting Proposition 2.14's principal-submatrix statement reuse the class predicate at the restricted index type directly. Quadratic forms are real-valued ((V → ℝ) → ℝ, using Matrix.mulVec and dotProduct) since this chapter's functions are always finite everywhere; the Legendre-Fenchel conjugate is EReal-valued via sSup, since a supremum over an infinite domain need not be finite in general even though it is finite here. Every min(0, \dots)-based condition in Theorem 2.12 and the exchange axioms is unfolded as the logically equivalent disjunction over the finitely many terms achieving the minimum, rather than reified via Finset.inf/WithTop machinery — a faithful, checked-equivalent simplification, not a narrowing (see MODERATION_NOTES.md). "Nonsingular" is Matrix.det ≠ 0. No numeric constant needs instantiation anywhere in this mission. The formalization does not trivialize: the goal's exchange property is stated for the specific combinatorial direction $\chi_i - \chi_j$ with $i\in \operatorname{supp}^+(x-y)$, $j \in \operatorname{supp}^-(x-y)\cup\{0\}$ — not an arbitrary direction, which would reduce the exchange property to a restatement of ordinary convexity and discard the entire combinatorial content the mission is about.

Infrastructure needed: Matrix.PosDef/Matrix.PosSemidef/Matrix.IsSymm (present in Mathlib); everything else (submodularity, translation submodularity, the exchange axioms, the sign-consistency conditions) is defined fresh in DiscreteConvex.CombinatorialB. A solution to the goal will likely want Proposition 2.9, Theorem 2.12, and the Farkas lemma (Proposition 2.13) as lemmas; contributions completing any of the seven milestones independently, or supplying the Schur-complement induction behind Proposition 2.4, are welcome.

Selected references

• K. Murota, Discrete Convex Analysis, SIAM, 2003, DOI 10.1137/1.9780898718508, Chapter 2.
• A. Berman, R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, 1994.
• M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes, De Gruyter, 1994.
• J. Farkas, Theorie der einfachen Ungleichungen, J. Reine Angew. Math. 124 (1902), 1–27.

Motivation

For a convex function on $\mathbb R^n$, a point is a global minimizer as soon as it is a local minimizer — this is one of the earliest and most consequential facts of convex analysis, and it underlies why local-search and gradient methods can certify global optimality in convex programs. The discrete analogue is not automatic: a function on the integer lattice $\mathbb Z^n$ can be "locally optimal" with respect to any fixed finite neighborhood system and still fail to be a global minimizer, unless the function's discrete structure is compatible with that neighborhood in the right way. Identifying exactly which classes of lattice functions admit a local-to-global optimality principle, and with respect to which neighborhood, is one of the organizing questions of discrete convex analysis.

Integrally convex functions, introduced by Favati and Tardella (1990) and developed systematically by Murota, are the most general class of $\mathbb Z^n$-valued functions for which such a principle holds. They are defined purely in terms of the classical convex closure of a real relaxation, which lets one import theorems from ordinary convex analysis, but the resulting notion of local optimality — checking only the $3^n - 1$ neighbors obtained by independently nudging each coordinate by $-1$, $0$, or $+1$ (excluding the trivial no-change case) — is a genuinely discrete, dimension-independent statement about functions whose domain can be arbitrarily large. Almost every discrete convex function class studied later in the book, including M-convex and L-convex functions, is a special case of integral convexity, and this mission's goal theorem is the direct ancestor of the optimality criteria (Theorems 6.26 and 7.14) that drive the algorithms in the rest of the book.

Setting

Let $f : \mathbb Z^n \to \mathbb R \cup \{+\infty\}$ be a function with nonempty effective domain $\operatorname{dom}_{\mathbb Z} f = \{x \in \mathbb Z^n : f(x) \ne +\infty\}$. The convex closure of $f$ is

$$\bar f(x) = \sup_{p \in \mathbb R^n,\, \alpha \in \mathbb R} \{\langle p,x\rangle + \alpha : \langle p,y\rangle + \alpha \le f(y)\ \forall y \in \mathbb Z^n\} \qquad (x \in \mathbb R^n),$$

the pointwise supremum of every affine function minorizing $f$ on all of $\mathbb Z^n$. If $\bar f$ agrees with $f$ on integer points, $f$ is convex extensible. The integral neighborhood of $x \in \mathbb R^n$ is

$$N(x) = \{y \in \mathbb Z^n : \lfloor x_i \rfloor \le y_i \le \lceil x_i \rceil,\ 1 \le i \le n\},$$

and the local convex extension $\tilde f$ relaxes $\bar f$'s definition by requiring the affine minorant condition only on $N(x)$ rather than on all of $\mathbb Z^n$. Always $\tilde f \ge \bar f$ pointwise, and the two agree on $\mathbb Z^n$. A function $f$ is integrally convex if $\tilde f = \bar f$ everywhere on $\mathbb R^n$ — equivalently, if $\tilde f$ is a convex function on all of $\mathbb R^n$ (it is automatically convex on every unit cube $[z, z+1]^n$ with $z \in \mathbb Z^n$, but need not be convex globally without this extra condition).

A discrete set $S \subseteq \mathbb Z^n$ is hole free if $S$ equals the set of integer points in its own real convex hull, and $\arg\min f[-p]$ denotes the minimizer set, over $\mathbb Z^n$, of the linearly perturbed function $f[-p](x) = f(x) - \langle p,x\rangle$.

Formalization targets

Goal: Theorem 3.21 (local optimality characterizes global optimality)

For integrally convex $f$ and $x \in \operatorname{dom}_{\mathbb Z} f$:

$$f(x) \le f(y)\ (\forall y \in \mathbb Z^n) \iff f(x) \le f(x + \chi_Y - \chi_Z)\ (\forall\, Y, Z \subseteq \{1,\dots,n\}),$$

where $\chi_Y \in \{0,1\}^n$ is the indicator vector of $Y$. The right-hand side is a check over at most $3^n - 1$ points (each coordinate independently unchanged, incremented, or decremented), regardless of how large $\operatorname{dom}_{\mathbb Z} f$ is; this uniform, dimension-only bound is the entire content of the theorem, and is the weakest correct formulation — restricting to a single $(Y,Z)$ or letting the right-hand side range over all of $\mathbb Z^n$ would trivialize or falsify the equivalence.

Milestones: Propositions 3.18 and 3.19

Proposition 3.18: $f$ convex extensible $\implies$ $\arg\min f[-p]$ hole free for every $p$ (and conversely, when $\operatorname{dom}_{\mathbb Z} f$ is bounded). Proposition 3.19: $f$ is integrally convex if and only if every restriction $f_{[a,b]}$ to a finite integer interval is integrally convex — integral convexity is detectable by looking at bounded pieces of $f$ one at a time.

Significance

The result itself. Theorem 3.21 is what makes integrally convex functions tractable: without it, verifying global optimality on an infinite or exponentially large integer domain would require checking every point. The theorem reduces this to a check whose size depends only on the dimension $n$, not on the size of the domain, and it does so for the widest class of lattice functions for which such a reduction is possible — the class is defined precisely so that this property holds and no wider natural class enjoys it. Every specialized local-optimality theorem later in the book (for M-convex, M$^\natural$-convex, L-convex, and L$^\natural$-convex functions) restricts this same neighborhood-checking principle to a class where the local check can be made even smaller (a single-element exchange rather than a full sign pattern) precisely because those classes are integrally convex plus more.

Formalizing it. No matching item exists on the platform: a direct search for "integrally convex" returns no results, and the theorem's own proof leans on results (Theorem 1.1's local-to-global principle for ordinary convex functions on $\mathbb R^n$, and an LP-duality-based alternate formula for $\tilde f$) that are either classical convex analysis or belong to a different chapter of this same book. The remaining work is therefore to give a complete, correct account of the definitional chain — convex closure, local convex extension, integral convexity — in a form a solver can build a proof from directly, and to state the finite local-check equivalence itself exactly at the strength the book proves it, not a plausible-looking weakening of it.

Difficulty

The natural first attempt is to try to prove the "$\Leftarrow$" direction of Theorem 3.21 by a direct induction on the $\ell^1$-distance to a global minimizer, moving one coordinate at a time. This fails in general lattice functions (a function that is only "coordinatewise convex" can have strict local minima that are not global), and the theorem's actual proof instead routes through the real relaxation: it shows the neighborhood-check hypothesis forces $x$ to be a local minimizer of the local convex extension $\tilde f$ restricted to the unit ball around $x$, then invokes ordinary convex analysis (local minimality implies global minimality for a convex function on $\mathbb R^n$) to conclude $x$ globally minimizes $\bar f$, and finally uses integral convexity ($\tilde f = \bar f$) to transfer this back to $f$ on $\mathbb Z^n$. The identification of $f$'s local behavior with $\tilde f$'s convexity on a single unit cube — rather than any coordinatewise or separable argument — is the step that makes the class of integrally convex functions exactly the right one for this theorem, and is where a naive combinatorial argument breaks down.

Formalization scope

The ground set is $\mathbb Z^n$, represented as Fin n → ℤ; $f$'s codomain is WithTop ℝ (exactly $\mathbb R \cup \{+\infty\}$), while the convex closure $\bar f$ and local convex extension $\tilde f$ take values in EReal (exactly $\mathbb R \cup \{\pm\infty\}$, a complete lattice, so their defining suprema are total functions with no side conditions). A trivializing formalization of the goal would quantify the right-hand side over a single fixed $(Y,Z)$ pair, or over all of $\mathbb Z^n$ instead of the sign-pattern neighbors; both are excluded by keeping $Y, Z$ universally quantified Finset (Fin n) ranging over the full $3^n$ sign-pattern space (minus the trivial case, which the equivalence still holds through vacuously).

Checked against the platform (GET /theorems?q=integrally convex, 0 hits) and against Mathlib's Analysis/Convex/ for the classical facts this chapter's proof would eventually need (ordinary convex-function local-to-global optimality, LP duality): these are broadly available in Mathlib's convex-analysis library in some form, but none of them is imported here, since none appears in the statement of any item this mission drafts — they belong to a proof this pass does not attempt. Contributions to a shared DiscreteConvex.IntegralConvexity definitions layer are welcome from chunks 06–09, which specialize integral convexity to M-convex and L-convex functions and will need the same convex-closure/local-extension vocabulary.

Selected references

• K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
• P. Favati, F. Tardella, "Convexity in nonlinear integer programming," Ricerca Operativa, 53, 1990, pp. 3–44.

Motivation

A classical convex function's global minimality is equivalent to its local minimality — the single fact that makes convex optimization tractable, since checking a small neighborhood suffices to certify a global guarantee. The discrete analogue is not automatic: a function on the integer lattice can fail to have any well-behaved "local" notion at all, and even when a discrete convexity-like property is imposed, the naive candidate (a function's values agreeing with its own convex-hull interpolation) does not by itself guarantee that local optimality implies global optimality. Murota's Discrete Convex Analysis (SIAM, 2003) isolates exactly the extra condition — integral convexity — that restores this guarantee, and shows it is general enough to contain every other discrete convexity notion the book studies (M-convex, L-convex, and their variants), making it the common ancestor of the book's entire hierarchy of classes. This mission completes the chapter's account of integral convexity: how it behaves under sums, restrictions, and linear perturbations, how it transfers between a function and its domain or minimizer sets, and a companion fact about hole-freeness under intersection and Minkowski sums that motivates why integral convexity, not mere hole-freeness, is the right notion to use.

Setting

For $f : \mathbb Z^n \to \mathbb R \cup \{+\infty\}$ with nonempty effective domain $\operatorname{dom}_{\mathbb Z} f$, the convex closure is $\bar f(x) = \sup_{p,\alpha} \{\langle p,x\rangle + \alpha : \langle p,y\rangle+\alpha \le f(y)\ \forall y \in \mathbb Z^n\}$. The integral neighborhood of $x \in \mathbb R^n$ is $N(x) = \{y \in \mathbb Z^n : \lfloor x_i \rfloor \le y_i \le \lceil x_i \rceil\}$, and the local convex extension $\tilde f$ replaces "for all $y \in \mathbb Z^n$" in $\bar f$'s definition with "for all $y \in N(x)$". A function is integrally convex if $\tilde f = \bar f$ everywhere on $\mathbb R^n$. A set $S \subseteq \mathbb Z^n$ is integrally convex if its indicator function is; it is hole free if $S = \bar S \cap \mathbb Z^n$, where $\bar S$ is the real convex hull of $S$. The discrete Minkowski sum is $S_1+S_2 = \{x_1+x_2 : x_1 \in S_1, x_2 \in S_2\}$. A function is separable convex if $f(x) = \sum_i f_i(x(i))$ for univariate functions $f_i$ satisfying the discrete convexity inequality $f_i(t-1)+f_i(t+1) \ge 2f_i(t)$. For $p \in \mathbb R^n$, $f[-p] (x) = f(x) - \langle p,x \rangle$ and $\arg\min f[-p]$ is its minimizer set.

Formalization targets

Goal (Theorem 3.29). For $f$ with nonempty bounded effective domain,

$$f \text{ is integrally convex} \iff \arg\min f[-p] \text{ is an integrally convex set for every } p \in \mathbb R^n.$$

This leaves the characterization at the level of the two named properties (integral convexity of the function versus of every minimizer set), the strongest statement of this kind that holds without extra hypotheses beyond boundedness of the domain.

Supporting milestones. Proposition 3.17 (four basic containment/equality relations between hole-free sets' intersections, Minkowski sums, and their real closures); Proposition 3.22 (for a periodic integrally convex function, global optimality reduces to a one-sided local check); Proposition 3.24 (an integrally convex function plus a separable convex function is integrally convex); Proposition 3.25 (separable convex functions are integrally convex, and integral convexity survives linear perturbation); Proposition 3.26 (an integrally convex set is hole free); Proposition 3.28 (the effective domain and every minimizer set of an integrally convex function are integrally convex sets — the forward direction of the goal); and Proposition 3.30 (for integer-valued integrally convex functions, a finite infimum is always attained).

Significance

Theorem 3.29 turns a statement about a function on all of $\mathbb R^n$ (integral convexity, a condition on $\tilde f$ and $\bar f$ that is a priori about uncountably many points) into a statement about a countable family of discrete sets (the minimizer sets $\arg\min f[-p]$), giving a genuinely different and often more tractable way to certify or refute integral convexity. Propositions 3.24–3.25 are the closure properties that make integral convexity useful in practice: without them, verifying integral convexity of a function built from simpler pieces (a sum with a separable cost, a linearly reweighted objective) would require re-deriving the property from scratch each time. Proposition 3.17, by contrast, is a cautionary result: Note 3.27 and Example 3.15 (the two hole-free sets whose Minkowski sum has a hole) show that hole-freeness alone does not inherit good behavior under set operations, which is exactly the gap integral convexity's stronger, locally-checkable condition is built to close — this mission's Proposition 3.17 documents the "obvious"/general-purpose relations that hold regardless, so that the reader can see precisely which inclusion is automatic and which requires more.

Difficulty

The naive approach to Theorem 3.29's converse direction (integral convexity of every minimizer set implies integral convexity of $f$) tries to check $\tilde f(x) = \bar f(x)$ directly at an arbitrary $x \in \operatorname{dom} f$; this is circular, since $\tilde f$ and $\bar f$ are themselves defined via suprema over affine minorants, not via minimizer sets. The book's actual proof instead sets up a primal-dual pair of linear programs whose optimal solutions witness $\bar f(x)$ and $\tilde f(x)$ respectively, uses LP duality's complementary slackness to show the dual optimal solution can be chosen supported inside $N(x)$, and only then concludes $\tilde f(x) = \bar f(x)$ — routing the entire argument through the integral convexity of the specific minimizer set $S = \arg\min f[-p^*]$ at the optimal dual price $p^*$. This is why Theorem 3.29's proof needs LP duality (Theorem 3.10, formalized in the previous mission in this series) as an ingredient, not just the closure-property machinery of Propositions 3.24–3.28.

Formalization scope

All apparatus (ConvexClosure, IntegralNeighborhood, LocalConvexExtension, IntegrallyConvex, ArgMinPerturbed, HoleFree, IntegrallyConvexSet, SeparableConvex, MinkowskiSumZ) is redeclared fresh in DiscreteConvex.IntegralConvexityC, mirroring chunk 03-integral-convexity's already-established constructions (Fin n-indexed, WithTop ℝ-valued functions, EReal-valued convex closures via sSup), since a draft mission cannot import another draft's definitions. IntegrallyConvexSet is defined via the book's own primary definition (indicator function integrally convex) rather than either of its two stated equivalent reformulations, since no result in this mission needs those forms as a named predicate. An integer-valued function (Proposition 3.30) is represented as Zⁿ → WithTop ℤ and cast to WithTop ℝ via a small casting map wherever the real-valued apparatus is needed — a faithful embedding. Boundedness of a discrete set is containment in a finite integer interval. The formalization does not trivialize: Theorem 3.29's hypothesis is exactly "nonempty bounded effective domain", not further restricted to, say, a fixed small dimension or a finite ground set with a fixed cardinality bound, and every milestone is stated at the same generality as Propositions 3.24–3.28 and 3.30 give it (arbitrary $n$, arbitrary integrally convex function). Infrastructure needed beyond Mathlib: all definitions are fresh; a contribution completing any milestone, or the LP-duality-based proof of Theorem 3.29's converse direction, would be a natural entry point, alongside chunk 03's Theorem 3.21 as background.

Selected references

• K. Murota, Discrete Convex Analysis, SIAM, 2003, DOI 10.1137/1.9780898718508, Chapter 3.
• K. Murota, A. Shioura, "M-convex function on generalized polymatroid," Mathematics of Operations Research 24 (1999), 95–105 (Lemma 6.13, cited for Proposition 3.30).

Motivation

Matroid intersection is one of the founding results of combinatorial optimization: given two matroids on a common ground set, the largest common independent set can be found in polynomial time, and its size equals the minimum of a natural upper bound ranging over all subsets — a min-max theorem in the spirit of König's theorem and Menger's theorem, but for a strictly richer combinatorial structure. Jack Edmonds proved this in 1970, and Jack Edmonds and Rick Giles's subsequent generalization to submodular flows, together with André Frank's discrete separation theorem for submodular and supermodular set functions (1982), placed matroid intersection inside a single unifying framework: submodular function duality. This framework explains, in one stroke, matroid intersection, the base-exchange structure of matroids, and a family of other combinatorial min-max theorems that had previously seemed unrelated.

Murota's Discrete Convex Analysis develops this framework as the theory of M-convex sets: sets of integer vectors satisfying a lattice-exchange axiom that turns out to be exactly equivalent to being the integer points of a base polyhedron of an integer-valued submodular set function. This mission formalizes the chapter's central results: the equivalence of four variant forms of the exchange axiom (Theorem 4.3), the M-convex set / submodular function correspondence (Theorem 4.15), Frank's discrete separation theorem (Theorem 4.17), and Edmonds's intersection theorem itself (Theorem 4.18) — the deepest duality result in the theory of submodular functions and the historical origin of the M-convexity concept that the rest of the book generalizes to real-valued functions.

Setting

Let $V$ be a finite ground set. A set function $\rho : 2^V \to \mathbb R \cup \{+\infty\}$ with $\rho(\emptyset) = 0$ and $\rho(V) < +\infty$ is submodular (the class $S[\mathbb R]$) if

$$\rho(X) + \rho(Y) \ge \rho(X \cup Y) + \rho(X \cap Y) \qquad (X, Y \subseteq V).$$

Its base polyhedron and submodular polyhedron are

$$B(\rho) = \{x \in \mathbb R^V : x(X) \le \rho(X)\ (\forall X \subseteq V),\ x(V) = \rho(V)\}, \qquad P(\rho) = \{x \in \mathbb R^V : x(X) \le \rho(X)\ (\forall X \subseteq V)\},$$

where $x(X) = \sum_{v \in X} x(v)$; a supermodular function $\mu$ is one with $-\mu$ submodular. A nonempty set $B \subseteq \mathbb Z^V$ is an M-convex set if it satisfies the exchange axiom (B-EXC[Z]): for $x, y \in B$ and $u$ in the positive support of $x-y$, there is $v$ in the negative support of $x-y$ with both $x - \chi_u + \chi_v \in B$ and $y + \chi_u - \chi_v \in B$, where $\chi_u$ is the characteristic vector of $u$. A polyhedron $P \subseteq \mathbb R^V$ is integral if $P = \operatorname{conv}(P \cap \mathbb Z^V)$.

Formalization targets

Goal: Theorem 4.18 (Edmonds's intersection theorem)

For submodular set functions $\rho_1, \rho_2 \in S[\mathbb R]$,

$$\max\{x(V) : x \in P(\rho_1) \cap P(\rho_2)\} = \min\{\rho_1(X) + \rho_2(V \setminus X) : X \subseteq V\},$$

with both sides attained. If $\rho_1, \rho_2$ are integer valued, $P(\rho_1) \cap P(\rho_2)$ is an integral polyhedron and the maximum is attained at an integer point. Dropping the integrality clause and stating only the real max-min equality would leave ordinary LP duality with no discrete content at all; this mission keeps it in the goal at every strength the book proves it.

Milestones: Theorems 4.3, 4.15, 4.17

Theorem 4.3: the exchange axiom (B-EXC[Z]) is equivalent to three variants that impose the exchange condition asymmetrically or only for distinct vectors — groundwork establishing that M-convexity does not depend on which variant is taken as primitive. Theorem 4.15: $B$ is M-convex if and only if $B = B(\rho) \cap \mathbb Z^V$ for some integer-valued submodular $\rho$ — M-convex sets and integer-valued submodular set functions are two descriptions of the same combinatorial object. Theorem 4.17 (Frank): if a submodular $\rho$ dominates a supermodular $\mu$ pointwise, a single vector $x^*$ separates them ($\rho \ge x^* \ge \mu$ pointwise on every subset), integrally when $\rho, \mu$ are integer valued — derived, in the book, as a direct corollary of the goal theorem.

Significance

The result itself. Edmonds's intersection theorem is the min-max theorem underlying polynomial-time matroid intersection (a matroid's rank function is submodular, so the classical matroid intersection theorem is the special case $\rho_1, \rho_2$ both matroid rank functions), and its generality — arbitrary submodular set functions, not just matroid ranks — is what lets Frank's discrete separation theorem, and through it a wide range of combinatorial duality results in network flows, scheduling, and matroid theory, be derived as corollaries rather than proved from scratch each time. The integrality clause specifically is the fact that makes these duality theorems combinatorial: it guarantees that optimal fractional solutions to the underlying linear program can always be taken integral, without which the connection to discrete optimization would be lost.

Formalizing it. No matching item exists on the platform (searches for "submodular set function", "base polyhedron", "matroid intersection" return no relevant hits; Mathlib's Combinatorics/Matroid/ develops matroid rank functions, a special case, but not general submodular set functions or their polyhedra). This mission gives the first formal statement of the theorem at its natural generality, together with the M-convex-set viewpoint that motivates the rest of the book, and Frank's separation theorem as an explicit worked corollary.

Difficulty

The real-valued half of Theorem 4.18 is ordinary LP duality applied to a cleverly chosen primal program (maximize $\langle p, x\rangle$ over $P(\rho_1) \cap P(\rho_2)$) and its dual — routine once the right LP is written down. The integrality half is where the combinatorics enters: an optimal dual solution can always be chosen supported on a chain in each $\rho_i$'s effective domain (an extremal argument maximizing a strictly convex potential over the optimal dual face), and the incidence matrix of a chain of subsets is totally unimodular — this is the fact, external to ordinary LP theory, that forces an integral optimal solution to exist whenever the data ($\rho_1, \rho_2$) are integral. A proof that stops at real-valued LP duality, however carefully done, misses this step entirely and cannot produce the integrality clause; total unimodularity of a chain's incidence matrix is the one piece of combinatorics doing all the discrete work in an otherwise classical convex-duality argument.

Formalization scope

The ground set $V$ is a Fintype with DecidableEq; subsets are Finset V, vectors are V → ℝ/V → ℤ. Submodular functions take values in WithTop ℝ (exactly $\mathbb R \cup \{+\infty\}$); supermodular functions in WithBot ℝ; comparisons across the two use an explicit embedding into EReal. The max/min in the goal are stated via IsGreatest/IsLeast sharing a common EReal witness, so that "both sides attained, at the same value" — not merely "sup equals inf" — is what the Lean statement asserts, which is essential since the integrality clause's whole content is about which point attains the maximum.

A trivializing formalization of the goal would drop the integrality clause (leaving unqualified LP duality) or replace IsGreatest/IsLeast with a bare supremum/infimum equality (losing the "is attained" content the second half of the theorem needs); both are avoided. Theorem 4.15 is stated as the existential "iff" (some integer submodular $\rho$ realizes $B$) rather than reifying the book's own named bijection $\Phi, \Psi$ explicitly — a deliberate, documented scope reduction of that one milestone (see MODERATION_NOTES.md), not of the goal. Contributions building the explicit $\Phi$ map, the Lovász extension (needed for Theorem 4.16, not drafted here), or M-convex-set infrastructure reusable by chunks 06–07 (M-convex functions, which build on this chapter's vocabulary) are welcome.

Selected references

• K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
• J. Edmonds, "Submodular functions, matroids, and certain polyhedra," in Combinatorial Structures and Their Applications, Gordon and Breach, 1970, pp. 69–87.
• A. Frank, "An algorithm for submodular functions on graphs," Annals of Discrete Mathematics, 16, 1982, pp. 97–120.

Motivation

The classical separating hyperplane theorem says that any two disjoint convex sets in $\mathbb R^n$ can be separated by a hyperplane with an arbitrary real normal vector. When the sets in question are not arbitrary convex sets but the integer points of specially structured discrete sets, one can sometimes ask for much more: not merely that a separator exists, but that it can be chosen from a small, structured, dimension-independent family regardless of the size or shape of the sets being separated. Results of this kind — "discrete separation theorems" — are a recurring and often surprising theme in combinatorial optimization, playing the role that the ordinary separation theorem plays in continuous convex analysis, but with genuinely combinatorial content beyond it.

L-convex sets, introduced by Murota as part of the discrete convex analysis framework, are one of the two dual families of well-behaved discrete convex sets studied in the book (the other being M-convex sets, chunk 04 of this series). They are defined by a lattice-closure axiom together with translation invariance, and they correspond one-to-one to integer-valued distance functions satisfying the triangle inequality — objects long familiar from network flow theory and shortest-path duality, even though the L-convexity terminology is not traditionally used there. This mission formalizes the chapter's central results, culminating in Theorem 5.9: two disjoint L-convex sets can always be separated by a vector with entries in $\{-1, 0, 1\}$, no matter how large or complicated the sets are.

Setting

Let $V$ be a finite ground set. A nonempty set $D \subseteq \mathbb Z^V$ is an L-convex set if it satisfies the sublattice axiom (SBS[Z]) — $p, q \in D \implies p \vee q,\ p \wedge q \in D$, where $\vee, \wedge$ are componentwise maximum and minimum — and the translation axiom (TRS[Z]) — $p \in D \implies p \pm \mathbf 1 \in D$, where $\mathbf 1$ is the all-ones vector. A distance function $\gamma : V \times V \to \mathbb R \cup \{+\infty\}$ satisfies $\gamma(v,v) = 0$ for every $v$; it satisfies the triangle inequality if $\gamma(v_1,v_2) + \gamma(v_2,v_3) \ge \gamma(v_1,v_3)$ for all $v_1, v_2, v_3$. The admissible-potential polyhedron of $\gamma$ is

$$D(\gamma) = \{p \in \mathbb R^V : p(v) - p(u) \le \gamma(u,v)\ (\forall u \ne v)\}.$$

The convex hull of a discrete set $D \subseteq \mathbb Z^V$ is written $\bar D \subseteq \mathbb R^V$.

Formalization targets

Goal: Theorem 5.9 (discrete separation for L-convex sets)

If $D_1, D_2 \subseteq \mathbb Z^V$ are disjoint L-convex sets, there exists $x^* \in \{-1,0,1\}^V$ such that

$$\inf\{\langle p, x^*\rangle : p \in D_1\} - \sup\{\langle p, x^*\rangle : p \in D_2\} \ge 1.$$

Dropping the $\{-1,0,1\}^V$ restriction and allowing an arbitrary real separator would recover the classical separation theorem for convex sets, which holds regardless of L-convexity and carries no discrete-convexity content; the three-valued restriction is the weakest correct strengthening and is kept in full.

Milestones: Theorems 5.2, 5.5, 5.7

Theorem 5.2: an L-convex set is hole free ($D = \bar D \cap \mathbb Z^V$) — its integer points are exactly the integer points of its own convex hull. Theorem 5.5: $D$ is L-convex if and only if $D = D(\gamma) \cap \mathbb Z^V$ for some integer-valued distance function $\gamma$ satisfying the triangle inequality — L-convex sets and such distance functions are two descriptions of the same object, the discrete analogue of chunk 04's M-convex-set / submodular- function correspondence. Theorem 5.7 (parts (1), (4)): L-convex sets are closed under intersection in the strongest sense — the convex hulls intersect exactly where the sets do, and a nonempty intersection of L-convex sets is again L-convex.

Significance

The result itself. Theorem 5.9 packs two claims into one, as the book itself points out: the separator is forced into $\{-1,0,1\}^V$ (explicit in the statement), and disjoint L-convex sets satisfy "convexity in intersection" — their convex hulls are already disjoint whenever the sets themselves are (implicit, and necessary for the stated inequality to be possible at all). The $\{-1,0,1\}$ structure connects directly to combinatorial duality in network flows: L-convex polyhedra are, without the name, a familiar object there, and a $\{-1,0,1\}$-separator corresponds to a signed cut or a negative-cost cycle in an associated graph. Theorem 5.5's correspondence is the L-convex mirror of chunk 04's M-convex/submodular correspondence, and the book explicitly flags that the two will be unified into a single conjugacy relationship in a later chapter (Note 5.6) — this mission's formalization of the L-side is a prerequisite for that later unification.

Formalizing it. No matching item exists on the platform (searches for "L-convex", "distance function", and "negative cycle" return only unrelated results — number-theoretic distance estimates, polytope graph metrics, shortest-path graph structures — none matching the combinatorial L-convexity/discrete-separation content here). This mission gives the first formal statement of L-convex sets and their central separation theorem. Notably, Theorem 5.9's own statement — unlike the analogous M-convex Theorem 4.18 — needs none of the distance-function machinery that its proof uses; only the L-convexity axiom itself appears in the goal, making its formal statement comparatively lean even though the underlying mathematics is just as deep.

Difficulty

The natural first attempt at Theorem 5.9 is to try to construct $x^*$ directly from the structure of $D_1, D_2$ — for instance, from a normal vector to a real separating hyperplane, rounded coordinatewise. This does not work: rounding an arbitrary real separator gives no control over its entries, and there is no reason a rounded vector should still separate. The book's actual proof instead represents $D_1, D_2$ via distance functions $\gamma_1, \gamma_2$ (Theorem 5.5), combines them into $\gamma_{12} = \min(\gamma_1, \gamma_2)$, and extracts the separator from a shortest negative cycle in the associated graph: the vertices of the cycle alternate between the two sets' "tight" arcs, and the alternating $\pm 1$ pattern around the cycle is exactly the $\{-1,0,1\}$ vector $x^*$ — with the cycle's negativity translating directly into the required gap of at least $1$. Locating the right combinatorial object (a shortest negative cycle, not an arbitrary one) is what pins the separator down to a vector supported on a single alternating cycle rather than an arbitrary $\{-1,0,1\}$ pattern, and is the step a naive rounding or linear-algebra argument has no analogue of.

Formalization scope

The ground set $V$ is a Fintype with DecidableEq; $D \subseteq \mathbb Z^V$ is a Set (V → ℤ). Distance functions take values in WithTop ℝ; the goal's infimum and supremum are taken in EReal (a complete lattice), since L-convex sets are always infinite (translation invariance along the all-ones direction), so an ℝ-valued supremum/infimum would silently return a junk value on an unbounded set. The conclusion is stated as $\sup_{D_2}\langle p,x^*\rangle + 1 \le \inf_{D_1}\langle p,x^*\rangle$, an addition-based reformulation of the book's subtraction inequality that avoids EReal's ⊤ - ⊤ ambiguity while remaining equivalent whenever both sides are finite.

A trivializing formalization of the goal would drop the $\{-1,0,1\}^V$ constraint on $x^*$ (recovering the classical, L-convexity-independent separation theorem) or fix a single coordinate pattern rather than asserting existence over the full three-valued family; neither is done here. Theorem 5.5 is stated existentially rather than via the book's named bijection $\Phi, \Psi$ (a documented scope reduction, parallel to chunk 04's treatment of Theorem 4.15), and Theorem 5.7 is drafted with only its two representation-independent clauses (parts (1) and (4); see MODERATION_NOTES.md). Contributions building the distance-function/admissible- potential apparatus needed for Theorem 5.7's remaining clauses, or the L-convex/integrally-convex bridge (Theorem 5.10, needing chunk 03's vocabulary), are welcome.

Selected references

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

Motivation

Shortest-path distances and network potentials are among the oldest objects in combinatorial optimization: a directed graph with arc lengths, its shortest-path distances, and the "feasible potentials" (vertex labels consistent with those lengths) underlie duality in min-cost flow, scheduling, and difference-constraint systems. Murota's Discrete Convex Analysis (SIAM, 2003) isolates the abstract structure behind these objects — distance functions satisfying the triangle inequality, and their associated sets of admissible potentials — and shows it is governed by exactly the same discrete-convexity machinery as submodular set functions: a one-to-one correspondence with a second family of well-behaved integer point sets, the L-convex sets. Where an M-convex set (chapter 4) is defined by an exchange axiom generalizing matroid base exchange, an L-convex set is defined by closure under coordinatewise lattice operations (∨, ∧) and translation by the all-ones vector — a genuinely different axiom system that nonetheless produces a parallel structural theory: hole-freeness, a polyhedral description via an induced distance function, and integral convexity.

Companion mission 05-lconvex-sets (Discrete Convex Analysis IV) covers this chapter's other half: the hole-free property (Theorem 5.2), the one-to-one correspondence between L-convex sets and integer-valued triangle-inequality distance functions (Theorem 5.5), the intersection properties (Theorem 5.7), and the chapter's discrete separation theorem (Theorem 5.9, 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 the chapter leaves for its second half: the fundamental facts connecting a distance function to its admissible potentials (Proposition 5.1), the two-way polyhedral correspondence's supporting propositions (5.3-5.4), Minkowski-sum convexity (Theorem 5.8), and — this mission's goal — the explicit description of an L-convex set's convex hull that establishes its integral convexity (Theorem 5.10).

Setting

Fix a finite ground set $V$. A distance function is a map $\gamma : V \times V \to \mathbb R \cup \{+\infty\}$ with $\gamma(v,v) = 0$; it may take negative finite values and need not be symmetric. It defines a directed graph $G_\gamma = (V, A_\gamma)$ with $A_\gamma = \{(u,v) : \gamma(u,v) < +\infty\}$, arc $(u,v)$ having length $\gamma(u,v)$. Write $\bar\gamma(u,v)$ for the shortest-path length from $u$ to $v$ in $G_\gamma$ ($+\infty$ if none exists); $\gamma$ is well defined ($\bar\gamma$ finite-valued wherever a path exists) exactly when $G_\gamma$ has no negative cycle. The triangle inequality $\gamma(v_1,v_2) + \gamma(v_2,v_3) \ge \gamma(v_1,v_3)$ defines the class $T[\mathbb R]$ (or $T[\mathbb Z]$ when integer-valued). A vector $p \in \mathbb R^V$ is an admissible potential of $\gamma$ if $p(v) - p(u) \le \gamma(u,v)$ for all $u \ne v$; write $D(\gamma)$ for the set of all such potentials.

A nonempty set $D \subseteq \mathbb Z^V$ is L-convex if it satisfies (SBS[Z]): $p, q \in D \implies p \vee q,\ p \wedge q \in D$ (coordinatewise max/min), and (TRS[Z]): $p \in D \implies p \pm \mathbf 1 \in D$. A set $S \subseteq \mathbb Z^V$ is integrally convex if every point of its convex hull $\overline S$ lies in the convex hull of $S$ restricted to that point's integral neighborhood $N(p) = \{y \in \mathbb Z^V : \lfloor p \rfloor \le y \le \lceil p \rceil\text{ coordinatewise}\}$ — a strong, local form of "no holes" saying every real point of the hull is explained by nearby integer points alone.

Formalization targets

Goal: integral convexity of L-convex sets

For an L-convex set $D \subseteq \mathbb Z^V$, writing $a = p - \lfloor p \rfloor$ for the fractional part of $p \in \mathbb R^V$, $\alpha_1 > \cdots > \alpha_m$ for the distinct nonzero values of $a$, and $U_i(p) = \{v : a(v) \ge \alpha_i\}$ (with $U_0 = \emptyset$):

$$\overline D = \{p \in \mathbb R^V : \lfloor p \rfloor + \chi_{U_i(p)} \in D\ \ (i = 0, 1, \ldots, m)\}, \qquad \text{hence } D \text{ is integrally convex}.$$

This is the weakest stable form available: it exhibits an explicit, finite set of at most $|V|+1$ integer witnesses for every point of the hull, which is what "integrally convex" asserts abstractly, rather than a numerical bound that a sharper construction could later shrink.

Supporting structural targets

Four further results build the correspondence this goal uses: the basic duality between a distance function's admissible potentials, its shortest-path closure, and negative-cycle freedom (Prop. 5.1); the induced-distance-function construction recovering a triangle-inequality distance function from any integer point set, and the convex hull of an L-convex set as its associated polyhedron (Prop. 5.3); the converse construction recovering an L-convex set from an integer-valued distance function (Prop. 5.4); and convexity in Minkowski sum (Thm. 5.8).

Significance

Theorem 5.10 is what makes "L-convex" a genuinely convex-analytic notion rather than a combinatorial curiosity: it shows the convex hull of an L-convex set is not merely a polyhedron (already known from the chapter's polyhedral-description results) but one with the strongest local integrality property discrete convex analysis considers, integral convexity — every real point's hull membership is certified by a small, explicitly constructed set of nearby lattice points, uniformly across the whole set. This is the L-convex counterpart of the corresponding M-convex fact (chapter 4's Theorem 4.24) and is used later in the book wherever L-convex functions (chapter 7) need their epigraphs' local structure. Proposition 5.1 is the combinatorial engine underneath: it is exactly the LP-duality statement between shortest paths and feasible potentials that appears, in various guises, throughout network flow theory, made precise here as the base case the L-convex correspondence rests on.

None of these results are open — Murota presents them as, in his own words, "fundamental facts well known in network flow theory" (Proposition 5.1) systematized into the discrete convex analysis framework. What this mission contributes is a faithful, machine-checked formal statement of each, in the shared Lean vocabulary (LConvexSet, AdmissiblePotentials, ShortestDist) the rest of the Discrete Convex Analysis series can build on; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The shortest-path closure $\bar\gamma$ is not a bookkeeping convenience but genuinely graph-theoretic content: proving Proposition 5.1 requires constructing an admissible potential from a shortest-path labeling and, conversely, deriving the negative-cycle-freeness of $G_\gamma$ from the mere existence of one admissible potential — a min-cost-flow-style LP duality argument, not a direct combinatorial check. Theorem 5.10's difficulty sits in a different place: the naive approach to "$D$ is integrally convex" would attempt an inductive argument peeling off one coordinate at a time, but the actual proof constructs a single, uniform family of $m+1$ witness points from the sorted fractional values of $p$ — a Carathéodory-style representation (Eq. (5.11)) that must simultaneously stay inside the integral neighborhood $N(p)$ and land in $D$ itself via the triangle inequality of $D$'s induced distance function, a construction with no one-coordinate-at-a-time shortcut.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; L-convex sets are Set (V → ℤ); distance functions are V → V → WithTop ℝ; admissible-potential sets are Set (V → ℝ). The shortest-path closure is formalized directly from finite walks (Fin (k+1) → V) rather than via a graph-library shortest-path predicate, matching the book's own construction. The Eq. (5.11) witnesses are built exactly as the book describes them — sorted distinct nonzero fractional values and their level sets — mirroring the Lovász-extension construction of the companion mission 20-ch04b-mconvexsets. No numeric constants are hard-coded anywhere in this mission (rule 7 is vacuous). The goal's explicit witness set (at most $|V|+1$ points) is not a trivializing special case: it holds for every L-convex set and every point of its hull, with no extra hypothesis narrowing the class. This mission's definitions (LConvexSet, AdmissiblePotentials, DistanceFunction, IsIntegrallyConvex) are redeclared from chunk 05-lconvex-sets (and, for IsIntegrallyConvex/IntegralNeighborhood, from chapter 3's own definitions) rather than imported, since sibling drafts in this series cannot yet reference one another; a later, published version of this book's namespace should consolidate them. Contributions completing any of the five sorrys are welcome; Proposition 5.1's LP-duality argument and the goal's Carathéodory-style construction are the two with the most independent proof content.

Selected references

• K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
• A. J. Hoffman, "On abstract dual linear programs," Naval Research Logistics Quarterly, 10 (1963), pp. 369-373 (feasible-potential duality in network flow theory).

Motivation

Scaling algorithms are one of the standard techniques for solving discrete optimization problems efficiently: instead of searching a huge integer domain directly, an algorithm first solves a coarsened version of the problem — checking optimality only against neighbors reached by a large step size $\alpha$ — and then refines the resulting approximate solution down to the true optimum. This strategy is only as good as the guarantee that a coarse-scale local optimum is provably close to a true, fine-scale global optimum; without such a guarantee, refinement could require an unbounded number of steps. Results providing this guarantee are called proximity theorems, and they are a standard tool across combinatorial optimization, from network flow scaling algorithms to submodular function minimization.

M-convex functions, the subject of this chapter, are exactly the class of discrete convex functions for which the classical local-optimality test of chapter 3 (checking a full neighborhood of up to $3^n-1$ sign patterns) sharpens to a much smaller, purely pairwise test: checking $f(x) \le f(x - \chi_u + \chi_v)$ for every pair of coordinates $u, v$. This mission formalizes the chapter's central definitional equivalence (Theorem 6.2), this pairwise optimality criterion (Theorem 6.26), a structural minimizer-cut lemma (Theorem 6.28), and the chapter's capstone, the M-proximity theorem (Theorem 6.37) — the result that makes M-convex scaling algorithms provably correct, with an explicit, dimension-and-scale-only distance bound between a coarse-scale local optimum and a true global minimizer.

Setting

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

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

An M$^\natural$-convex function is one whose lift $\tilde f$ to the extended ground set $\tilde V = \{0\} \cup V$ — defined by $\tilde f(x_0, x) = f(x)$ when $x_0 = -x(V)$, and $+\infty$ otherwise — is M-convex; equivalently (Theorem 6.2, below) $f$ satisfies the axiom (M$^\natural$-EXC[Z]), a variant of (M-EXC[Z]) that additionally allows a single-coordinate move ($u$ alone, with no compensating $v$). Every M-convex function is M$^\natural$-convex, but not conversely. For $\alpha$ a positive integer, a point satisfies the scaled local optimality condition at scale $\alpha$ if $f(x_\alpha) \le f(x_\alpha + \alpha(\chi_v - \chi_u))$ for all relevant $u, v$ — a check against neighbors $\alpha$ steps away rather than adjacent ones.

Formalization targets

Goal: Theorem 6.37 (the M-proximity theorem)

Assume $\alpha$ is a positive integer and $n = |V|$.

$$\text{(1) } f \text{ M-convex, } f(x_\alpha) \le f(x_\alpha + \alpha(\chi_v-\chi_u))\ \forall u,v \implies \exists x^* \in \arg\min f,\ \|x_\alpha - x^*\|_\infty \le (n-1)(\alpha-1),$$ $$\text{(2) } f \text{ M}^\natural\text{-convex, same hypothesis over } u,v \in V \cup \{0\} \implies \exists x^* \in \arg\min f,\ \|x_\alpha - x^*\|_\infty \le n(\alpha-1).$$

Both bounds are exact and specific to their hypothesis class; replacing either with an unspecified function of $n$ and $\alpha$ would discard exactly the content chapter 10's algorithms rely on.

Milestones: Theorems 6.2, 6.26, 6.28

Theorem 6.2: M$^\natural$-convexity (defined via the lift) is equivalent to the direct exchange axiom (M$^\natural$-EXC[Z]) — the chapter's central definitional theorem, needed to work with M$^\natural$-convex functions without repeatedly invoking the lift construction. Theorem 6.26 (the M-optimality criterion): global optimality of $f$ at $x$ is equivalent to a purely pairwise local check, $f(x) \le f(x-\chi_u+\chi_v)$ for all $u,v$ (plus, in the M$^\natural$ case, $f(x) \le f(x\pm\chi_v)$). Theorem 6.28 (the M-minimizer cut): from any point and any coordinate pair minimizing a one-step exchange, one can certify a coordinate-wise bound that some global minimizer must satisfy — the structural fact underlying both the domain-reduction algorithm and, via the same proof technique, the proximity theorem itself.

Significance

The result itself. Theorem 6.26 already sharpens chapter 3's local-to-global criterion (checking a full $3^n-1$-point neighborhood) to an $O(n^2)$-size pairwise check — the minimum spanning tree optimality criterion is a direct special case. The proximity theorem builds on this to control what happens when the local check is only performed at a coarse scale $\alpha$: it guarantees that scaling-based algorithms, which alternate between coarse-scale local search and scale reduction, terminate with a guaranteed-close approximation at every stage, with an explicit linear-in-$n$, linear-in-$\alpha$ error bound rather than a qualitative "eventually converges" guarantee.

Formalizing it. No matching item exists on the platform for M-convex functions, the exchange axiom, or a discrete proximity theorem of this kind. This mission gives the first formal statement of the M-optimality criterion and the M-proximity theorem, together with the exchange-axiom / lift-based-definition equivalence (Theorem 6.2) that the rest of the M-convex function theory (chunks 07, and indirectly 10–14) is built on.

Difficulty

The natural first attempt at Theorem 6.37 is to try a direct coordinatewise argument: since the scaled hypothesis holds for every pair $u, v$, one might hope to bound $|x_\alpha(v) - x^*(v)|$ coordinate by coordinate independently. This does not work, because a single application of the exchange axiom only ever improves $f$ by trading one coordinate down and one other coordinate up simultaneously — there is no way to move a single coordinate toward a minimizer in isolation without accounting for where the compensating mass goes. The actual proof instead fixes a target coordinate $v$, constructs a chain of strictly decreasing function values $y_0 = x_\alpha, y_1, \ldots, y_k$ by repeatedly applying (M-EXC[Z]) against a fixed near-optimal point $x^*$ (exactly the technique of Theorem 6.28's proof), and then bounds how far each other coordinate can move along this chain using the scaled hypothesis itself, before summing those bounds via the M-convex domain's hyperplane constraint $x(V) =$ constant to recover the bound on $v$. The chain construction, not a per-coordinate estimate, is what makes the linear-in-$n$ bound provable at all.

Formalization scope

The ground set $V$ is a Fintype with DecidableEq; $f : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ is (V → ℤ) → WithTop ℝ. M$^\natural$-convexity is represented via an explicit lift to Option V (none standing for the extended ground set's new element $0$), matching the book's own primary definition; the direct exchange-axiom form is a separate predicate related to it by Theorem 6.2, not conflated with it. `‖x_\alpha - x^*|_\infty \le c$ is stated pointwise.

A trivializing formalization of the goal would replace either exact bound, $(n-1)(\alpha-1)$ or $n(\alpha-1)$, with an unspecified asymptotic bound, or merge the two hypothesis classes into a single weaker statement; neither is done here. Propositions establishing dom f as an M-convex set, the M/M$^\natural$ relationship (Theorem 6.3), and several structural closure properties are cut from this mission's scope (not needed by the chosen items' statements — see MODERATION_NOTES.md) but are natural targets for a follow-on mission or for chunk 07, which builds directly on this chunk's exchange-axiom vocabulary. Contributions building the arg min f M-convexity corollary (Proposition 6.29) or the scaled minimizer cut (Theorem 6.39, the direct generalization of Theorem 6.28 drafted here) are welcome.

Selected references

• K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
• D. S. Hochbaum, "Lower and upper bounds for the allocation problem and other nonlinear optimization problems," Mathematics of Operations Research, 19(2), 1994, pp. 390–409.

Motivation

An M-convex function is defined on the integer lattice, but chapter 6's earlier results (companion missions 06-mconvex-functions-i, 22-ch06b-mconvexfunctions, 23-ch06c-mconvexfunctions) show it always extends to a genuine convex function on real space. Once that extension exists, every tool of classical convex analysis — directional derivatives, subdifferentials, positive homogeneity — becomes available, and the natural question is whether these classical objects remain combinatorially special when applied to an M-convex function's extension. This mission answers that question at its sharpest: the directional derivative of an M-convex function at any point is again a positively homogeneous M-convex function, its subdifferential is exactly the admissible-potential set of a distance function satisfying the triangle inequality, and this correspondence between positively homogeneous M-convex functions and triangle-inequality distance functions is itself a clean one-to-one correspondence. This closes the loop between chapters 4-5 (M-convex and L-convex sets, distance functions) and the continuous convex-analytic machinery chapter 8 needs for its duality theory.

Companion missions 06-mconvex-functions-i, 22-ch06b-mconvexfunctions, and 23-ch06c-mconvexfunctions cover this chapter's optimality theory, algebraic toolkit, and convex-extensibility characterization. This mission builds the vocabulary those results also need (redeclared here, since sibling drafts cannot yet import one another) and proves the chapter's real-variable capstones: the transfer of M-convexity's basic operations, optimality criterion, and supermodularity to the polyhedral (real-variable) setting, the identification of positively homogeneous M-convex functions with distance functions satisfying the triangle inequality, and — this mission's goal — the full directional-derivative/subdifferential correspondence.

Setting

Fix a finite ground set $V$. A polyhedral convex function $g : \mathbb R^V \to \mathbb R \cup \{+\infty\}$ is (polyhedral) M-convex if it satisfies the real-variable exchange axiom (M-EXC[R]): for $x,y \in \operatorname{dom}_{\mathbb R} g$ and $u \in \operatorname{supp}^+(x-y)$, some $v \in \operatorname{supp}^-(x-y)$ and $\alpha_0 > 0$ make the exchange inequality hold on $\alpha \in [0,\alpha_0]$; M♮-convex if its lift to one extra coordinate is M-convex. The directional derivative of $g$ at $x \in \operatorname{dom}_{\mathbb R} g$ in direction $d$ is $g'(x;d) = \inf_{t>0} (g(x+td) - g(x))/t$. A function is positively homogeneous if $g(tx) = t \cdot g(x)$ for all $t > 0$; write $0M[\mathbb R \to \mathbb R]$ for the positively homogeneous polyhedral M-convex functions. A distance function $\gamma$ satisfying the triangle inequality and its set of admissible potentials $D(\gamma)$ were introduced in chapter 5; the subdifferential $\partial_{\mathbb R} f(x) = \{p : f(y) - f(x) \ge \langle p, y-x \rangle\ \forall y\}$ generalizes this to any function $f$ at a point $x$ in its domain.

Formalization targets

Goal: the directional-derivative/subdifferential correspondence

For $f \in M[\mathbb R \to \mathbb R]$ and $x \in \operatorname{dom}_{\mathbb R} f$, setting $\gamma_{f,x}(u,v) = f'(x;-\chi_u+\chi_v)$:

$$\gamma_{f,x} \text{ satisfies the triangle inequality}, \quad \partial_{\mathbb R} f(x) = D(\gamma_{f,x}) \ne \emptyset, \quad f'(x;\cdot) = \widehat{\gamma_{f,x}}(\cdot),$$

with the analogous statement for $f \in M[\mathbb Z \to \mathbb R]$ at an integer point $x$, using $\gamma_{f,x}(u,v) = f(x-\chi_u+\chi_v)-f(x)$ (Theorem 6.61). This is the weakest stable form: it identifies the subdifferential exactly, as a set, rather than bounding its size or complexity, and holds at every point of the domain uniformly.

Supporting structural targets

Ten further results build the real-variable toolkit and the positive-homogeneity correspondence this goal completes: the transfer of M♮-convexity, the basic operations, the optimality criterion, supermodularity, and weighted-minimizer polyhedrality to the real-variable setting (Theorems 6.48-6.52, Proposition 6.53), the identification of the classes $0M[\mathbb Z|\mathbb R \to \mathbb R]$ and $0M[\mathbb R \to \mathbb R]$ and the compatibility of convex extension with positive homogeneity (Proposition 6.56), the two directions of the correspondence between positively homogeneous M-convex functions and triangle-inequality distance functions (Propositions 6.57-6.58, Theorem 6.59), and the fact that a directional derivative of an M-convex function is itself positively homogeneous and M-convex (Proposition 6.60).

Significance

Theorem 6.61 is the technical bridge that lets discrete convex analysis borrow the entire apparatus of classical convex duality: because the subdifferential of an M-convex function is always the admissible-potential set of a chapter-5 distance function, every fact already proved about $D(\gamma)$ (its polyhedral structure, its own L-convexity, its relationship to shortest paths) transfers immediately to subdifferentials of M-convex functions. This is exactly the mechanism the book calls out as essential for Chapter 8's separation theorem for M♮-convex functions. The $0M \leftrightarrow T$ correspondence (Theorem 6.59) is independently significant: it says the positively homogeneous special case of M-convex function theory — which is what directional derivatives of any M-convex function reduce to, by Proposition 6.60 — is exactly as rich as ordinary shortest-path distance function theory, no more and no less, so nothing new needs to be built to understand local behavior at a point.

None of these results are open — they are Murota's account of how the discrete exchange axiom interacts with directional differentiation and subgradients, a bridge chapter between the purely combinatorial theory of chapters 4-6 and the duality theory of chapter 8. What this mission contributes is a faithful, machine-checked formal statement of each, extending the shared Lean vocabulary (MExchangeAxiomR, DirDeriv, GammaHat) the Discrete Convex Analysis series builds on; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The naive approach to Theorem 6.61 would try to compute $\partial_{\mathbb R} f(x)$ directly from the definition of subgradient and separately verify it happens to equal some $D(\gamma)$; the book's actual proof instead derives the equality of sets from the M-optimality criterion (Theorem 6.52) applied pointwise: $p \in \partial_{\mathbb R} f(x)$ is shown, via a chain of logical equivalences, to be exactly the condition defining $D(\gamma_{f,x})$, so no separate verification of polyhedrality or nonemptiness is needed beyond what Theorem 6.52 and Proposition 6.60 already supply. The genuine difficulty is upstream, in Proposition 6.60 itself: showing a directional derivative is M-convex requires exploiting the local validity of the identity $f(x+d)-f(x) = f'(x;d)$ for small $\|d\|_1$ (Eq. (6.85)) and then extending the exchange property from that neighborhood to all of $\mathbb R^V$ using positive homogeneity — a two-step argument with no single-step shortcut, since the exchange axiom's defining inequality is not obviously homogeneous-invariant on its own.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; real-domain functions are (V→ℝ)→WithTop ℝ. The directional derivative is built directly as an infimum of difference quotients over $t>0$, matching the book's own local characterization (Eq. (6.85)) without a separate limit construction. Positive homogeneity and the classes 0M[R→R]/0M[Z→R] are stated exactly as the book defines them (the latter via positive homogeneity of the convex extension, not of f itself, since f is undefined off Zⱽ). Theorems 6.49-6.50 restate 4 of their 8 operations (matching the identical scope decision for chunk 22-ch06b-mconvexfunctions's Theorem 6.13); Theorem 6.61 omits the dual-integral refinement clauses for the M[R→R|Z]/ M[Z→Z] sub-classes. Both reductions are documented, not trivializing omissions — see Difficulty above and HARD.md/MODERATION_NOTES.md. No numeric constants are hard-coded anywhere in this mission. This mission's definitions are redeclared from chunks 06-mconvex-functions-i, 21-ch05b-lconvexsets (for the distance-function/admissible-potential vocabulary), 22-ch06b-mconvexfunctions, and 23-ch06c-mconvexfunctions 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 and Proposition 6.60 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 (the polyhedral M-convex function theory this mission's real-variable results are drawn from).

Motivation

Every characterization of M-convexity so far in this series — the exchange axiom, the optimality criterion, convex extensibility, the directional-derivative/subdifferential correspondence — has been an equivalence with M-convexity itself: a function either is M-convex or it is not. Section 6.14 asks a different question: what happens when the exchange axiom's defining inequality is relaxed to only the sign patterns it actually forces? The answer is a hierarchy of "quasi M-convex" conditions — weaker than M-convexity, strong enough to keep the optimality criterion and the proximity/minimizer-cut theorems intact — and, at the top of that hierarchy, a genuinely new characterization of M-convexity itself: a function is M-convex if and only if every one of its linear perturbations is quasi M-convex in the weakest sense. This mission formalizes that entire hierarchy and its capstone, plus two further characterizations of polyhedral M-convexity (via directional derivatives, subdifferentials, and weighted-minimizer polyhedra) that complete the real-variable theory chunk 24-ch06d-mconvexfunctions began.

Setting

Fix a finite ground set $V$ and $f : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$. Write $\Delta f(z;v,u) = f(z + \chi_v - \chi_u) - f(z)$. Relaxing the exchange axiom's inequality $\Delta f(x;v,u) + \Delta f(y;u,v) \le 0$ to the sign patterns it forces gives quasi M-convexity (QM) and semistrict quasi M-convexity (SSQM), and requiring only some pair $(u,v)$ rather than every $u$ gives their weaker variants (QMw), (SSQMw); the minimization-only variants (SSQM$\ne$), (SSQM$\ne_w$) replace "$x,y \in \operatorname{dom} f$" with "$f(x) \ne f(y)$". The set-level analogue (Q-EXC)/(Q-EXCw) relaxes the M-convex-set exchange axiom the same way. For $\alpha \in \mathbb R$, the level set $L(f,\alpha) = \{x \in \mathbb Z^V : f(x) \le \alpha\}$. A polyhedral convex function $f : \mathbb R^V \to \mathbb R \cup \{+\infty\}$ is (real-variable) M-convex, $f \in M[\mathbb R \to \mathbb R]$, if it satisfies the real exchange axiom (M-EXC[R]) from chunk 24-ch06d-mconvexfunctions; $L_0[\mathbb R]$ denotes polyhedra realized as $D(\gamma)$ for a triangle-inequality distance function $\gamma$, and $M_0[\mathbb R]$ denotes real M-convex polyhedral cones.

Formalization targets

Goal: the quasi M-convexity hierarchy (Theorem 6.68)

For $f : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$: (1) the implication diagram (M-EXC[Z]) $\Rightarrow$ (SSQM) $\Rightarrow$ (QM), (M-EXCw[Z]) $\Rightarrow$ (SSQMw) $\Rightarrow$ (QMw), (M-EXC[Z]) $\Leftrightarrow$ (M-EXCw[Z]), (SSQM) $\Rightarrow$ (SSQMw), (QM) $\Rightarrow$ (QMw); (2) $f$ satisfies (M-EXC[Z]) if and only if $f[p]$ satisfies (QMw) for every $p \in \mathbb R^V$. This theorem is not named in the chunk's own extraction table — the automated extractor, which requires a result's label to start a text line, misses it because it opens mid-paragraph directly after an ASCII-rendered implication diagram — but it is the capstone of the section: its own proof is "combining Theorems 6.72 and 6.74," both formalized here as milestones, and part (2) answers exactly the question a reader of this stretch would ask: what does the entire apparatus of quasi M-convexity ultimately say about M-convexity itself?

Supporting structural targets

Fourteen further results build the hierarchy and complete the real-variable theory. Proposition 6.62 checks the two halves of chunk 24-ch06d-mconvexfunctions's Theorem 6.61 agree at integer points. Theorems 6.63–6.64 add two more characterizations of polyhedral M-convexity — via positively homogeneous directional derivatives and $L_0[\mathbb R]$-valued subdifferentials, and via $M_0[\mathbb R]$/$M_0[\mathbb Z|\mathbb R]$-valued weighted-minimizer polyhedra — to the convex-extensibility characterization chunk 23-ch06c-mconvexfunctions proved. Theorem 6.67 gives two equivalent reformulations of (QMw) as pointwise inequalities; Propositions 6.69–6.70 and Theorems 6.72–6.74 build the level-set/perturbation machinery the goal needs. Theorem 6.75 is the (SSQM$\ne_w$) analogue of Theorem 6.67. Theorem 6.76 is the quasi M-optimality criterion (optimality still characterized by local non-improvement, under only the weak quasi-convexity hypotheses). Theorems 6.77–6.79 show the M-minimizer-cut and M-proximity theorems (from missions 06-mconvex-functions-i and 23-ch06c-mconvexfunctions) hold verbatim under the strictly weaker (SSQM$\ne$) hypothesis.

Significance

The hierarchy's practical payoff is immediate: Theorems 6.77–6.79 mean the algorithms of chapter 10 that rely on minimizer cuts and proximity bounds do not actually need the full exchange axiom to run correctly on nonlinearly rescaled M-convex functions (Example 6.66 shows any nondecreasing scaling $\phi \circ f$ of an M-convex $f$ is quasi M-convex, yet nonlinear scalings are common in practice and destroy M-convexity itself). The goal, Theorem 6.68, is significant independently: it says the exchange axiom — a condition that looks irreducibly combinatorial, quantifying over pairs of points and directions — is equivalent to a purely ordinal, perturbation-based condition (every linear tilt of $f$ has no strict local improvement that a level set can't witness), giving a genuinely different lens on why M-convexity is the right discrete analogue of convexity. Theorems 6.63–6.64 close out chunk 24-ch06d-mconvexfunctions's program of characterizing polyhedral M-convexity in every classical convex-analytic vocabulary at once (directional derivatives, subdifferentials, weighted minimizers), completing the bridge to Chapter 8's duality theory that chunk builds toward.

None of these results are open — they are Murota's account of how far the exchange axiom's defining inequality can be relaxed while keeping optimization theory intact. What this mission contributes is a faithful, machine-checked formal statement of each, including four theorems (6.68, 6.76, 6.77, 6.78) the platform's own automated extractor missed entirely, extending the shared Lean vocabulary (DeltaF, QMw, LevelSet) the Discrete Convex Analysis series builds on; no comparable formalization exists on the platform (see Formalization scope).

Difficulty

The naive approach to the goal would try to prove the implication diagram's six arrows and the perturbation equivalence as six independent facts; the book's own proof of part (2) instead derives it in one step from Theorems 6.72 and 6.74 — themselves nontrivial (Theorem 6.74's proof strengthens the local exchange axiom equivalence (Theorem 6.4) to hold whenever the domain merely satisfies (Q-EXCw), then runs a bipartite-matching argument on a 4-point neighborhood to verify the resulting local condition). The difficulty is genuinely upstream of the goal's own statement: everything the goal needs is already proved by the time Theorem 6.68 is reached, so the formalization work is in stating the sixteen distinct axioms and their level-set reformulations precisely enough that "combining 6.72 and 6.74" is literally how a Lean proof would proceed.

Formalization scope

Ground-set elements are a Fintype V with DecidableEq; integer-domain functions are (V→ℤ)→WithTop ℝ, real-domain ones (V→ℝ)→WithTop ℝ. All fifteen numbered results — including the four (6.68, 6.76, 6.77, 6.78) the automated extractor missed because their labels open mid-paragraph — are placed as milestone or goal, and every clause of every one is stated in full; no partial-coverage scope reduction was needed in this chunk (contrast chunks 22-ch06b-mconvexfunctions/24-ch06d-mconvexfunctions, which restated 4-of-8-part operations theorems). Two formalization choices are recorded in HARD.md/MODERATION_NOTES.md: "$\inf f[-p] > -\infty$" is replaced by the equivalent (ArgMinOn ...).Nonempty hypothesis (WithTop ℝ has no $-\infty$ element), matching mission 23-ch06c-mconvexfunctions's identical substitution; and $M_0[\mathbb R]$/$M_0[\mathbb Z|\mathbb R]$ are realized via the book's own indicator-function device rather than a freestanding cone axiom. This mission's definitions are redeclared from chunks 06-mconvex-functions-i, 22–24-ch06*-mconvexfunctions rather than imported, since sibling drafts in this series cannot yet reference one another. Contributions completing any of the fifteen sorrys are welcome; the goal and Theorem 6.74 carry the most independent proof content.

Selected references

• K. Murota, Discrete Convex Analysis, SIAM, 2003. DOI: 10.1137/1.9780898718508.
• M. Avriel, W. E. Diewert, S. Schaible, and I. Zang, Generalized Concavity, Plenum Press, 1988 (the continuous quasi-convexity theory this chapter's discrete analogue generalizes).

Motivation

Submodularity — the diminishing-returns property $g(p) + g(q) \ge g(p \vee q) + g(p \wedge q)$ on a lattice — is one of the most useful structural hypotheses in combinatorial optimization, underlying efficient algorithms for network flows, matroid theory, and set-function minimization. Chapter 7 studies L-convex functions: functions on the integer lattice $\mathbb Z^V$ that are submodular and linear along the all-ones direction. This is the "dual" notion, under the conjugacy developed later in the book, to chunk 06's M-convex functions, and it inherits the same strong minimization theory — a purely local optimality criterion and a proximity theorem with an explicit distance bound — while additionally supporting a genuinely new characterization with no M-convex counterpart: discrete midpoint convexity, the direct lattice analogue of the classical real-valued midpoint convexity condition. This mission formalizes the chapter's definitional theorem, its midpoint-convexity characterization, the L-optimality criterion, and the L-proximity theorem itself.

Setting

Let $V$ be a finite ground set. A function $g : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ with nonempty effective domain is an L-convex function if it satisfies (SBF[Z]): $g(p) + g(q) \ge g(p \vee q) + g(p \wedge q)$ for all $p, q$ ($\vee, \wedge$ componentwise max/min), and (TRF[Z]): there is $r \in \mathbb R$ with $g(p + \mathbf 1) = g(p) + r$ for all $p$, where $\mathbf 1$ is the all-ones vector. An L$^\natural$-convex function is one whose lift to the extended ground set $\{0\} \cup V$ is L-convex; equivalently (Theorem 7.1), $g$ satisfies the translation-submodularity axiom (SBF$^\natural$[Z]): $g(p) + g(q) \ge g((p - \alpha\mathbf 1) \vee q) + g(p \wedge (q + \alpha\mathbf 1))$ for all $p, q$ and all nonnegative integers $\alpha$. Discrete midpoint convexity asks $g(p) + g(q) \ge g(\lceil (p+q)/2 \rceil) + g(\lfloor (p+q)/2 \rfloor)$ componentwise. For $\alpha$ a positive integer, a point satisfies scaled local optimality if $g(p_\alpha) \le g(p_\alpha \pm \alpha \chi_Y)$ for every $Y \subseteq V$.

Formalization targets

Goal: Theorem 7.18 (the L-proximity theorem)

Assume $\alpha$ is a positive integer and $n = |V|$. (1) If $g$ is L-convex with $g(p) = g(p+\mathbf 1)$ for all $p$, and $p_\alpha \in \operatorname{dom} g$ 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

$$p_\alpha \le p^* \le p_\alpha + (n-1)(\alpha-1)\mathbf 1.$$

(2) If $g$ is L$^\natural$-convex and $p_\alpha$ satisfies the two-sided version, then there is $p^*$ with $p_\alpha - n(\alpha-1)\mathbf 1 \le p^* \le p_\alpha + n(\alpha-1)\mathbf 1$. The bound is a genuine vector (lattice-order) inequality, not an $\ell^\infty$-norm bound — the form later chapters' applications need.

Milestones: Theorems 7.1, 7.7, 7.14

Theorem 7.1: L$^\natural$-convexity (defined via the lift) is equivalent to the direct translation-submodularity axiom. Theorem 7.7: this same class is also characterized by discrete midpoint convexity — a three-way equivalence with the approach property (L$^\natural$-APR[Z]) as a bridge — giving L-convexity a genuinely different, more geometric face than anything available on the M-convex side. Theorem 7.14 (the L-optimality criterion): global optimality reduces to a purely local check against the sign-pattern neighbors $p \pm \chi_Y$, mirroring chunk 06's Theorem 6.26 but with the plain L-convex case additionally requiring the periodicity condition $g(p) = g(p+\mathbf 1)$.

Significance

The result itself. Discrete midpoint convexity (Theorem 7.7) is philosophically important: it shows the lattice-submodularity definition of L-convexity is not an arbitrary discretization choice but coincides exactly with the most direct discrete analogue of ordinary midpoint convexity, the classical characterization of convex functions via $f((p+q)/2) \le (f(p)+f(q))/2$. The L-optimality criterion and L-proximity theorem give L-convex minimization the same algorithmic footing as M-convex minimization (chunk 06): scaling algorithms for L-convex objectives — which arise naturally from network flow and submodular-function duality — inherit a provable, dimension-and-scale-explicit distance guarantee between a coarse-scale local optimum and the true minimizer.

Formalizing it. No matching item exists on the platform for L-convex functions, discrete midpoint convexity, or the L-optimality/proximity theorems. This mission gives the first formal statement of these results, completing (alongside chunk 06's M-convex-function results) both halves of the exchange-axiom-based theory that chapter 8's conjugacy duality later unifies.

Difficulty

A natural shortcut, given the structural parallel to chunk 06, is to assume the L-proximity theorem's proof is a mechanical relabeling of the M-proximity theorem's proof. It is not: the M-convex proof (chunk 06) crucially uses the exchange axiom's additive four-term inequality to build a chain of strictly improving points, whereas the L-convex proof instead exploits (TRF[Z])'s periodicity directly — it reduces to the case $p_\alpha = 0$ using translation invariance, then constructs a minimal (with respect to the lattice order) point among all sufficiently good solutions and shows this minimality, combined with submodularity (SBF[Z]), forces the componentwise bound. The vector (rather than norm) form of the conclusion is not cosmetic: it is exactly what this lattice-order argument naturally produces, and is the form needed by later chapters' applications.

Formalization scope

The ground set $V$ is a Fintype with DecidableEq; $g : \mathbb Z^V \to \mathbb R \cup \{+\infty\}$ is (V → ℤ) → WithTop ℝ. Unlike chunk 06's M-convex axiom, (SBF[Z]), (TRF[Z]), and (SBF$^\natural$[Z]) are stated for all of $\mathbb Z^V$, not restricted to $\operatorname{dom} g$, so no explicit import of chunk 05's L-convex-set vocabulary was needed for dom g's structure (unlike the corresponding note in chunk 06's BRIEF.md, which flagged the same concern for dom f). L$^\natural$-convexity is represented via an explicit lift to Option V, matching the book's own primary definition, with the direct axiom (SBF$^\natural$[Z]) kept as a separate object related to it by Theorem 7.1.

A trivializing formalization of the goal would convert its componentwise vector bound into an $\ell^\infty$-norm bound (losing the direction-of-approach information the vector form carries) or drop Part (1)'s periodicity hypothesis $g(p) = g(p+\mathbf 1)$; neither is done here. Propositions establishing dom g as an L-convex set, the L/L$^\natural$ relationship (Theorem 7.3), the submodular-set-function embedding (Proposition 7.4), and several structural closure properties are cut from this mission's scope (see MODERATION_NOTES.md) but are natural targets for a follow-on mission or for chunk 09, which builds directly on this chunk's exchange-axiom vocabulary, mirroring chunks 06→07.

Selected references

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