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This result is braic numbers. Most of these notions were needed for the formal- known as the fundamental theorem of symmetric polynomials. This ized proof of Feit and Thompson’s odd-order theorem [9]. However, theorem is instrumental in our formal proof. multivariate polynomials were not described in this context and our Before stating and giving a precise proof to this theorem, we work is filling this gap. need to introduce a last important notion, the notion of weight. For In this paper, we first give an overview of the proofs, expressed a given monomial M, the weight w(M) of this monomial is the in mathematical terms. We then illustrate the techniques used to weighted sum of the degrees in each variable, weighted by the rank make the proofs verifiable by the Coq proof assistant [6]. of this variable. The formal development is available at the following address: n n k1 k http://marelledocsgit.gforge.inria.fr. The multinomial w(X1 · · · Xn )= i × ki. 1 library is available as a separate component on github . i=1 X Intuitively, the weight of a monomial makes it possible to compute 2. Mathematical context the total degree of the result when applying this monomial to the In this section, we first give more mathematical background sequence of symmetric polynomials. 2 about multivariate polynomials and we then concentrate on the For instance, w(X1 X2X3) = 1 × 2 + 2 × 1 + 3 × 1 = 7 and 2 proofs of transcendence, showing that there is central lemma that applying the monomial X1 X2X3 to the symmetric polynomials can be specialized for e and π. leads to this result: 2 2 2.1 Multivariate and symmetric polynomials σ3,1σ3,2σ3,3 =(X + Y + Z) × (XY + XZ + Y Z) × XY Z Multivariate polynomials are polynomial expressions where sev- This polynomial indeed has total degree 7. eral indeterminates appear. Common mathematical notations rely on writing the indeterminates as X, Y , and Z when there are no LEMMA 1. For every symmetric polynomial P in n variables there exists a polynomial T in n variables so that more than three indeterminates and as Xi where i ranges between 1 and n in the general case. The set of multivariate polynomials for a P [X1; · · · ; Xn]= T [σn.1[X1; · · · ; Xn]; · · · ; σn,n[X1; · · · ; Xn]]. given set of indeterminates and coefficients in a given ring also has the structure of a ring. We define monomials as expressions of the Moreover, the weight of polynomial T is smaller than or equal to k1 kn the total degree of polynomial P . form X1 · · · Xn where ki ∈ N is the degree in variable Xi and the sum k1 + · · · + kn is called the total degree. For a polynomial, Proof. This proof is performed by well-founded induction follow- the total degree is the maximum total degree of its monomials. We ing the lexicographic order of the leading coefficient. can sort the monomials of a multivariate polynomial lexicographi- Because P is symmetric, its leading monomial necessarily has k1 kn cally, so that the first monomial is the one with the highest degree the shape X1 · · · Xn with k1 ≥ k2 ··· ≥ kn. This monomial in X1, and among the monomials of highest degree in X1, the one naturally has a total degree which is smaller than or equal to the with the highest degree in X2, etc. In what follows, we will call this total degree of P . This monomial is also the leading monomial of first monomial in lexicographical order the leading monomial. the expression: The polynomials that are unchanged when permuting the vari- k1−k2 k2−k3 k σ σ · · · σ n ables are called symmetric polynomials. For instance, the following n,1 n,2 n,n k1−k2 k2−k3 kn polynomial is symmetric. Let us note temporarily M = X1 X2 · · · Xn (M is 3 3 3 3 3 3 an n-variable polynomial), then the above expression can also be X Y + X Z + XY + XZ + Y Z + Z Y written as follows: Among the symmetric polynomials, the polynomials with only M[σn,1; · · · ; σn,n] the permutations of a single monomial, with degree at most one in if c is the coefficient of the leading monomial in P , we can consider each variable are called elementary symmetric polynomials. For n- the polynomial variable polynomials, there are exactly n+1 elementary symmetric polynomials, one for each total degree between 0 and n. In this pa- Q = P − c × M[σn,1; · · · ; σn,n] per, we shall write σ for the elementary symmetric polynomial n,k If Q is constant, then the required polynomial is T = cM − Q of n variables with degree k. For three variables, the non-trivial and the weight of P is the weight of cM. If Q is non constant, it is elementary symmetric polynomials are: symmetric (because it is the subtraction of a symmetric polynomial σ3,1 = X + Y + Z σ3,2 = XY + Y Z + XZ σ3,3 = XY Z. from a symmetric polynomial) and its leading coefficient is lexico- graphically less than the leading coefficient of P . We can deduce The elementary symmetric polynomials are especially interesting ′ in proofs of transcendence because of Vieta’s formula. When work- by induction hypothesis that there exists a polynomial T such that ′ ing in an algebraically closed field, considering a polynomial P Q[X1; · · · ; Xn]= T [σn,1[X1; · · · ; Xn]; · · · ; σn,n[X1; · · · ; Xn]]. whose roots are αi (i ∈ {1 ...n}), noting c the leading coefficient, The weight of T ′ is less than or equal to the total degree of Q, and noting α the vector whose components are the roots αi, this polynomial can be written as which is less than or equal to the total degree of P . The required polynomial is n n ′ n−i i T = T + c × M P = c (X − αi)= c( (−1) σn,n−i(α)X ) i=1 i=0 The weight of this polynomial is less than or equal to the maximum Y X ′ Thus, the elementary symmetric polynomials applied to the roots of the weights of T and M, which is enough to guarantee the of P give the coefficients of P , after multiplication by the leading weight property. coefficient and a sign factor. 2.2 Common lemmas for proofs of transcendence Elementary symmetric polynomials make it possible to gener- ate all symmetric polynomials using multiplication, addition and The two transcendence proofs are quite similar, so to factorize the formalization work, we have proved a generalized lemma, that can 1 https://github.com/strub/multinomials-ssr then be applied for both cases (e and π). The proof revolves around an analytic part and an algebraic part. Thanks to Lemma 3, the coefficients of Gp will inherit from T the The analytic part relies on the following integral: property that the coefficients are integers. 1 −αx IP (α)= αe P (αx)dx 2.2.3 Main lemma Z0 We can now state our main lemma, which relies on the concepts of We also consider an operation on polynomials, which adds up T , Fp, and Gp, which we just introduced. all the derivatives of this polynomial: degP LEMMA 4. If T has integral coefficients, then for every sequence ∗ (i) γi of integers and any k ∈ Z , if the following equality holds Pd = P , i=0 n−1 X αi k + γie = 0 where P (i) denotes the i-th derivative of P and degP is the degree i=0 of P . X then for p prime and large enough, 2.2.1 Preliminary results on integrals n−1 np LEMMA 2. For every polynomial P and every complex number α, c γiGp(αi) 6∈ Z the following identity holds: i=0 X −α IP (α)= Pd(0) − e Pd(α) (1) Proof. Let us assume that T has integral coefficients and that Proof. Note that P (0) = P , P (i) = 0 for degP < i, and eα×0 = 1. n−1 αi k + (γie ) = 0 (3) −αx ′ (Pd(αx)e ) = i=0 −αx −αx ′ X αi = −αe Pd(αx)+ αe Pd(αx) Consider the sum −γie IFp (αi) and re-use the result from = αe−αx(P ′ (αx) − P (αx)) (1). d d P degP +1 degP n−1 n−1 −αx (i) (i) αi αi −αi = αe P (αx) − P (αx) −γie IFp (αi) = −γie (Fpd(0) − e Fpd(αi)) ! i=0 i=0 i=1 i=0 X X −αx (degPX +1) (0)X n−1 n−1 = αe (P (αx) − P (αx)) αi −αx = −γie Fpd(0) + γiFpd(αi) = −αe P (αx) i=0 i=0 X X αi Hence we have: by (3) we have −γie = k, and therefore we get the following 1 result: −αx −αx 1 −α αe P (αx)dx = [−Pd(αx)e ]0 = Pd(0) − e Pd(α) n−1 P n−1 0 Z αi −γie IFp (αi)= kFpd(0) + γiFpd(αi) (4) 2.2.2 Main construction i=0 i=0 X X In what follows, we work with a sequence of n non-zero complex We will now consider a new equation, where both sides are ∗ np numbers αi ∈ C and a non-zero integer c. We first consider the multiplied by c . following polynomial: n−1 n−1 np αi np np T = c(X − α0)(X − α1) · · · (X − αn−1) (2) c −γie IFp (αi)= c kFpd(0)+c γiFpd(αi) (5) i=0 i=0 We then introduce, for an arbitrary p, (which we shall later choose X X ′ to be very large and prime), the polynomial Fp : We will call Ep the left hand side and Ep the right hand side. We will now show that, for p large enough, the |Ep| is less than (p−1)!. p−1 p Let be an upper bound of the finite sequence . The follow- Fp = X T . A |αi| ing inequalities hold: And as in formula (1), we consider IFp and the sum of deriva- 1 tives of Fp, which we shall note Fpd. We also consider an interme- −αix d diate lemma concerning solely the derivatives of order larger than |IFp(αi)| = αie Fp(αix) x Z0 or equal to p. 1 −αix ≤ αie Fp(αix) dx LEMMA 3. When the coefficients of T are integers, the coefficients Z0 of 1 A degFp ≤ Ae |Fp(αix)|dx (i) 0 Fp Z 1 i=p A p−1 p X ≤ Ae A |T (αix)| dx (6) are divisible by p! Z0 Proof. For every polynomial Q with integral coefficients, all coeffi- by definition of Fp. (m) Let M be an upper bound of |T (αix)| on [0, 1]. Then we get cients of Q are divisible by m! So in particular, all coefficients p A p (i) |IFp (αi)|≤ A e M and there exist K and L (independent of p) of Fp are divisible by i! and if p ≤ i, they are divisible by p! np αi p−1 such that |Ep = c −γie IFp (αi)|≤ KL . Divisibility can then be inherited by the sum n We know that limn x /n! = 0, so when p is large enough We introduce a family of polynomials Gp defined as follows: →∞ |Ep| < (p − 1)! P degFp (i) i=p Fp From now on, we want to show that for p prime and large Gp = ′ enough, |Ep| is larger than or equal to (p − 1)! P p! The numbers αi are roots of Fp with multiplicity p, and there- degree nπ with rational coefficients so that Bπ(iπ) = 0. Without (m) fore F (αi) = 0 for every m p−1 degFp Bπ = (X − βi). (j) (j) i=0 Fpd(αi) = Fp (αi)+ Fp (αi) Y j=0 j=p where none of the βi is zero and iπ = β0. X X ′ degFp To build the sequence α, we first build a sequence α and ′ (j) then remove the zero elements. The sequence α is built from the = 0+ F (αi) p sequence β by taking all the sums of non-empty subsets of the j=p ′ X sequence β. Thus α contains β0, β1, · · ·, β0 + β1, · · · We should ′ Similarly for the value of Fpd in 0, we know that 0 is a root of note that iπ appears in the sequences α and α. (m) iπ multiplicity p−1 of Fp, and thus Fp (0) = 0 for every m n−1 n−1 α ′ np αi k + e i = 0 Ep = c −γie IFp (αi) i=0 i=0 X X n−1 Taking γi = 1 for every i we have: np np = c kFpd(0) + c γiFpd(αi) n−1 i=0 αi np X p k + γie = 0. = c × k × (p − 1)! × T (0) + p!Gp(0)) i=0 n X np We can now instantiate the main lemma in the following fashion: + p!c γiGp(αi) i=0 1. α is a sequence of non-zero complex numbers, X np When c γiGp(αi) is an integer, the right hand side is 2. γ0 = · · · = γn = 1, the γi are integers, k is a positive integer. obviously divisible by (p−1)!. But when p is a large enough prime 3. The polynomial (larger than c, Pk, and T (0)) only the last two terms can be divided ′ by p! This expression is an integer divisible by (p − 1)! but not by T =(X − α0)(X − α1) · · · (X − αn−1) p!. the absolute value must be larger than or equal to (p − 1)! is such that ′ ′ ′ k−1 ′ 2.3 Case of e (X − α0)(X − α1) · · · (X − α2nπ −2)= X T k−1 ′ If e is an algebraic number, let Ae be a polynomial with integer By Vieta’s formula, the coefficients of X T are obtained ′ coefficients such that Ae(e) = 0. Let us note ai the coefficients of by applying elementary symmetric polynomials on α , but the ′ Ae components of α are stable modulo permutation of the vari- ne ne ables βi and so these coefficients are also obtained by applying i Ae = aiXAe(e)= aie symmetric polynomials on β. Using the fundamental theorem i=0 i=0 of symmetric polynomials for a first time, these coefficients are X X Without loss of generality, we can assume that 0 is not a root of obtained by applying multivariate polynomials with integer co- Ae (i.e. a0 6= 0). We instantiate the main lemma in the following efficients to the elementary symmetric polynomials of β, which ′ fashion: are the coefficients of Bπ . Thus, the coefficients of T are all rational and we can exhibit a constant c so that 1. α0 = 1, α1 = 2 · · · αne−1 = ne. The alphas are non-zero ′ complex numbers, T = cT = c(X − α0)(X − α1) · · · (X − αn−1) 2. γ0 = a1 · · · γne−1 = ane . The numbers γi are integers; has integral coefficients. k = a0 is a non-zero integer. 4. Because γi = 1 for every i, the following formula is actually 3. the polynomial T = (x − 1) · · · (x − ne) trivially has integral symmetric in the αi coefficients, we take c = 1. n−1 np np 4. The expression c γiGp(αi) trivially is an integer because c γiGp(αi) the numbers c, γi, and αi are integers and the polynomial Gp i=0 P X has integer coefficients. Since we know that Gp is a polynomial with integral coeffi- cients, and using the fundamental theorem of symmetric poly- 2.4 Case of π nomials a second time, we can deduce that this expression can For π, we work with iπ instead of π because it is equivalent to be computed by applying a polynomial with integer coefficients prove that one is algebraic or the other. Let Bπ be a polynomial of on the coefficients of c (X − αi), which also are integers. Q 3. Formalization properties of the lower rungs (and hence the theorems can be writ- Our description of the formal development is organized into five ten once and for all). The Mathematical Components library then parts: we describe the working context provided by the Mathemati- ensures that the abstract properties can be recognized and used for cal Components library, the development of multivariate polynomi- a wide class of concrete instances. Thus, integers can be recognized als, the connection to the Coquelicot library, the proof of the main as a ring structure. Then many theorems are expressed in terms of lemma, and the specialization to π. morphisms. Actually, the ring of integers is even recognized as the initial ring, so that the notation %:~R is defined as a morphism from 3.1 The common infra-structure: Mathematical Components the type of integers to any ring (for instance to the ring of complex The mathematical Components library draws from a tradition that numbers, or to the ring of polynomials with integer coefficients). was started with the formal proof of the four-color theorem in 2004 The Mathematical Components library defines the notion of [8] and continued for the proof of the Feit-Thompson theorem [9]. algebraic number, by considering two fields and a morphism from the first to the second field. A number a in F is algebraic over E This library is built on top of the Coq proof assistant, an inter- i active theorem proving tool that relies on type theory: most the- and φ, if there is a polynomial with coefficients in E, i piX i orem statements are types in a functional programming language such that the equality i φ(pi)a = 0 holds. So the notion of P and theorems are functions from one type to the other. One of the algebraic number is not intrinsic, it relies on two fields and a key functionalities of Coq that we use intensively is the mechanism morphism. In our experiment,P the common meaning of algebraic of canonical structures. numbers correspond to numbers that are algebraic over Q in C, When compared to other libraries for the Coq system, the Math- using the natural morphism of rational numbers into the field of ematical Components library also differs in its use of small-scale re- complex numbers as the morphism. So we write algebraicOver flection, where most decidable logical expressions are represented ratr x, where ratr is the morphism from rational numbers to any by boolean formulas. When compared with libraries developed for field. other interactive proof systems, we should note that classical logic- Our work on multivariate polynomials integrates in this context. based interactive theorem provers, like Isabelle [17] or HOL-Light These polynomials are described by choosing a data-structure and [11] do represent logical statements with boolean formulas, but on then providing the operations of addition, opponent, multiplication, the other hand, they have practically no tool to distinguish between that make this type a ring, together with the operations that are a constructive proof and a non-constructive one. expected from a polynomial: decomposing into the monomial basis For mathematical practice, the Mathematical Components li- (and thus observing the coefficients), evaluating at a multiplet of brary provides an extensive collection of notations, which are com- values, finding the leading monomial, etc. We show that many patible with the level of abstraction. For instance a + b represents operations actually are morphisms. For instance, the function for the addition of two elements in any ring. So that we will write evaluating at a given multiplet is a ring morphism from the ring of the same formula a + b, whether a and b represent one-variable multivariate polynomials to the ring of coefficients. or multivariate polynomials, even though these are two different types. In the same vein, the Mathematical Components library also 3.2 Formalizing Multivariate Polynomials provides support for big iterated operations such as In this section we describe the construction of multivariate polyno- n−1 mials. In the univariate case, SSReflect concretely represents poly- αi γie nomials as a sequence of coefficients. For instance, the univariate i i=0 polynomial N αiX , where {αi}i is a null set, is represented X i∈ by the sequence [α0,...,αk], with αk 6= 0 being the last non-null which represent the repeated iteration of the addition operator of P any ring structure. In our formal development, this formula is coefficient when ordering the αis by their indices. This representation can be generalized to a finite number of in- written as: 1 n determinates, using an enumeration {(ǫi ,...,ǫi )}i∈N of the count- n \sum_(0 <= i < n) (gamma i)%:~R ∗ Cexp (alpha i) able set N . In this case, the multivariate polynomial The tag %:~R is needed here to express that the numbers γi which 1 n are integers should be injected into the type of complex numbers. i1 in ǫ ǫ 1 i i αi1,...,in X · · · Xn = α n X · · · Xn This is a first illustration that in terms of notations, not everything 1 ǫi ,...,ǫi 1 i1,...,in i∈N is seamless. Another illustration is that we need to accommodate X X βi different notions of exponentiation. In rings, exponentiation is only defined when the exponent is a natural number, by iterating the would be represented by the sequence [|β0,...,β{z } k], where βk is multiplication operation, in plain fields (like Q) it is also defined again the last non null coefficient when taken in the ordering im- for negative integral exponents, by simply taking multiplicative plied by the enumeration of Nn. However, although this represen- inverses, but in complete fields like R or C, it is defined as the tation is quite canonical and effective in many respects in the uni- result of completely different processes, by taking the limit of a variate case (e.g. the degree and the coefficients of a polynomial power series (for which a convergence proof is actually provided). are easily definable from its sequence of coefficients), it seems less For this reason, we use a different notation for each of these four effective in the multivariate case. For instance, the link between cases and it is sometimes tricky to reconcile two formulas that use the βis and αi1,...,in s is now non trivial and depends on the enu- different notations. meration of Nn, which may lead to overly complicated definitions. The Mathematical Components library is designed to alleviate Moreover, this representation does not lift canonically to the case of the difficulties arising from the distinction between various kinds of an infinite number of indeterminates, and would require an ad-hoc numbers. The first step is to describe the various levels of proper- construction, for example by taking a structure limit. ties that a set and operations on this set can satisfy. At the first level, Other options are available to represent multivariate polynomi- the notion of monoid (only one binary law, which is associative and als. For example, one could iterate the univariate case, representing has a neutral element), then groups, then commutative groups, then k[X1 · · · Xn] as k[X1] · · · [Xn]. Yet again, this representation does rings, integral domains, fields, up to algebraically closed and com- not lift canonically to an infinite number of variables, and equip- plete fields. Each structure at a higher level in the ladder satisfies the ping this representation with the canonical structures of the SSRe- flect algebra hierarchy requires some contortions. Indeed, one can && (all [pred zx | zx.2 != 0] g). not simply define the type of multivariate polynomials as: Record prefreeg : Type := mkPrefreeg { Fixpoint mpoly (R : ringType) (n : nat) := seq_of_prefreeg : seq (T ∗ G); if n is p.+1 then {poly (mpoly R p)} else R. _ : reduced seq_of_prefreeg }. as {poly (mpoly R p)} requires (mpoly R p) to be already The intent of prefreeg is to give a unique representation of equipped with a ringType structure at the time of definition. A a free-group expression, up to the order of the coefficients. For proper definition is: instance, if g = k1 (x1)+ · · · + kn (xn) (with all the xis pair- wise distinct and all the kis in G), then the reduced sequence Fixpoint mpoly (R : ringType) (n : nat) : ringType := s = [:: (x_1, k_1), ..., (x_n, k_n)], or any sequence if n is p.+1 then equal to s up to a permutation, is a valid representation of g. The [ringType of {poly (mpoly R p)}] type freeg of free abelian groups is then obtained by taking the else R. quotient of prefreeg by the perm_eq equivalence relation. Such However, this construction triggers a complexity explosion in a construction requires the axiom of choice over seq (T ∗ G), the Coq unification algorithm, and we never succeeded in equip- which amounts to both T and G equipped with the choiceType structure, as provided in the Mathematical Components. ping (mpoly (R : idomainType) p) with an integral domain structure. We now show that our representation is faithful. Given a se- Last, the SSReflect trunk archive comes with an embryo theory quence (s : seq (T ∗ G)) = [(ki,xi)]i (not necessarily re- for multivariate polynomials [9] that relies on the quotient of a free duced), we define precoeff x s as: algebra (built from the indeterminates, the ring constants and the uninterpreted operators + and ∗) by an interpretation relation, using Definition precoeff x s : G := univariate polynomials as the domain of interpretation. This defi- \sum_(k <− s | k.1 == x) k.2. nition directly handles the case of an infinite number of indetermi- nates and is appealing as it allows the definition of basic functions The function precoeff computes the coefficient of x in the via the manipulation of ring expressions — once these manipula- formal sum i ki (xi). (Note that if the sequence s is reduced, this tions are proved to be compatible with the quotient relation. How- amounts to looking up in s, defaulting to 0 is x cannot be found). P ever, the quotient relation is non-trivial and we expect these proofs We then prove that two freeg inhabitants are equal if and only if of compatibility to be harder that necessary. they agree on their coefficients: For our formalization, we take the angle of representing a mul- Lemma freegP (g1 g2 : freeg T G) : reflect tivariate polynomial literally as a monoid ring, i.e. as a formal sum (forall x, precoeff x g1 = precoeff x g2) (g1 == g2). 1 n ki ki of the form αi(X1 · · · Xn ). For that purpose, we develop an On the other hand, we show that we can, from any association independent structure for free abelian groups. We then obtain a sequence, not necessarily reduced, construct a reduced one s.t. P structure for k[X1,...,Xn] by instantiating the one of free abelian precoeff agrees on both. groups, taking the coefficients from the multivariate polynomials base ring k and using a free commutative monoid over {1,...,n} Definition reduce (s : seq (T ∗ G)) := ... as the monoid of generators. This gives us the basic structure for Lemma rdce_reduce s : reduced (reduce s). multivariate polynomials in n variables. We then develop an ex- Lemma rdce_eq s x : precoeff x s = precoeff x (reduce s). tensive library, including polynomials derivation, evaluation, mor- phisms, and prove that our representation is isomorphic (as a ring) From there, we equip the type freeg with a group structure, to the iterated representation. We then use our development to prove the representative of 0 being the empty list, and the representative the fundamental lemma of symmetric polynomials, proving that the of the sum being the reduced concatenation of the representatives ring of symmetric polynomials in n variables is isomorphic to the of the operands. We also define all the usual notions related to free ring of polynomials in the elementary symmetric polynomials. groups (domain, coefficient, degree, ...), most of them being defined using the following group morphism: 3.2.1 Free Abelian Groups Definition fglift The core structure of our multivariate polynomials library is the (M : lmodType R) (f : T −> M) (g : freeg T G) : M := one of free abelian groups. The latter is based on the quotient \sum_(k <− repr g) k.1 ∗: (f k.2). libraries of SSReflect. Assume T to be a set of generators and G Definition deg (g : freeg T int) : int := to be a group. The free abelian group with coefficients in G and fglift (fun x => 1) D. generators in T is the set of formal sums of the form x∈T αx (x), T where fglift stands for the group homomorphism defined by: where {αx}x T ∈ G is a null set, equipped with a group structure ∈ P where 0 = x 0(x) and x αx (x)+ x βx (x) = x(αx + fglift (f, x αx (x)) = x αx f(x) βx)(x). P P P P 3.2.2 Multivariate PolynomialsP P The formal sum x∈T αx (x) can be represented by the finite map {x 7→ αx | αx 6= 0}, and, in term of formalization, by the We construct our type of multivariate polynomials in n variables association list [(x, αPx) | x ∈ T, αx 6= 0]. We define prefreeg by instantiating the type of free abelian groups, taking for the as the type of all valid association lists, i.e. as the collection of all coefficients the base ring and using a free commutative monoid sequences s of type seq (T ∗ G) s.t. no pair of the form (_, 0) over {1,...,n} as the set of generators. occurs in s and for any (x : T), a pair of the form (x, _) appears We use the type of finite functions from {1,...,n} to N for at most once in s. representing the free commutative monoid over {1,...,n}. This type is then equipped with the adequate monoid structure. Definition reduced (g : seq (T ∗ G)) := (uniq [seq zx.1 | zx <− g]) Inductive multinom := Multinom of {ffun ’I_n −> nat}. Coercion fun_of_multinom m := let: Multinom m := m in m. Following Section 2.1, the proof of the fundamental lemma Definition mzero := Multinom [ffun _ => 0]. constructively builds the decomposition polynomial, using some Definition madd m m’ := Multinom [ffun i => m i + m’ i]. fixed well-founded monomial ordering for enforcing termination. In our formalization, we explicitly exhibit this function, and prove The type mpoly R n of multivariate polynomials in n variables that its terminates and is correct w.r.t. the fundamental lemma over the ring R is then defined as freeg R (multinom n). It statement. For that, we define3 in Algorithm 1 a procedure SYMF, directly inherits the basic functions (domain, coefficients, ...) and that given a multivariate polynomial p, computes, if it exists, the structures (decidable equality, group) of freeg. It remains to prove decomposition t of p s.t. p = t ◦ {Xi 7→ σn,i}. As written, this that our structure is a ring, defining the ring laws as follows: procedure may not terminate, and this is indeed the case when p is not symmetrical. Definition pone := << 1 ∗g 0%M >> Definition pmul (p q : mpoly R n) := \sum_(k1 <− dom p) \sum_(k2 <− dom q) Algorithm 1 << (coeff p k1) ∗ (coeff p k2) ∗g (k1 + k2)%M >>. 1: function SYMF1(p : polynomial in n variables) where << k ∗g x >> is the free group notation for k (x) and 2: if p is the zero polynomial then return (0, 0) 0%M and (_ + _)%M are the free monoid operations. Looking at 3: else 4: k1 kn these definitions, we see that our definitions follow closely the α (X1 · · · Xn ) ← the leading monomial of p k1⊖k2 kn−1⊖kn kn ones that can be found in textbooks. The subtle difference relies in 5: m ← X1 · · · Xn−1 Xn the ranges of the summations. Most textbooks make implicit that 6: return (αm,p − α (m ◦ {Xi 7→ σn,i})) these sums are finite, whereas in our case, this is made explicit 7: end if by iterating over the domains of p and q. Besides this explicit 8: end function management of summation supports, the proofs follow closely the 9: ones of textbooks. Moreover, we tackle the difficulty of manual 10: function SYMF(p : polynomial in n variables) handling of summations supports by adapting the big enough[3] 11: (t,p) ← (0,p) mechanism to our case, allowing us to defer and to compute a 12: while p is not the zero polynomial do posteriori the domain on which the involved summations must be 13: (q,p) ← SYMF1(p) done. 14: t ← t + q From there, we develop an extensive library, including polyno- 15: end while mial differentiation, evaluation, morphisms. The development fol- 16: return t lows closely the one of the univariate case, and we equip our con- 17: end function structions with the relevant algebraic structures that can be found in the SSReflect library. In the next section, we illustrate our library Our main result in this section is the formalization of the termi- by describing the formal proof of the fundamental lemma of sym- nation and correctness of the SYMF procedure. We start by trans- metric polynomials. lating the pseudo-code of the decomposition to some Coq function. Unsurprisingly, the unbounded while loop cannot be defined in Coq 3.2.3 Formalizing the Fundamental Lemma as-is. Instead, we code a variant symfn of SYMF that takes an extra The fundamental lemma of symmetric polynomials states, for any (fuel) parameter n and returns the pair (t,p) that results after the symmetric polynomial p in n variables, the existence and unique- execution of exactly n + 1 iterations of SYMF: ness of a decomposition polynomial t s.t. p is equal to t ◦ {Xi 7→ σn,i} where the σn,is are the n elementary symmetric polynomials Definition symf1 (p : {mpoly R[n]}) := #|s|=k 2 if p == 0 then (0, 0) else defined by σ = Xi . We detail in this n,k s∈P({1,...,n}) i∈s let (a, A) := (coeff p (mlead p), mlead p) in section the existence part of the proof of the fundamental lemma, P Q let m := [multinom i < n | (A i − A i.+1)%N] in as stated by the following Coq statement: (a ∗: ’X_[m], p − a ∗: (’X_[m] \mPo S)). Definition symmetric := [forall s, msym s p == p]. Fixpoint symfn (k : nat) (p : {mpoly R[n]}) := if k is k’.+1 then Definition mesym (k : nat) : {mpoly R[n]} := let (t1, p) := symf1 p in \sum_(h : {set ’I_n} | #|h| == k) \prod_(i in h) ’X_i. let (t2, p) := symfn k’ p in (t1 + t2, p) else symf1 p. Let S := [tuple (mesym i) | i < n]. Beside the mlead p construct, that returns the maximum (lead- Lemma sym_fundamental (p : {mpoly R[n]}) :d ing) monomial of p for a fixed monomial ordering, symf1 uses only p \is symmetric −> { t | t \mPo S }. standard functions of the multivariate polynomials library. For the definition of mlead, we develop a library for orders. Notably, we de- where msym s p stand for p where the indeterminates are permuted fine the lexicographic lift of orders to fixed size tuples, and proved by the permutation s, and p \mPo S stands for the composition that totality and well-foundness are preserved. This allows use to of p with the n elementary symmetric polynomials (mesym i). define some monomial ordering as: All these definitions follow closely the mathematical ones and rely essentially on facilities already present in the SSReflect library and Definition mnmc_le (m1 m2 : ’X_{1..n}) := in our multivariate polynomials library. Due to the lack of space, lex [posetType of nat] (mdeg m1 :: m1) (mdeg m2 :: m2). we do not expand on these definitions and we immediately shift to the proof formalization. where lex is the lexicographic lift of an order (here the one of natu- ral numbers) and m1, m2 are cast, via a coercion, to their respective 2 The notation p ◦ {Xi 7→ qi} is the multivariate polynomial composition, 3 returning the polynomial obtained by the formal substitution of the Xis by The notation ⊖ stands for the truncated subtraction, returning 0 if the the qis in p. result of the subtraction is negative. n-tuple indeterminate powers. This order differs from the one de- composition polynomial by a simple induction over the fuel argu- fined in Section 2.1 by prepending the degree of the monomials to ment of symfn — the fact that the decomposition polynomial has the sequence of indeterminates powers. It is known as the degrevlex a weight smaller that the degree of the input polynomial is proved monomial order and guarantees that a monomial is always strictly this way. larger than any monomial of strictly lower degree. Being the lexico- graphic lift of a total well-founded order, we obtain that mnmc_le is also a total well-founded order over monomials, and equip this 3.3 Analysis part with the relevant structure of the order library. Notably, this allows For the analysis part, we rely on the axiomatic real numbers pro- us to define a new induction principle for polynomials, based on vided by the Coq distribution and the extension provided by the the monomial ordering: Coquelicot library [2]. The main advantage of this library is that it treats integration and differentiation in a smooth way. Lemma mleadrect (P : {mpoly R[n]} −> Prop) : (forall p, (forall q, (mlead q < mlead p)%O −> P q) −> P p) −> forall p, P p. 3.3.1 Bridges between various formalized complex numbers Complex numbers were developed independently in Coquelicot We are now ready to define the leading coefficient of a polyno- and the Mathematical Components. In the Coquelicot library, com- mial p as its greatest monomial (i.e. with a non-null coefficient) for plex numbers are used for real analysis. In the Mathematical Com- the monomial order we just defined: ponents, the construction of complex numbers is studied from a more abstract point of view as the result of field extensions on top of < Definition mlead p := (\max_(m − msupp p) m)%O. arbitrary real closed fields. In particular, the construction in Mathe- The leading coefficient denomination here makes sense as matical Components makes it possible to consider the field of com- mlead p returns one of the monomials of p of maximal degree. plex algebraic numbers (which is not topologically complete but where equality is decidable) independently from the field of com- Correctness and completeness We then prove that symf1 is cor- plex numbers (which is topologically complete, but where equality rect w.r.t. the fundamental lemma, and that this function progresses is not decidable). w.r.t. the monomial ordering. The first property is needed to prove We need to re-use many of the notions from both libraries: for the correctness of the final decomposition function, whereas the lat- instance, the concept of algebraic number, inherited from Mathe- ter is used when proving the termination of the iteration of symf1: matical Components and the concept of integrating a function from R to C inherited from Coquelicot. Lemma symf1P (p : {mpoly R[n]}) : p \is symmetric −> In a first module, which we call Rstruct, we instantiate most [&& ((symf1 p).2 == 0) || (mlead (symf1 p).2 < mlead p) of the structures of Mathematical Components on the type of real , (symf1 p).2 \is symmetric & p == (symf1 p).1 \mPo S + (symf1 p).2]. numbers, whose description comes from the Coq distribution. In our Rstruct module, we show that as a consequence of the set In essence, the property states that, when given a symmetric of axioms that describe the real numbers, this type also satisfies polynomial p, symf1 returns a pair (t, q) composed of a partial many of the characteristics required for Mathematical Components decomposition polynomial t and a remainder q smaller than p, i.e. types. For instance, the Mathematical Components has a notion of such that p = t ◦ {Xi 7→ σn,i} + q, and s.t. q is symmetric and is eq_type, a type where equality is mirrored by a boolean test func- either null or with a leading monomial strictly lower than the one tion. The existence of this boolean test function is a consequence of p. Moreover, in case of a null remainder, we see that the problem of the axioms defining the real numbers. is solved, as we obtain a polynomial t s.t. p = t ◦ {Xi 7→ σn,i}. It should be noted that Mathematical Components structures Having symf1 returning a symmetric polynomial, this property require ring structures to satisfy a choice property. This choice can be directly lifted to symfn by induction on the fuel argument. property is not a consequence of the axioms defining R in the Coq Again, as for symf1, if the remainder returned by symfn is null, distribution, so we do not only use these axioms, but also the axiom then the problem of decomposing the input polynomial is solved. provided in the Epsilon package, whose strength is similar to the We prove that symfn is complete, i.e. that it returns a null remainder axiom of choice. For a similar reason, we also rely on functional as long as it is given enough fuel, and give a concrete bound for the extensionality. needed fuel: Thus, we are studying the consequences of the set of axioms that describe the real numbers in Coq’s standard library. Even though Lemma symfnS (p : {mpoly R[n]}) : Coq’s logic is constructive, our proof is classical because it relies {n:nat|p\is symmetric −> (symfn n p).2 = 0 }. on axioms that are only valid in classical logic. The Mathematical Components library provides a type construc- The property is directly proved using the mleadrect induction tor named complex, which takes as input any type equipped with a principle and by application of the progress property. At this point, real closed field structure and returns a new type with a field struc- we have all the ingredients to define the final decomposition func- ture, which is algebraically closed and a field extension of the for- tion (by hiding the fuel argument of symfn) and to give a formal mer. We applied this constructor on the type of real numbers and proof of the fundamental lemma: obtained a type we called complexR. This type satisfies the prop- erties of being a numClosedFieldType, in other words an alge- Definition symf (p : {mpoly R[n]}) := braically closed type with a norm satisfying triangular inequalities (symfn (tag (symfnS p)) p).1. and stability with respect to multiplication. Lemma symfP (p : {mpoly R[n]}) : We then needed to establish many correspondences between this p \is symmetric −> p = (symf p) \mPo S. type of complex numbers and the similar construction present in the Coquelicot library. In the end, we manage to combine the concepts Making the decomposition function explicit has one major ben- of limits, derivability, continuity, integrability from the Coq distri- efit: the decomposition algorithm is not hidden inside some proof bution and the Coquelicot library with the algebraic properties of term, and we can prove, a posteriori, extra properties on the de- the complex numbers from the Mathematical Components library. 3.3.2 Subsets of complex-integers and complex-naturals Crderive (fun y => Cexp(a ∗ y%:C)) x = a ∗ Cexp(a ∗ x%:C). For our proof, we also need to define the predicates Cnat and Cint that recognize exactly the complex numbers which are integers and The first of these two lemmas states that the function y 7→ eay is natural numbers, respectively. Once these predicates are defined, differentiable everywhere. The second lemma gives the value of the the Mathematical library provides a natural notion of polynomial derivative. over a predicate: it is polynomial such that all coefficients satisfy For integration, we only define notions related to integration of the predicate. For instance, to express that polynomial P has all its functions from R to C, by considering integration independently on coefficients in Cint one simply writes: each component. Unfortunately, we were unable to benefit from the Coquelicot library when developing this part, because integration P \is a polyOver Cint in vector space was less smoothly designed than integration of In the same manner that mathematical structures can be attached real-valued functions. For instance, the Coquelicot library only to types, they can also be attached to predicates. In the case of the provides theorems linking integrals and antiderivatives for real- Cint predicate, we show that 0 and 1 satisfy this predicate, and that valued functions, and not for more general functions with values it is stable for all the ring operations. This in turn makes it possible in arbitrary normed spaces. As an illustration, we had to prove the to invoke general theorems provided once and for all for all stable theorem that links integration and antiderivative: predicates. For instance, our development contains the following Lemma RInt_Crderive f a b: declarations about Cint: (forall x, Rmin a b <= x <= Rmax a b −> ex_derive f x) −> Fact Cint_subring : subring_closed Cint. (forall x, Rmin a b <= x <= Rmax a b −> Proof. ... Qed. Crcontinuity_pt (Crderive f) x) −> CrInt (Crderive f) a b = f b − f a. Canonical Cint_opprPred := OpprPred Cint_subring. Canonical Cint_addrPred := AddrPred Cint_subring. In this statement, CrInt represents the integral operator for func- ... Canonical Cint_subringPred := tions from R to C, and Crderive f represents the derivative of SubringPred Cint_subring. f. A first hypothesis expresses that f must be differentiable every- where in the integration interval, a second hypothesis expresses that For example, any statement of the form (x + y) \in Cint can the derivative must be continuous everywhere in this interval. be transformed into the two statements x \in Cint and x \in 3.3.4 Details on a proof at the frontier between libraries Cint using a lemma rpredD which was defined once for all predi- cates compatible with addition. A necessary elementary fact about exponentials is that for any nat- ural numbers a and b, for n large enough, abn < n!. This is a fact 3.3.3 Formally defining complex exponential about exponentials because the power series xn/n! converges to x for any , and therefore n converges towards 0. We man- Complex exponential is defined from real exponential by using the e x x /n! age to perform this proof by re-using a generalP theorem about the following definitions, when x and y are both real numbers. generic term of a converging series (provided by Coquelicot), and ex+iy = ex × (cos y + i sin y) then the formal definition of exponential. As an illustration of the formalization work, we can follow the In our formal text, this definition is written in the following steps of the formal proof for the statement manner: ∀a,b ∈ N, ∃M,M In this lemma, the notation *+ is used to represent multiplication This proof reduces to showing that applying elementary symmetric by a natural number. Another companion lemma expresses that if P polynomials on the sequences pre_alpha’ k returns a symmetric has integer coefficients, then P^‘N(k) also does. The polynomial polynomial expression on betaseq. For this we rely on a general G is then defined as the sum of all derivatives of Fp^‘N(p) up to lemma from the multivariate library about composing symmetric order pn and G has integer coefficients. polynomials with multiplets of polynomials that are stable by per- np mutation. After adding the assumption that c γiG(αi) is an integer, we prove the final steps of Section 2.2.3 by representing them P Lemma msym_comp_poly k as specific lemmas. After discharging all assumptions, the main (p : {mpoly R[n]})(t : n.−tuple {mpoly R[k]}): lemma actually has the following general statement: p \is symmetric −> (forall s : ’S_k, main_contradiction : perm_eq t [tuple (msym s t‘_i) | i < n]) forall n : nat, n != 0 −> forall k : int, k != 0 −> −> p \mPo t \is symmetric. forall (gamma : nat −> int) (c : nat), c != 0 −> forall alpha : nat −> complexR, In this statement, msym s t‘_i represents the polynomial t‘_i (forall i : ’I_n, alpha i != 0) −> where the variables have been permuted according to the permu- \prod_(i < n) (’X − (alpha i)%:P) ∗+ c tation s. Proving this premise in the case of pre_alpha’ is quite \is a polyOver Cint −> (k%:~R + tedious. \sum_(0 <= i < n) Other key steps in the formalization are the uses of the funda- (gamma i)%:~R ∗ Cexp (alpha i))%R = 0 −> mental theorem of symmetric polynomials. This theorem applies ~ (\sum_(0 <= i < n) for every symmetric polynomial with coefficients in a commutative (gamma i)%:~R ∗ (G n k gamma c alpha).[alpha i] ∗+ ring, and returns a polynomial with coefficients in that ring. c ^ (n ∗ p n k gamma c alpha))%R \in Cint In our case, the polynomial we are working on is a polynomial with complex coefficients which satisfy the Cint predicate. If we use directly the fundamental theorem on this polynomial, we ob- 3.5 Instantiating the common lemma: cases of e and pi tain a resulting polynomial with complex coefficients and we lose crucial information. The proof of transcendence of e is a very simple instantiation of We needed to prove a stronger version of the theorem: the common lemma and the formal development does not pose any difficult problem. For instance, the following line describes the α Lemma mpolysym nu (p : {mpoly complexR[nu]}) : sequence for e: p \is a mpolyOver Cint −> p \is symmetric −> p != 0 −> {q | [&& (p == q \mPo [tuple ’s_(nu, i.+1) | i < nu]) , ((mweight q) <= msize p)%N Definition alpha i := (i.+1%:R : complexR) & (q \is a mpolyOver Cint)]}. We plan to extend the multivariate polynomials library to the case of monoid algebras. While we expect minor modifications This proof is done by applying a morphism from polynomials on the base definitions and proofs —notably on the ones for free with complex-integer coefficients to polynomials with integer co- groups that are already abstracting over the set of generators— this efficients, using the fundamental theorem in the type {poly int} should bring to the library the study of polynomials over of infinite to obtain a polynomial with integer coefficients, that can then be number of indeterminates, as well as the study of free modules. translated into a polynomial with complex-integer coefficients. This extension will allow us to fill gaps on some on-going formal The final transcendence statement has the following shape: developments, notably in the proof of algorithms in the field of algebraic combinatorics. Theorem pi_transcendental : ~algebraicOver ratr PI%:C. In the longer run, the infra-structure to combine the algebra li- brary Mathematical Components and the calculus library Coqueli- cot should be improved. Even though these libraries are devel- 4. Conclusion and Future Work oped with the same system, we could probably re-use some of the tools that were developed to communicate proofs from one theorem 4.1 Related work prover to the other. To our knowledge, the earliest formalized descriptions of multivari- ate polynomials is proposed by Jackson [13]. In a sense, this early References work already contains many of the ingredients that are found in our description, with reliance on a free abstract monoid (while we rely [1] Jesse Bingham. Formalizing a proof that e is transcendental. Journal on a free group). However, this early development contains very of Formalized Reasoning, 4(1):71–84, 2011. few proofs. [2] Sylvie Boldo, Catherine Lelay, and Guillaume Melquiond. Coquelicot: Multivariate polynomials were also used by Théry [20] to prove A user-friendly library of real analysis for Coq. Mathematics in that Buchberger’s algorithm was correct. This work also relies on Computer Science, 9(1):41–62, 2015. an abstract description where each polynomials is a sum of terms, [3] Cyril Cohen. Formalized algebraic numbers: construction and first- where each term is the multiplication of a coefficient and a mono- order theory. PhD thesis, École polytechnique, Nov 2012. mial, the sums are represented as ordered list of terms with non- [4] Cyril Cohen, Maxime Dénès, and Anders Mörtberg. Refinements for zero coefficients. Most of the operations, like addition or multiplica- free! In Georges Gonthier and Michael Norrish, editors, Certified tion, take care of maintaining the ordering and non-zero properties. Programs and Proofs, CPP 2013, Melbourne, Australia, Springer, This made many of the proofs rather complicated. By comparison, 2013. our development stays further away from implementation details. [5] Julian Lowell Coolidge. The number e. The American Mathematical Another contribution fine-tuned for a specific algorithm is the work Monthly, 55(9):591–602, Nov 1950. by Mahboubi on an efficient gcd algorithm [15]. Apparently, this [6] Coq development team. The coq proof assistant, 2008. description relies on a recursive approach, where a n-variable poly- http://coq.inria.fr. nomial is actually encoded as 1-variable polynomial with - (n − 1) [7] Maxime Dénès, Anders Mörtberg, and Vincent Siles. A refinement- variable polynomials as coefficients. based approach to computational algebra in coq. In Lennart Beringer More recently Muñoz and Narkawicz used multivariate polyno- and Amy Felty, editors, Interactive Theorem Proving, ITP2012, vol- mials in optimization problems [16]. The work by Haftmann et al. ume 7406 of LNCS, pages 83–98, Princeton, USA, Springer, 2012. [10] is especially interesting because it enumerates the various pos- [8] Georges Gonthier. Formal proof—the Four Color Theorem. Notices sible choices for the implementation of multivariate polynomials of the AMS, 55(11):1382–1393, 2008. and insists on the benefits of first implementing an type of abstract polynomials. It then shows how implementations can be derived [9] Georges Gonthier, et al. A Machine-Checked Proof of the Odd Order Theorem. In Sandrine Blazy, Christine Paulin, and David Pichardie, through refinements. In a sense, the formal development that we editors, Interactive Theorem Proving, ITP2013, volume 7998 of LNCS, described here fulfills the requirements of an abstract type of poly- pages 163–179, Rennes, France, Springer, 2013. nomials. When considering implementations, we also intend to rely [10] Florian Haftmann, Andreas Lochbihler, and Wolfgang on refinement described by Dénès, Cohen, and Mörtberg [4, 7]. Schreiner. Towards abstract and executable multivari- None of the work we have been able to find on formalized multi- ate polynomials in isabelle. Isabelle Workshop 2014, variate polynomials includes any significant results about symmet- http://www.infsec.ethz.ch/people/andreloc/publications/haftmann14iw.p ric polynomials and for this reason we believe that our work is the 2014. only one to contain a proof of the fundamental theorem of symmet- [11] John Harrison. HOL Light: A Tutorial Introduction. In FMCAD, pages ric polynomials. 265–269, 1996. For formalized proofs of transcendence, the only prior work that [12] Charles Hermite. Sur la fonction exponentielle. Gauthier-Villars, Paris, we are aware of is the formalized proof that e is transcendent de- 1874. veloped in Hol-light by Bingham [1]. The more complex proof of transcendence for π was not formalized yet. This is not a surprise, [13] Paul Jackson. Exploring abstract algebra in constructive type theory. since most known proofs use the fundamental theorem of symmet- In Proceedings of the 12th International Conference on Automated De- duction, CADE-12, pages 590–604, London, UK, UK, 1994. Springer- ric polynomials, for which we also provide the first formalization. Verlag. [14] Ferdinand von Lindemann. Über die Zahl π. Mathematische Annalen 4.2 Future work 20 (1882): pp. 213-225. Springer-Verlag. This work started with an explicit aim of studying precisely the [15] Assia Mahboubi. Proving formally the implementation of an efficient proof of transcendence for π. We were able to isolate a common gcd algorithm for polynomials. In Ulrich Furbach and Natarajan lemma that applies both for the proof of transcendence of e and π. Shankar, editors, Automated Reasoning, IJCAR 2006, volume 4130 As a continuation we are now considering more general transcen- of LNCS, pages 438–452, Seattle, WA, USA, Springer. 2006. dence theorems, like Lindemann’s theorem. This theorem states re- [16] César Muñoz and Anthony Narkawicz. Formalization of a representa- sults about linear and algebraic independence of numbers and their tion of Bernstein polynomials and applications to global optimization. exponentials. Journal of Automated Reasoning, 51(2):151–196, August 2013. [17] Tobias Nipkow, Lawrence C. Paulson, and Markus Wenzel. Is- abelle/HOL — A Proof Assistant for Higher-Order Logic, volume 2283 of LNCS. Springer, 2002. [18] Ivan Niven. The transcendence of π. American Mathematical Monthly, 46(8):469–471, October 1939. [19] Ivan Niven. A simple proof that π is irrational. Bulletin of the AMS, 53(6):509, 1947. [20] Laurent Théry. A machine-checked implementation of buchberger’s algorithm. Journal of Automated Reasoning, 26(2):107–137, 2001.