Hugo Herbelin
email: hugo.herbelin@inria.fr

A -calculus Structure Isomorphic to Gentzen-style Sequent Calculus Structure

We consider a -calculus for which applicative terms have no longer the form (:::((u u1) u2)::: un) but the form (u u1; :::;un]), for which u1; :::;un] is a list of terms. While the structure of the usual -calculus is isomorphic to the structure of natural deduction, this new structure is isomorphic to the structure of Gentzen-style sequent calculus. To express the basis of the isomorphism, we consider intuitionistic logic with the implication as sole connective. However we do not consider Gentzen's calculus LJ, but a calculus LJT which leads to restrict the notion of cut-free proofs in LJ. We need also to explicitly consider, in a simply typed version of this -calculus, a substitution operator and a list concatenation operator. By this way, each elementary step of cutelimination exactly matches with a -reduction, a substitution propagation step or a concatenation computation step. Though it is possible to extend the isomorphism to classical logic and to other connectives, we do not treat of it in this paper.

Introduction

By the Curry-Howard isomorphism between natural deduction and simply-typed -calculus, and using Prawitz's standard translation 11] of cut-free LJ into natural deduction, we get an assignment of LJ proofs by -terms. Zucker 14] and Pottinger 10] have studied the relations between normalisation in natural deduction and cut-elimination in LJ. They were considering normalisation without paying special attention to the computational cost of the substitution of a proof in place of an hypothesis. But in sequent calculus, among the di erent uses of the cut rule, there is one which stands for an explicit operator of substitution and among the elementary rules for cut-elimination, there are rules to compute the propagation of substitution. Therefore, Zucker and Pottinger were led to consider proofs up to the equivalence generated by these substitution propagation computation rules.

Here, we consider a -calculus with an explicit operator of substitution and with appropriated substitution propagation rules. This allows to have a more precise correspondence with the elementary rules for cut-elimination. However, there are two problems. The rst one is that several cut-free proofs of LJ are associated to the same normal simply-typed -terms. An answer to this problem is to rather consider a restriction of LJ, called LJT, having the same structure and same strength as LJ but for which there is a one-to-one correspondence with normal simply-typed terms. The second problem is that Gentzen-style sequent calculus and -calculus (or natural deduction) have not the same structure. Consequently, the reduction rules in one and the other calculi do not match. An answer to this second problem is to consider an alternative syntax for -calculus of which, this time, the simply-typed fragment is isomorphic to LJT.

Note that a radically di erent approach of the computational content of Gentzen's sequent calculus appears in Breazu Tanen et al 1], Gallier 4] and Wadler 13]. Each of them interprets the left introduction rules of sequent calculus as pattern construction rules.

A Motivated Approach to LJT and -calculus 2.1 The Sequent Calculus LJ

We consider a version of LJ with the implication as sole connective. The formulas are de ned by the grammar A::= X j A!A where X ranges over V F , an in nite set of which the elements are called propositional variable names. In the sequel, we reserve the letters A, B, C, ... to denote formulas.

Sequents of LJ have the form ? `A. To avoid the need of a structural rule we de ne ? as a set. To avoid confusion between multiple occurrences of the same formula, this set is a set of named formulas. We assume the existence of an in nite set of which the elements are called names. Then, a named formula is just the pair of a formula and a name. Usually, we do not mention the names of formulas (anyway, no ambiguity occurs in the sequents we consider here).

Under the condition that A, with its name, does not belong to ?, the notation ?; A stands for the set-theoretic union of ? and fAg.

To avoid the need of a weakening rule, we admit irrelevant formulas in axioms.

The rules of LJ are:

?; A `A Ax ?; A; A `C ?; A `C Cont ? `A ?; B `C ?; A!B `C I L ?; A `B ? `A!B I R ? `A ?; A `B ? `B Cut

The Usual Interpretation of LJ Cut-free Proofs by Normal -terms

There is a standard way to interpret cut-free proofs of LJ as -terms, see for instance Prawitz 11] or, for a more formal presentation, Zucker 14], Pottinger 10] or Mints 9]. To express the interpretation, it is cumbersome to choose the set of -variables names as set of names. We then mention explicitly the name x of a formula A under the form x:A. The interpretation is by induction on the proofs and we mention the associated -terms on the right of the symbol `.

?; x:A `x:A Ax ?; x:A; y:A `u:C ?; x:A `ufy := xg:C Cont ? `u:A ?; y:B `v:C ?; x:A!B `vfy := (x u)g:C I L ?; x:A `u:B ? ` x:u:A!B I R for which vfx := ug denotes the term v in which each occurrence of x has been replaced by u.

Towards the Calculus LJT

However di erent proofs may be associated to the same -term. For instance: We decide to restrict LJ in order to get a bijective correspondence between normal simply-typed -terms and cut-free proofs. For this purpose, we restrict the use of the I L rule in order to forbid the second proof. The calculus we obtain has two kind of sequents. We call it LJT, since it appears as the intuitionistic fragment of a calculus called LKT and de ned by Danos, Joinet, Schellinx 2].

A; C `A Ax A; C; B `B Ax A!B; A; C `B I L A!B; A `C!B I R
A sequent of LJT has either the form ?; `A or the form ?; A `B. In both cases, ? is de ned as a set of named formulas. The semi-colon delimits a place on its right. A uniform notation for sequents of LJT is the following one: ?; `B where is a notation to say that the place on the right of the semi-colon may be either empty or lled with one (not named) formula. The idea of using these kinds of sequents comes from Girard 5] who called \stoup" the special place between the symbols \;" and \`".

The rules of LJT are ?; A `A Ax ?; A; A `B ?; A; `B Cont ?; `A ?; B `C ?; A!B `C I L ?; A; `B ?; `A!B I R Remarks: 1) With these rules, the rst proof above is not directly a proof in the restriction: the axiom rule of LJ has to be encoded in the restriction by an axiom rule followed by a contraction rule.

2)This calculus appears also in Danos et al 2] with a slight di erence in the treatment of structural rules. Like its classical version LKT, it has been considered by Danos et al for its good behaviour w.r.t. embedding into linear logic. The calculus LJT appears also as a fragment of ILU, the intuitionistic neutral fragment of uni ed logic described by Girard in 6]. The calculus ILU is itself a form of LJ constrained with a stoup, for which Girard pointed out that \the formula in the stoup] (if there is one) is the analogue of the familiar head-variable for typed -calculi".

Recently, Mints de ned in 9] a notion of normal form for cut-free proofs of LJ which also coincides with the notion of cut-freeness in LJT.

We have also to mention the de nition of a cut-free sequent calculus similar to the cut-free LJT in the paper of Howard 12] on the interpretation of natural deduction as a -calculus. Howard mentions that the proofs of this cut-free calculus are in one-to-one correspondence with the normal simply-typed -terms.

The proofs of the cut-free LJT are e ectively in one-to-one correspondence with the normal simply-typed -terms. For instance, a normal term of the form x 1 ::: x n :(y u 1 :::u p ) and of type A 1 ! ::: ! A n ! B, in which y is of type D = C 1 !:::!C p !B, is unambiguously associated to a proof of the form ?; `u1 : `(y u 1 :::u p ):B Cont ?nfx n g; `

x n :(y u 1 :::u p ):A n !B I R . . . . ?nfx 2 ; :::; x n g; ` x 2 ::: x n :(y u 1 :::u p ):A 2 !:::!A n !B ?nfx 1 ; :::; x n g; ` x 1 ::: x n :(y u 1 :::u p ):A 1 !::

:!A n !B I R
where ? contains y:D; x 1 :A 1 ; :::; x n :A n .

Remark: The construction of the applicative part of the term starts from u p and ends with u 1 in contrast with the usual way of building a term (u u 1 ::: u p ) incalculus. This is why we have not an exact correspondence with the substitution operator when we consider the cut rule.

Cut and Reduction Rules: Towards the -calculus

According to the place of the cut formula (in the stoup or not), there are two kinds of cut rules in LJT: head-cut rule mid-cut rule ?; `A ?; A `B ?; `B C H ?; `A ?; A; `B ?; `B C M for which means one or zero formula in the stoup. The mid-cut rule is naturally interpreted as an operator of explicit substitution: ?; `v:A ?; x:A; `u:B ?; `u x := v]:B C M A standard way to eliminate cuts is to apply rewriting rules to proofs in order to propagate the cuts towards smaller proofs. Here is an example of such a rewriting rule (we let C = A 2 !:::!A n !B):

?; `v:A ?; x:A; `u1 :A1 ?; x:A; y :C `(y u2 ::: un):B ?; x:A;y :A1 !C `(y u1 ::: un):B IL ?; y :A1 !C `(y u1 ::: un) x := v]:B CM reduces to ?; `v:A ?; x:A;`u1 :A1 ?; `u1 x := v]:A1 CM ?; `v:A ?; x:A; y :C `(y u2 ::: un):B ?; y :C `(y u2 ::: un) x := v]:B CM ?; y :A1 !C `(y u1 x := v] ::: un) x := v]:B IL It seems \natural" that such a rewriting rule is in correspondence with a rule of substitution propagation. But it is not the case. Indeed it corresponds to the reduction of (y u 1 ::: u n ) x := v] into (y u 1 x := v] ::: u n ) x := v] while we would like to get ((y u 1 :::

u n?1 ) x := v] u n x := v]).
This is because the structure of a proof in sequent calculus is di erent from the structure of the associated -term and this suggests to consider an alternative formalismfor the -calculus in which an applicative term is no longer of the form ((u u 1 ):::u n ), but of the form (u u 1 ; :::; u n ]), i.e. considered as the application of a function to the list of its arguments. We call -calculus this alternative formalism for -calculus.

Digression: How to Recover LJ ?

LJT is as strengthful as LJ since a proof of a sequent ? `A in LJ can be compositionnally translated into a proof of ?; `A in LJT. To express the translation, it is more convenient to consider a variant of LJ with the I L rule and the Cont rules mixed (i.e. we assume that A!B is already in ? for the second item). We note ; the translation. Thus we have an interpretation of LJ into LJT. However, following this interpretation, cut-free proofs in LJ may no more be cut-free in LJT. [START_REF] Dragalin | Mathematical Intuitionism: Introduction to Proof Theory[END_REF] The -calculus

The -expressions

We assume the existence of an in nite set V of which the elements are called term variables names and here denoted by the letters x, y, z, ... The set of -expressions, including the -terms (or shortly terms) and the lists of arguments are mutually de ned by the following grammar for which x ranges over V Terms: t::= (x l) j ( x:t) j (t l) j (t x := t]) Argument lists: l::= ] j t :: l] j (l @ l) j l x := t] We use the letters t, u, v,... to denote terms and the (possibly quoted) letter l to denote lists of arguments.

The notation ] stands for the empty list of arguments and t :: l] stands for the adjunction of the term t to the list of arguments l, while (l @ l 0 ) stands for the explicit concatenation of the lists l and l 0 of arguments.

The syntax (t x := u]) stands for an operator of explicit substitution in terms (a \let x=u in t" operator) and (l x := u]) stands for an operator of explicit substitution in lists of arguments.

We usually abbreviate an argument list t 1 :: ::: :: t n :: ]]:::]] by t 1 ; :::; t n ]. Terms such as ((:::(t t 1 ) :::) t n ) are abbreviated (t t 1 ::: t n ). Sometimes (x ]) is shortened into x. Also, the expressions ( x:t), (t x := u]) and (l @ l 0 ) may be written respectively x:t, t x := u] and l @ l 0 when there is no ambiguity.

Subexpressions of -expressions are de ned as usual, but, in our case, by a simultaneous recursion on terms and arguments lists.

Bound variables are de ned as usual. We say that two -expressions are -equal if they di er only in the names (assumed distinct the one from the others) of bound variables. This notion of equality does not a ect the structure of expressions and, in the sequel, we consider -expressions up to this -equality.

Normal -expressions

A -expression is normal if and only if it does not contain any operator of explicit concatenation or explicit substitution and if all applicative subterms are of the form (x l) with l normal.

Otherwise said, a -expression, is normal if it is construed using this restricted grammar:

t::= (x l)j( x:t) l::= ]j t :: l] An approximation of normality is weak normality. A -expression is called weakly normal if it is of the form (x l) or x:t or ] or t :: l], where t and l denotes respectively any term and any list of arguments.

Remark: Usual -calculus can be embedded in -calculus, since there is, incalculus, the possibility to consider terms of the form (:::(x u 1 ]) ::: u n ]) having a structure similar to the structure of applicative terms in -calculus. However, such a -term is not normal. Indeed, its normal form is (x u 1 ; :::; u n ]).

Reduction Rules

The presence of explicit substitution and concatenation operators entails the presence of appropriated reduction rules:

{ -reduction ! v, then u is called a redex. We note 1 ! the one step reduction obtained from r ! by congruence. Since the system of reduction rules is left linear (if one takes an in nite family of rules S yes , S no and S , one for each possible combination of distinct x and y in V) and without critical pairs, according to Huet 7], 1 ! is con uent. We stay unprecised about the -equality problem stemming from the rule S . Solutions exist, for instance by adding an extra explicit renaming rule to the rewriting system.

( x:u v :: l]) r ! (u x := v] l) cons ( x:u ]) r
Remark: The absence of critical pairs may be quite restricting. For instance, it is not possible to simulate usual -reduction using these rules for the reason that substitutions are not allowed to go through -redexes. However, the set of rules is enough to reach a normal form, when this one exists.

4 Cut-elimination in the Calculus LJT

We say that two proofs are equal if they di er only by the names of formula in the proved sequent or by addition of irrelevant formulas to the left part of the proved sequents. We consider proofs up to this notion of equality. In particular, if p is a proof of ?; `A, then, for any named formula B not in ?, p is a proof of ?; B; `A, even if it becomes necessary to change the name of another similarly named occurrence of B throughout p.

Cut-elimination Proposition1. (Strong and con uent cut-elimination)

There exists a con uent system of rewriting rules which allows to derive a cutfree proof of ?; `A from any proof of the same sequent.

Such a system of rewriting rules is listed hereafter. It is easy to see that it is complete, since it exhausts all possible patterns having a cut rule as head symbol. Its con uence comes from its left linearity (if one takes one di erent rule for each di erent variable name) and from the absence of critical pairs, as for the system of reduction rules of the -calculus. As for its strong termination, the proof is done in a next section. Proofs of LJT are isomorphic to -expressions. We show it by rst assigning -expressions to proofs of LJT. It remains just to check that, through this assignment, the reduction rules for -expressions are in exact correspondence with the rewriting rules for proofs of LJT.

To describe the assignment, we identify the set of formula names with the set of -term variable names and we write the named formulas under the form x:A. It is also cumbersome to consider arguments lists as applicative contexts:

An applicative context is a list of arguments written under the form (: l) where : is a special notational symbol. Also, we call hole declaration a formula written under the form : :A.

We express the assignment by judgments.

A judgement is something of the form ?; `t : A. In this writing is either nothing, in which case t is a term, or a hole declaration in which case t is an applicative context.

Otherwise said, in the assignment, proofs of sequents with an empty stoup are interpreted by terms while proofs of sequents with a non empty stoup are interpreted by applicative contexts.

Applicative context formation

Term formation A -expression e such that we have x 1 : A 1 ; :::; x n : A n `e:A for some term variable names x 1 , ..., x n and for some formulas A 1 , ..., A n , A is said simplytyped of type A, or shortly, typable by A.

Strong Termination

By the isomorphism,the strong termination of cut-elimination for LJT (using the above rewriting system) and the strong termination of reduction for typableexpressions are equivalent. We show hereafter the strong termination for typable -expressions.

Proof of Strong Termination. Let e be a -expression and R ! a notion of reduction. We say that e is strongly normalisable w.r.t. R ! in the following cases: { e is not reducible w.r.t. R ! { for all e 0 such that e R ! e 0 , we have e 0 strongly normalisable Let e be a -expression. If e is typable, then it is strongly normalisable w.r.t. the reduction 1 !. To prove that, we prove something stronger, the strong Enormalisability.This latter is preserved by the various operations of -expressions construction.

We de ne a notion of reduction h ! which removes the head constructor of a -expression. The reduction h ! is de ned by the following cases:

x:u h ! u u :: l] h ! u (x l) h ! l u :: l] h ! l
where u ranges over the set of -terms and l over the set of argument lists.

We note E !, and we call E-reduction, the notion of reduction de ned by e E ! e 0 either because e 1 ! e 0 or because e h ! e 0 (without considering the closure of h ! by congruence). We say that e is strongly E-normalisable (shortly SEN)

if it is strongly normalisable w.r.t. E !.

Lemma 2. If the -term u and the argument list l are SEN then x:u, (x l) and u :: l] are SEN.

Proof. By induction on the proof that u is SEN then by induction on the proof that l is SEN. Let us treat of the case u :: l]. If u :: l] E ! e 0 then, either e 0 is u or l, in which case, by hypothesis, e 0 is SEN, or e 0 is u 0 :: l] with u 1 ! u 0 , or u :: l 0 ] with l 1 ! l 0 in which cases e 0 is SEN by induction hypothesis. Therefore, in any case, e reduces to a SEN -expression. This implies that e is itself SEN.

Lemma 3. Let e and u be SEN -expressions. If, for all l SEN, the typability of (u l) implies that (u l) is SEN, then, also the typability of e x := u] implies that e x := u] is SEN.

Proof. It works by induction on the proof that e is SEN then by induction on the proof that u is SEN. Let us assume that e x := u] 1 ! w. If the reduction touchs a redex in u then w has the form e x := u 0 ] with u 1 ! u 0 . The proof of SEN for u 0 is smaller than the one for u, thus, by induction hypothesis, e x := u 0 ] is SEN. Similarly, if the reduction is in e. It remains the case where e x := u] is itself a redex and where it is this redex which is reduced. We look at the di erent possible forms for e.

{ The case where e is (x l 0 ) { in which case w denotes (u l 0 x := u]) { is the more delicate one. But since e h ! l 0 , the proof of SEN for l 0 is smaller than the one for e. Therefore, by induction hypothesis, l 0 x := u] is SEN. And since we have assumed that for all l SEN, (u l) was SEN, we infer that (u l 0 x := u]) is SEN.

{ If e is (y l) then w is (y l x := u]). Here again, l x := u] is SEN by induction hypothesis. Then, by lemma 2, we get that w is SEN.

{ If e is the term y:v, up to a change of the variable name y in y:v { and this does not change the structure of the proof of SEN {, we may assume that y and x are distinct variable names. We may then a rm that w is y:(v x := u]). Since y:v E ! v, by induction hypothesis, (v x := u]) is SEN and by lemma 2, w is SEN.

{ If e is v :: l] then w denotes v x := u] :: l x := u]]. But we have both v :: l] E ! v and v :: l] E ! l. Therefore, by induction hypothesis, we have that v x := u] and l x := u] are SEN. Then, by lemma 2, we get that w is also SEN.

{ If e is ] then w is ] which is directly SEN.

Thus, whatever the form of e, the reducts of e x := u] are all SEN. This is enough to say that e x := u] is SEN. Lemma 4. Let A be a formula. Let e be a -expression, SEN and typable by A.

Let l be a SEN arguments list. If the expression (e l) (if e is a -term) or the expression e @ l (if e is an arguments list) is typable, then it is SEN.

Proof. We proceed by induction on A, then on the proof that e is SEN, then on the proof that l is SEN.

Let us assume that (e l) 1 ! w (if e is a -term) or e @ l 1 ! w (if e is an arguments list).

If the reduction a ects a redex in e then w has the form (e 0 l) or e 0 @ l with e 1 ! e 0 . Since the proof of SEN for e 0 is smaller that the one for e, by induction hypothesis, w is SEN. Similarly if the reduction is in l.

It may also happen that (e l) or e @ l is a redex and that this redex is the reduced one.

{ The more delicate case is when e has the form x:u while l has the form v :: l 0 ]. In this case, the type of A has the form B ! C, the -term v is typable by B and w denotes (u x := v] l 0 ). Since B is smaller than A, by induction hypothesis, the typability of (v l 00 ) implies that it is SEN whatever l 00 SEN. It is then possible to use lemma 3 in order to infer that u x := v] is SEN. But this latter is typable by C which is also smaller than A. By induction hypothesis, again, (u x := v] l 0 ) is SEN.

{ If e is (x l 0 ) then w denotes (x (l 0 @ l)). But (x l 0 ) E ! l 0 , therefore, by induction hypothesis, (l 0 @ l) is SEN. By lemma 2, w is SEN.

{ If e is x:u and l denotes ] then w is e which, by hypothesis, is SEN. { If e is ] then w denotes l which is directly SEN. { If e is v :: l 0 ] then w denotes v :: (l 0 @ l)]. But v :: l 0 ] E ! l 0 , therefore, by induction hypothesis, l 0 @ l is SEN. As for v, it is also SEN by induction hypothesis. Then, by lemma 2, w is SEN. Thus, whatever reduction of (e l) or e @ l we consider, we get a SENexpression. This means that (e l) (if e is a -term) or e @ l (if e is an arguments list) is SEN.

Proposition5. Typable -expressions are SEN.

Proof. Let e be a typable -expression. The proof works by induction on e. The cases v, (p l) and (v :: l) come directly from the lemma 2. The cases (u l) and (l @ l 0 ) come from the lemma 4. As for the cases v x := u] and l x := u], they come from the lemma 3 applied to the lemma 4.

The strong E-normalisability directly implies the strong normalisability. Corollary 6. Simply-typed -expressions are strongly normalizable. Remarks: 1) A similar proof has been done by Dragalin 3] for the system of reduction rules given in the seminal paper of Gentzen on the cut-elimination theorem for LK. The di erence is that Dragalin's proof does not work by structural induction on the proof of strong E-normalisability, but rather by induction on the length of these proofs. Our proof has been done independently, extending a proof from Coquand that the elimination of cuts according to an outermost strategy of reduction terminates.

Note that this kind of strong cut-elimination proof applies also to noncon uent systems of reduction rules (it is the case of Gentzen's system of reduction rules) but not to system including rules a ecting the order of cuts. This is contrast with the cut-elimination procedures that Zucker or Pottinger and have considered.

2) An interesting result would be to prove the strong normalisation of the simply-typed -calculus with the additional reduction rule ( x:t u) y := v] r ! (( x:t) y := v] u y := v]). As a corollary of this result, we would get the strong normalisation of the usual simply-typed -calculus and even the strong normalisation for the simply-typed -calculus with an explicit \let in "-like substitution operator (see for instance Lescanne 8]).

Conclusion

The isomorphism known as the Curry-Howard isomorphism expresses a structural correspondence between Hilbert-like axiomatic systems and combinatory logic and between natural deduction and -calculus. The isomorphism between LJT and the -calculus can be seen as the extension of this correspondence into the framework of sequent calculi and this shows that sequent calculus is no less related to functional features than natural deduction.

Among the di erent forms of sequent calculi, the calculus LJT has clearly a special place. Since the Modus Ponens rule of intuitionistic natural deduction can be split into a head-cut rule and an implication left introduction rule, LJT can even be seen as a strict re nement of natural deduction. Similarly thecalculus can be seen as a strict re nement of the usual -calculus, but, in order to make more precise this embedding relation, it would be necessary to extend the strong normalisation of the simply-typed -calculus by considering the extra reduction rule ( x:t u) y := v] r ! (( x:t) y := v] u y := v]).

  and A `A Ax A; C; B `B Ax A; B `C!B I R A!B; A `C!B I L are both associated to the Church-like typed -term z :C:(x y):C !B for a context in which x:A!B and y:A.

C 1 ?

 1 ; `up :Cp ?; y :B `y:B Ax ?; yp :Cp !B `(yp up):B IL . . . . ?; y 2 :C 2 !:::!C p !B `(y 2 u 2 :::u p ):B ? ; y 1 :D `(y 1 u 1 :::u p ):B I L ? ;

  substitution through weakly normal terms (x l) x := v] r ! (v l x := v]) S yes (y l) x := v] r ! (y l x := v]) S no ( y:u) x := v] r ! y:(u x := v]) S warning to a possible variable capture in rule S { propagation of substitution through weakly normal arguments ] x := v] r ! ] S nil u :: l] x := v] r ! u x := v] :: l x := v]] S cons If u r

?

  ; : :A `(: ]):A Ax ?; x:A; : :A `(: l):B ?; x:A; `(x l):B Cont ?; `u:A ?; : :B `(: l):C ?; : :A!B `(: u :: l]):C I L ?; x:A; `u:B ?; ` x:u:A!B I R ?; : :C `(: l):A ?; : :A `(: l 0 ):B ?; : :C `(: (l @ l 0 )):B C H ?; `u:A ?; : :A `(: l):B ?; `(u l):B C H ?; `u:A ?; x:A; : :C `(: l):B ?; : :C `(: l x := u]):B C M ?; `u:A ?; x:A; `v:B ?; `v x := u]:B C M Remark: The rules with an non empty stoup are polymorphic in the role of the formula in the stoup. So, there is a strong relation between a judgement ?; : :A 1 !:::!A n !B `(: u 1 ; :::; u n ]):B and a judgement ? ` u 1 ; :::; u n ]:A 1 ^:::^A n where A 1 ^::: ^An is de ned as 8B:(A 1 !:::!A n !B)!B (encoding of tuples in second order -calculus).

Acknowledgements

Simpli cations in the proof of strong normalisation are due to Thierry Coquand. I thank also the Paris 7 computer science logic group, Phil Wadler and Viviana Bono for echoes on this work.

? This research was partly supported by ESPRIT Basic Research Action \Types for

Proofs and Programs" and by Programme de Recherche Coordonn ees \M ecanisation du raisonnement".


A Beral

C Girousse

J Le Gouis

V Allard

G A Slafer

PHYSIOLOGICAL BASES OF CULTIVAR DIFFERENCES IN AVERAGE GRAIN WEIGHT IN WHEAT: SCALING DOWN FROM PLOT TO INDIVIDUAL GRAIN IN ELITE MATERIAL

Keywords: Wheat, grain number per m², average grain weight, trade-off, individual grain weight; genetic determinism

In recent decades, increases in wheat yield have been achieved mainly through increases in grain number per m 2 (GNM2) rather than through increases in average grain weight (AGW). Using AGW as a lever to increase yield would require avoidance of the negative relationship between GNM2 and AGW. It is usually supposed that this trade-off arises from an increase in the proportion of small grains as GNM2 rises. The proportional increase in small grains being the result either of (1) an increase in the proportion of secondary tillers in the spike population or (2) of an increase in the proportion of grains located in distal positions within each spike. Either or both of these two populational effects would tend to mask any true genotypic differences in AGW. The existence of these constitutive differences has already been proposed, but without considering the full extent of the populational confounding effects. Identification of a component of the constitutive genetic determinism of AGW -one that is truly independent of GNM2 -could contribute to cultivar developments that would lead to further increases in grain yield under future target environments.

To address this question, we analysed populational effects on AGW in four, modern, well-adapted bread-wheat cultivars. The four chosen cultivars show similar grain yields but contrasting AGWs.

The analysis of populational effects was carried out at three hierarchical levels (the plot, the spike and the single grain) and under two contrasting environmental conditions (well-watered vs waterdeficit conditions).

Regardless of the environment, no (or only slight) differences in individual spike size were observed between cultivars. Furthermore the weak relationship between spike size and AGWdemonstrates that AGW differences between cultivars cannot be attributed to spike-level populational effects. Meanwhile, the analysis of individual grain mass distributions, showed that the differences in AGW between cultivars, originated from shifts in the whole grain-mass distribution, rather than from shape changes in the grain-mass distribution. This clearly indicates that AGW differences between cultivars cannot be attributed to populational effects at the individual grain level.

The analysis carried out at both spike and individual grain levels indicates that the AGW differences between cultivars are largely constitutive, so that increases in grain yield through AGW may be considered independently of the GNM2 : AGW trade-off.

Taken together, these findings offer a new perspective for the genetic improvement of wheat, and one that should lead to further increases in yield.

Introduction

Yield of wheat can be decomposed into two major components: grain number per m² (GNM2) and average grain weight (AGW), often expressed as the thousand grain weight. The GNM2 is the main determinant of grain yield variation resulting from variations in environmental and/or genetic factors. Over the last fifty years grain yield increases have been achieved mainly through increases in GNM2 -rather than through increases in AGW (e.g., [START_REF] Fischer | The importance of grain or kernel number in wheat: A reply to Sinclair and Jamieson[END_REF][START_REF] Mondal | Fifty years of semi-dwarf spring wheat breeding at CIMMYT: Grain yield progress in optimum, drought and heat stress environments[END_REF][START_REF] Slafer | Coarse and fine regulation of wheat yield components in response to genotype and environment[END_REF]. Further increases in GNM2 are probably still possible under projected future climate conditions but the specific targeting of AGW is also a worthwhile option under climate conditions that may become less favourable and especially if these include greater levels of heat stress [START_REF] Calderini | Grain Position Affects Grain Macronutrient and Micronutrient Concentrations in Wheat[END_REF][START_REF] Quintero | Trade-off between grain weight and grain number in wheat depends on GxE interaction: A case study of an elite CIMMYT panel (CIMCOG)[END_REF].

Support for the strategy of obtaining a genetic yield increase through direct manipulation of AGW is found in the observation that grain weight has a strong and heritable genetic determinism [START_REF] Sadras | Evolutionary aspects of the trade-off between seed size and number in crops[END_REF]. Indeed, some molecular-physiological studies have identified genes involved in the various processes that govern grain growth and grain filling and the effects of their allelic variations on final grain size and weight [START_REF] Laudencia-Chingcuanco | Transcriptional profiling of wheat caryopsis development using cDNA microarrays[END_REF][START_REF] Nadaud | Proteomic and morphological analysis of early stages of wheat grain development[END_REF][START_REF] Simmonds | A splice acceptor site mutation in TaGW2-A1 increases thousand grain weight in tetraploid and hexaploid wheat through wider and longer grains[END_REF][START_REF] Brinton | Ubiquitin-related genes are differentially expressed in isogenic lines contrasting for pericarp cell size and grain weight in hexaploid wheat[END_REF][START_REF] Golan | GNI-A1 mediates trade-off between grain number and grain weight in tetraploid wheat[END_REF]. Moreover, numerous quantitative trait loci (QTLs) for grain weight have been identified (e.g., [START_REF] Breseghello | QTL analysis of kernel size and shape in two hexaploid wheat mapping populations[END_REF][START_REF] Gegas | A Genetic Framework for Grain Size and Shape Variation in Wheat[END_REF][START_REF] Farré | Application of a library of near isogenic lines to understand context dependent expression of QTL for grain yield and adaptive traits in bread wheat[END_REF][START_REF] Simmonds | A splice acceptor site mutation in TaGW2-A1 increases thousand grain weight in tetraploid and hexaploid wheat through wider and longer grains[END_REF]. However, for an increase in AGW to translate into a yield increase, requires a better understanding of how the mechanisms involved in grain weight determinism [START_REF] Brinton | A reductionist approach to dissecting grain weight and yield in wheat[END_REF] and, in particular, of those allowing avoidance of the critical trade-off between GNM2 and AGW under variations in both environment and genotype [START_REF] Slafer | Fruiting efficiency: an alternative trait to further rise wheat yield[END_REF][START_REF] Slafer | Source-sink relationships and grain mass at different positions within the spike in wheat[END_REF]. This trade-off could arise from limitations imposed by the source organs and/or by the sink organs.

A source limitation (i.e., competition for assimilates between sinks) could occur either before or after anthesis. Before anthesis, reduced assimilate availability can restrict individual grain ovary size, generating a physical constraint to growth [START_REF] Ugarte | Grain weight and grain number responsiveness to pre-anthesis temperature in wheat, barley and triticale[END_REF]) and thus reducing final grain size [START_REF] Calderini | Effect of temperature and carpel size during pre-anthesis on potential grain weight in wheat[END_REF][START_REF] Calderini | The Importance of the Period Immediately Preceding Anthesis for Grain Weight Determination in Wheat[END_REF][START_REF] Calderini | Overcoming the trade-off between grain weight and number in wheat by the ectopic expression of expansin in developing seeds leads to increased yield potential[END_REF]. After anthesis, during the grain filling phase, a grain's final size may also be constrained by its share of a limited amount of assimilate -competition between grains [START_REF] Sinclair | Grain number, wheat yield, and bottling beer: An analysis[END_REF]. Even though the exact nature on the relationship is uncertain [START_REF] Calderini | Overcoming the trade-off between grain weight and number in wheat by the ectopic expression of expansin in developing seeds leads to increased yield potential[END_REF] the idea that source limitation should affect final grain size through a limitation in ovary size seems unlikely, since the relation between ovary size and final grain size appears mainly to be correlative, not causative [START_REF] Benincasa | The relationship between grain and ovary size in wheat: An analysis of contrasting grain weight cultivars under different growing conditions[END_REF]. On the other hand, source limitation during grain filling has usually been dismissed as a main driver of average final grain size, at least in temperate environments (Borras et al., 2004;[START_REF] Borrill | Wheat Grain Filling Is Limited by Grain Filling Capacity rather than the Duration of Flag Leaf Photosynthesis: A Case Study Using NAM RNAi Plants[END_REF][START_REF] Serrago | Understanding grain yield responses to source-sink ratios during grain filling in wheat and barley under contrasting environments[END_REF]Elias et al., 2016 and references therein;[START_REF] Reynolds | A wiring diagram to integrate physiological traits of wheat yield potential[END_REF].

The idea of a trade-off originating from sink limitation, is based on the observation that an increase in GNM2 usually leads to a higher proportion of grains of low potential weight [START_REF] Acreche | Grain weight response to increases in number of grains in wheat in a Mediterranean area[END_REF][START_REF] Chen | The Proportion of Superior Grains and the Sink Strength are the Main Yield Contributors in Modern Winter Wheat Varieties Grown in the Loess Plateau of China[END_REF][START_REF] Feng | Contribution of Proximal and Distal Grains Within Spikelets in Relation to Yield and Yield Components in the Winter Wheat Production Region of China From 1948 to 2012[END_REF]. The higher proportion of small grains could be driven by processes occurring at two different levels. First, it could result from an increase in the number of secondary tillers and, consequently, an increase in the proportion of small spikes, that carry smaller grains [START_REF] Xu | Physiological basis for the differences of productive capacity among tillers in winter wheat[END_REF] and, second, it could result from an increase in the number of grains per spike. These additional grains may be located in the basal or the apical spikelets, or in the more distal positions within each spikelet. All these locations are associated with lower potential weights and, thus, will tend to contribute to an increase in the proportion of small grains within the spike (e.g., [START_REF] Baillot | Different grain-filling rates explain grain-weight differences along the wheat ear[END_REF][START_REF] Calderini | Grain Position Affects Grain Macronutrient and Micronutrient Concentrations in Wheat[END_REF][START_REF] Ferrante | Yield determination, interplay between major components and yield stability in a traditional and a contemporary wheat across a wide range of environments[END_REF][START_REF] Xie | Carpel size, grain filling and morphology determine individual grain weight in wheat[END_REF]. These relationships seem to offer the most plausible explanation for the apparent trade-off produced by the introgression of Rht dwarfing genes (Miralles et al., 1995a). Both the processes described above, would lead to a higher proportion of small grains and so will be referred to below as 'populational' effects. The important roles of these populational effects on the trade-off between GNM2 and AGW is supported by a large number of studies [START_REF] Acreche | Grain weight response to increases in number of grains in wheat in a Mediterranean area[END_REF][START_REF] Chen | The Proportion of Superior Grains and the Sink Strength are the Main Yield Contributors in Modern Winter Wheat Varieties Grown in the Loess Plateau of China[END_REF][START_REF] Feng | Contribution of Proximal and Distal Grains Within Spikelets in Relation to Yield and Yield Components in the Winter Wheat Production Region of China From 1948 to 2012[END_REF][START_REF] Ferrante | Yield determination, interplay between major components and yield stability in a traditional and a contemporary wheat across a wide range of environments[END_REF][START_REF] García | Grain Yield Potential Strategies in an Elite Wheat Double-Haploid Population Grown in Contrasting Environments[END_REF][START_REF] Li | Plant Density Effect on Grain Number and Weight of Two Winter Wheat Cultivars at Different Spikelet and Grain Positions[END_REF]Miralles and Slafer, 1995b;[START_REF] Quintero | Trade-off between grain weight and grain number in wheat depends on GxE interaction: A case study of an elite CIMMYT panel (CIMCOG)[END_REF].

Given the importance of the trade-off, it appears necessary to distinguish between the part of the AGW genetic determinism that occurs via the populational effects described above and the part having a constitutive nature that affects each grain individually. Most quantitative genetic studies focusing on AGW do not explicitly consider the trade-off between GNM2 and AGW (e.g., [START_REF] Gegas | A Genetic Framework for Grain Size and Shape Variation in Wheat[END_REF][START_REF] Brinton | Ubiquitin-related genes are differentially expressed in isogenic lines contrasting for pericarp cell size and grain weight in hexaploid wheat[END_REF]. The analytical framework of only one recent study seeks to take this trade-off into account [START_REF] Quintero | Trade-off between grain weight and grain number in wheat depends on GxE interaction: A case study of an elite CIMMYT panel (CIMCOG)[END_REF]. This suggests the genetic determinism of individual grain weight is at least partly independent of the populational effects. However, their demonstration is incomplete: (i) because the study was restricted to particular spike and grain models (i.e., proximal grain (G2) and distal grain (G4) sampled on the middle spikelet of five median spikes per experimental treatment) and (ii) because neither the number of grains per spike/spikelet nor the proportions of small vs. large grains were explicitly considered.

Thus, the aim of our study was to confirm the existence of a constitutive variability in AGW due only to genetic determinism, working with different well-characterised spike and grain populations. This requires we characterise the precise distribution of individual grain weights and their association with AGW at (1) plot scale, (2) spike scale and (3) single-grain scale. With this aim, we compared four modern, well-adapted, high-yielding wheat cultivars, that express robust differences in AGW. To help ensure that any differences we might observe in AGW and its putative determinism were not specific to any particular cultivar × environment interactions, our analyses were carried out under two contrasting environmental conditions, (a) water-sufficient and (b) waterdeficient.

Materials and methods

Field experiment

Choice of cultivars

Four cultivars expressing either low AGW (hereafter LCvs) or high AGW (hereafter HCvs) were chosen from among a range of registered, modern, awned and high-yielding winter-wheat cultivars. This choice was made by analysing a dataset compiled from a large number of post-registration field trials (J.-C. Deswartes, P. du Cheyron, Arvalis-Institut du végétal, personal communication, 2016) in which the mean yield components were calculated for each cultivar. Additional precautions were taken to minimise phenological differences between cultivars. However, it was not possible to identify cultivars that were within the exact same precocity class and still commercially available. Therefore, we selected Altigo (Nickerson, 2007) and Solehio (Momont, 2009) that exhibit higher AGWs and lower grain numbers per m² than Cellule (Florimond Desprez, 2012) and SY Moisson (Syngenta, 2012) (Table 1). and Z69) and one insecticide application (at Z69).

In the four water-deficit blocks, the rain-out shelters of the Pheno3C facility were set in action from 1 March 2018 to divert all incoming rainfall, except for one 17 mm rain event occurring during the last week of April. From sowing to harvest, the well-watered and water-deficit plots received 396 and 175 mm of rainfall, respectively. Calculated from the start of the drought treatment, cumulative rainfall was 238 and 17 mm in the well-watered and water-deficit blocks, respectively (Fig. 1A). In the water-deficit blocks, the water stress was well established from the start of April, and deepened in intensity thereafter until harvest (Fig. 1B). The well-watered blocks were maintained at the onset of water stress (or above) by irrigation. We emphasise that the aim of the study was not to analyse responses to water deficit but merely to create two contrasting background environments to explore the consistency of any differences that might appear in AGW and its determinants, so bringing a greater degree of confidence and robustness to our conclusions.

Plant sampling and measurements

Within each plot, the number of spikes per m² (SNM2) was determined after anthesis, at about anthesis + 500°C days (base 0°C) by counting all spikes within two, 100 cm long, transects in the two central rows.

At physiological maturity, in July 2018, 100 spikes per plot were sampled from the central 1.5 m of the two central rows to avoid possible edge effects. To avoid bias in the selection of spikes, the 100 spikes were sampled as four separate samples of 25 adjacent spikes. All spikes were oven-dried to constant weight (60°C, 48 h) and for each individual spike, average grain weight (AGW spike ) and the number of grains per spike (NGS spike ) were determined with an automatic seed-weighing/-counting device (Opto Agri2, Optomachine, France). The weight of each individual grain within each spike was then measured with an automatic seed-weighing device developed in our laboratory. This device comprises a six-axis robotic arm (Yaskawa MH3, Yaskawa, Japan) integrated with a precision weight module (WMC15-SH, Mettler-Toledo, Switzerland).

Additional variables were calculated at the plot level. Mean number of grains per spike per plot (NGS plot ) and AGW per plot (AGW plot ) were calculated as the means of the 100 spikes of the sample.

GNM2 was estimated as the product of SNM2 and NGS plot and grain yield (GY) was calculated as the product of GNM2 and AGW plot .

Statistical analyses

Results were analysed at different scales depending on the variable considered. Cultivar effects on mean yield components (SNM2, NGS plot , GNM2, AGW plot and GY) were analysed using mean variable values at the plot scale. Average grain weight per spike (AGW spike ) and the individual grain weight per spike were analysed using individual spike values considering the nesting effect of plots.

All analyses were carried out separately for the two environmental conditions.

Cultivar effects on mean yield components at plot scale

Within each environmental condition, comparison of the mean yield components (SNM2, NGS plot , GNM2, AGW plot , GY) between cultivars was carried out using Tukey's post-hoc test following an ANOVA. For each environmental condition, the ANOVA model corresponded to a randomised complete block design with cultivar and block as fixed effects (n=4):

Y = μ + X 1 Cv + X 2 B + ϵ (Eq.1)
where Y is the vector of each mean yield component value, µ is the mean effect, Cv the fixed effect of each cultivar, B the fixed effect of the block, X i the incidence matrix and is the residual ϵ following a normal distribution (0, σ² e ).

Cultivar effects on average grain weight (AGW spike ) at spike scale

To compare the differences in average grain weight per spike (AGW spike ) between cultivars at spike scale, a linear model was developed including the effect of the covariable NGS spike . The experimental design was also considered. To test the effect of the AGW genotypic level on AGW spike , the four cultivars were pooled into two groups: HCvs (i.e., Altigo and Solehio) and LCvs (i.e., Cellule and SY Moisson).

The following linear model was then used with the package ASREML [START_REF] Gilmour | Average Information REML: An Efficient Algorithm for Variance Parameter Estimation in Linear Mixed Models[END_REF]:

Y = μ + X 1 Cov + X 2 G + X 3 Cv + X 4 B + X 5 Cov×G ×B + ϵ (Eq.2)
where Y is the vector of average grain weight per spike (AGW spike ) values, µ is the intercept, Cov the fixed effect of the covariable (NGS spike ), G the fixed effect of the group of cultivars, Cv the fixed effect of each cultivar nested within each group of cultivars, B the fixed effect of the block, X i the incidence matrix and is the residual following a normal distribution (0, σ² ϵ e ). A Wald test was computed on this linear model to estimate the sizes of the different effects. Based on coefficients calculated with the linear model, slopes and intercepts were calculated for each cultivar and under each environmental condition.

Cultivar effects on individual grain weight

To analyse the putative populational effects associated with AGW differences requires going beyond just the differences in means, to try to understand how the distributions actually differ.

Comparing quantiles is well suited to this purpose as they provide informative, robust and intuitive information [START_REF] Rousselet | Beyond differences in means: robust graphical methods to compare two groups in neuroscience[END_REF]. [START_REF] Rousselet | Beyond differences in means: robust graphical methods to compare two groups in neuroscience[END_REF] proposed plotting a 'shift function' that represents the differences between the quantiles of the distributions of two groups as a function of the quantiles of one of the groups. Quantiles are estimated with the Harrell-Davis estimator (Harrell andDavis, 1982 in Wilcox, 1995) and then quantiles differences between groups and their associated confidence interval are estimated using a bootstrap method (as described in [START_REF] Rousselet | Beyond differences in means: robust graphical methods to compare two groups in neuroscience[END_REF]. In our study we extended this approach to provide a graphical representation of each cultivar distribution to compare with the distribution of a pooled population comprising all of the four cultivars. This offered a visual representation of the cultivar differences in spike size (NGS spike ) and individual grain weight. The same approach was used to compare individual grain weight distribution of the two groups of cultivars (HCv and LCv).

To complete the graphical comparison, we carried out an analysis using the deciles as a categorical variable with the following linear model using the package ASREML [START_REF] Gilmour | Average Information REML: An Efficient Algorithm for Variance Parameter Estimation in Linear Mixed Models[END_REF]:

Y = μ + X 1 GrD + X 2 G + X 3 Cv + X 4 B + X 5 GrD×G×Cv×B + ϵ (Eq. 4)
where Y is the vector of individual grain weight values, µ is the intercept, GrD the fixed effect of the grain decile, G the fixed effect of the group of cultivars, Cv the fixed effect of each cultivar nested within each group of cultivars, B the fixed effect of the block, X i the incidence matrix and ϵ is the residual following a normal distribution (0, σ² e ). A Wald test was computed on this linear model to estimate the sizes of the different effects.

All statistical analyses were carried out with R statistical software (R version 3.5.0, R Development Core Team ( 2005)).

Results

The four cultivars achieved similar grain yields through different combinations of AGW per plot and GNM2

Within each environmental condition (well-watered vs water-deficit), there were no differences in grain yield among the four cultivars (Table 2). These results agree with those of the multi-environment trials used to select the cultivars for the experiment (Table 1). As expected, water-deficit led to a decrease in grain yield (~29% averaged among the four cultivars; Table 2).

The expected small differences in cultivar precocity (Table 1) were also observed. Under wellwatered conditions, the anthesis date difference between the earliest and the latest cultivars was only four days (Table 2). Under water-deficit conditions, anthesis dates of the four cultivars occurred over a seven-day period (Table 2). The homogeneity among cultivars for yield, masked strong differences in the individual yield components between cultivars. Again, as expected from the cultivar choices, strong differences in AGW between cultivars were observed irrespective of the environmental conditions.

Based on AGW plot , under well-watered conditions, two classes of cultivars could be distinguished:

on the one hand Altigo and Solehio (HCvs) had an average AGW plot of ~51 mg grain -1 and on the other hand Cellule and SY Moisson (LCvs) had an average AGW plot of ~41 mg grain -1 (Table 2).

This distinction was emphasised under the water-deficit conditions with AGW plot averages of ~48.7 mg grain -1 for HCvs and ~39.0 mg grain -1 for LCvs (Table 2).

Given the trade-off between AGW and GNM2, strong differences in GNM2 were also expected between genotypes. Even though exhibiting a tendency, these differences in GNM2 were not always significant within the groups created on a AGW plot basis. For example, within the wellwatered experimental conditions, Solehio (HCv) and Cellule (LCv) showed similar values of GNM2 but different ones of AGW plot . Similarly, under water-deficit condition, SY Moisson (LCv)

showed GNM2 values not significantly different from those of Altigo or Solehio (HCvs) (Table 2).

No clear trend was observed for the source of cultivar variation in GNM2. The differences could be attributed either to genotypic variation of NGS at plot scale (NGS plot ) that varied between 34.5 and 40.5 grains per spike under well-watered conditions or to SNM2 depending on the genotype and the environmental conditions (Table 2).

Therefore, with similar grain yields but contrasting AGW plot , NGS plot and SNM2 under both environmental conditions, the four cultivars allow exploration of the main sources of variability in cultivar AGW plot and, in particular, the putative populational effects on variability of cultivar AGW.

AGW spike cultivar differences were not explained by differences in the distributions of spike size

To explore spike population effects on AGW our methodology requires information at the individual spike level. To this end, 100 individual spikes were sampled for each experimental unit, and NGS spike and AGW spike were determined for each spike sampled. This information obtained at the single spike scale allows us to study the contributions of spike population to cultivar AGW plot following a two-step evaluation. First, the distributions of spike size between cultivars were compared using NGS spike . Second, we examined whether between-spike variations in NGS spike significantly affected AGW spike . For the well-watered condition, all distributions of NGS spike were close to symmetrical (i.e., exempt from strong negative skewness) and were relatively similar except Cellule, for which NGS spike values were shifted towards higher values (Fig. 2A) for all deciles (Fig. 2B). Regardless of Cellule specificity, no link between SNM2 differences and NGS spike distributions could be inferred.

For example, SNM2 was 32% higher in Altigo (HCv) than in SY Moisson (LCv) (Table 2) whereas the NGS spike distributions of these two cultivars were nearly identical (Fig. 2A).

Similar conclusions can be drawn under the water-deficit condition (Fig. 2C and2D). The distributions of NGS spike were very similar for the four cultivars with only a small shift of Altigo towards smaller spikes. This small difference does not appear to be correlated with NSM2, as differences among the genotypes were marginal and not significant (Table 2). There were no relationships (Fig. 3A) between AGW spike and NGS spike for the well-watered conditions and only weak positive (Fig. 3B) ones for the water-deficit conditions. For both conditions, an analysis of variance (Table 3) reveals that the differences in AGW spike are explained mainly by the a priori classification of the cultivars into HCv or LCv with 61.3% of total variance explained for the well-watered and 47.9% for the water-deficit conditions. Although highly significant, the other effects never explained more than 4.2% of the variance in AGW spike (Table 3).

In particular, the interaction term between NGS spike and the group of cultivars (NGS×G) explained only 0.4 and 0.2% under well-watered and water-deficit conditions, respectively. Taken together, these results indicate that the cultivar differences in AGW spike were not due to populational effects at spike level.

Cultivar differences in individual grain weight were observed, irrespective of grain decile

The second populational effect proposed as a possible driver of cultivar differences in AGW plot , occurs within the spike, at the individual grain level. Under this assumption, cultivars with larger spikes (higher NGS) will present higher proportions of small grains within spikes and this will negatively affect AGW plot . Assessing individual grain weight distributions per spike is made difficult due to (i) the limited number of grains within a spike and (ii) the large variation in the numbers of grains between spikes. Given the absence of clear a relationship between NGS spike and AGW spike , the individual grain weight distribution was assessed only at the whole population level.

As expected, given that the choice of the four cultivars was based on their a priori differences in AGW, clear differences in individual grain weight could be observed between Altigo and Solehio (HCv cultivars) on the one hand and SY Moisson and Cellule (LCv cultivars) on the other hand under both environmental conditions (Fig. 4A and4D). All distributions were negatively skewed, with an excess of small grains. The shift function allowed comparison of the individual grain weight decile values, on the one hand of the whole-grain population made by pooling the four cultivars and, on the other hand, with each cultivar independently. This representation allows to visualisation of both the strong differences between the HCv and the LCv cultivars and of the relative homogeneity of the cultivars within each of these groupings, in spite of the divergent behaviours of the first four deciles (Fig. 4C and4F) that relate to small skewness differences among cultivars. Individual grain weight differences between the HCv and LCv cultivars are even more striking when averaged per group of cultivars. Under both conditions, individual grain weight from all deciles is higher for HCv than for LCv, regardless of the decile considered (Fig. 5A and5C).

Remarkably, these differences between cultivar groups tended to increase with grain weight (Fig. 5B and5D) and these relationships were very similar under both environmental conditions. The conclusions drawn from the graphical analysis of the shift function are supported by an analysis of variance (Table 4). Indeed, apart from the large effect of grain decile, the main factor affecting individual grain weight was the group of cultivars (14.2 for the well-watered conditions and 16.9% for the water-deficit conditions). This result is independent of grain weight decile, as indicated by the small interaction term (<1% of the variance explained in both environmental conditions). In other words, HCvs had higher AGW than LCvs due to their having constitutively heavier grains.

Discussion

Relevance of the selected cultivars to exploring the grain number/AGW relationship

Most studies comparing wheat cultivars and showing differences in grain number and AGW have focused on the dichotomy between old cultivars and new ones, or between tall cultivars and semi-dwarf ones [START_REF] Chen | The Proportion of Superior Grains and the Sink Strength are the Main Yield Contributors in Modern Winter Wheat Varieties Grown in the Loess Plateau of China[END_REF][START_REF] Ferrante | Yield determination, interplay between major components and yield stability in a traditional and a contemporary wheat across a wide range of environments[END_REF]Miralles and Slafer, 1995b), or they have explored cultivars showing wide-ranging yield potentials (e.g., [START_REF] Peltonen-Sainio | Grain number dominates grain weight in temperate cereal yield determination: Evidence based on 30 years of multi-location trials[END_REF][START_REF] Quintero | Trade-off between grain weight and grain number in wheat depends on GxE interaction: A case study of an elite CIMMYT panel (CIMCOG)[END_REF]. All these studies, together with an abundance of studies that have focused on yield responses to different management practices (e.g., [START_REF] Acreche | Grain weight response to increases in number of grains in wheat in a Mediterranean area[END_REF][START_REF] Feng | Contribution of Proximal and Distal Grains Within Spikelets in Relation to Yield and Yield Components in the Winter Wheat Production Region of China From 1948 to 2012[END_REF][START_REF] Fischer | Irrigated spring wheat and timing and amount of nitrogen fertilizer. II. Physiology of grain yield response[END_REF][START_REF] Li | Plant Density Effect on Grain Number and Weight of Two Winter Wheat Cultivars at Different Spikelet and Grain Positions[END_REF] have concluded that yield variations are mainly related to variations in GNM2 (e.g., Slafer et al., 2014 and references therein). Given the negative relationship that commonly exists between GNM2 and AGW [START_REF] Slafer | Coarse and fine regulation of wheat yield components in response to genotype and environment[END_REF][START_REF] Slafer | Source-sink relationships and grain mass at different positions within the spike in wheat[END_REF], studies that rely on a high variability in GNM2, only highlight the dominant effect of GNM2 on AGW. Therefore, this limits their ability to determine the component of AGW genetic variation, that can be attributed to constitutive genetic differences.

One way to resolve this problem is to drastically restrict the range of variation in GNM2, so as to allow a focus on AGW. In the studies cited above [START_REF] Ferrante | Yield determination, interplay between major components and yield stability in a traditional and a contemporary wheat across a wide range of environments[END_REF][START_REF] Peltonen-Sainio | Grain number dominates grain weight in temperate cereal yield determination: Evidence based on 30 years of multi-location trials[END_REF][START_REF] Quintero | Trade-off between grain weight and grain number in wheat depends on GxE interaction: A case study of an elite CIMMYT panel (CIMCOG)[END_REF], the range of values of GNM2 varied by more than ten-fold across the cultivars and/or conditions they tested. For example, [START_REF] Ferrante | Yield determination, interplay between major components and yield stability in a traditional and a contemporary wheat across a wide range of environments[END_REF] observed GNM2 values that varied between 2 500 and 25 000, [START_REF] Peltonen-Sainio | Grain number dominates grain weight in temperate cereal yield determination: Evidence based on 30 years of multi-location trials[END_REF] between 2 300 and 19 000 and [START_REF] Quintero | Trade-off between grain weight and grain number in wheat depends on GxE interaction: A case study of an elite CIMMYT panel (CIMCOG)[END_REF] between 11 000 and 25 000 grains m -2 . In our study, the range in GNM2 was much narrower i.e., between 13 000 and 17 000 grains m -2 under well-watered conditions and between 10 000 and 13 000 grains m -2 under water-deficit conditions (Table 2). Nevertheless, significant differences in AGW were still evident (Table 2). This suggests the observed differences in AGW were not driven solely by GNM2 differences. The four modern, high-yielding winter bread wheat cultivars we used, expressed consistently high (Altigo and Solehio) and low (Cellule and SY Moisson) values of AGW. However, these strong differences in yield components are also accompanied by earliness differences between LCv and HCv cultivars (Table 2). Earliness differences between cultivars are suspected to have a strong confounding effect on the establishment of yield components, in particular when developmental differences generate differences in exposure to climatic stresses. In our experiment, no acute uncontrolled abiotic stress event occurred during the crop cycle (Fig. 1, Fig. S1). Therefore, we do not believe that the limited precocity differences observed will have caused significant differences in yield component establishment.

We thus conclude that the four cultivars we study here will allow us to determine if a proportion of the determination of the AGW differences could be constitutive. In addition, the two contrasting growing conditions (well-watered and water-deficit) will allow us to determine if our conclusions are reasonably independent of the growing environment. In this case it will allow a degree of generalisation for our results.

AGW cultivar differences are poorly explained by populational effects reflecting their constitutive nature

The results of numerous studies published in the last few years support the idea that a large portion of the trade-off between GNM2 and AGW originates from populational effects. In other words, an increase in GNM2 will be associated with an increase in the relative contributions of grains having a lower weight potential than the rest of the grain population [START_REF] Acreche | Grain weight response to increases in number of grains in wheat in a Mediterranean area[END_REF]Miralles and Slafer, 1995a) and this will directly and negatively affect AGW.

From the putative populational effects suggested above, the one acting at the spike population level has often been mentioned but rarely studied directly. [START_REF] Xu | Physiological basis for the differences of productive capacity among tillers in winter wheat[END_REF] showed that tillers of higher order, produce spikes with lower grain numbers per spike and lower AGW. In particular, spikes from the first primary tillers have (depending on genotype) between 4 and 12% less grains per spike than spikes from the main stem. This effect is even stronger for tillers of higher order. The same observation was highlighted by [START_REF] Thorne | Effects of Radiation and Temperature on Tiller Survival, Grain Number and Grain Yield in Winter Wheat[END_REF] with a much higher (28%) reduction in NGS in tillers compared to main stems. Such results imply that any increase in SNM2 made through an increased proportion of tillers (of any order) would induce a higher proportion of smaller spikes in the population. In the present study, the number of grains per spike (NGS spike ) of individual spikes was recorded but information on the tiller order that produced these spikes was not. Nevertheless, given that the plant density at sowing under our conditions (250 plants m -2 ) and the observed SNM2, are within the range 344-485 spikes m -2 (Table 2), our spike populations would have been composed mainly of spikes originating from the main stem with the calculated number of spikes per plant within the narrow range of 1.4 to 1.9 spikes per plant, depending on genotype and condition. The low SNM2 observed in our experiment probably explains the absence of clear genotypic differences in the NGS distribution (Fig. 2). This absence of populational effect observed at the spike level is likely to explain the absence of a relationship between NGS and AGW spike in spite of the large range of NGS spike observed under both conditions and in all genotypes (Fig. 3). This implies that the hypothesis that spike populational effects on AGW probably relies on the effects of tiller order on AGW, rather than on a direct effect of NGS spike on AGW. To fully validate this proposition, the characterisation of both the individual NGS (such as in the present experiment) and the order of the tillers carrying each spike (as made by [START_REF] Xu | Physiological basis for the differences of productive capacity among tillers in winter wheat[END_REF][START_REF] Thorne | Effects of Radiation and Temperature on Tiller Survival, Grain Number and Grain Yield in Winter Wheat[END_REF][START_REF] Thorne | Effects of Radiation and Temperature on Tiller Survival, Grain Number and Grain Yield in Winter Wheat[END_REF] would require additional experimental work.

The second populational effect suggested in the literature having a putative effect on AGW, operates at the within-spike level. From this point of view, increasing grain number per m 2 through an increase in NGS is accompanied by a higher proportion of small grains positioned in basal or apical spikelets within the spike (e.g., [START_REF] Feng | Contribution of Proximal and Distal Grains Within Spikelets in Relation to Yield and Yield Components in the Winter Wheat Production Region of China From 1948 to 2012[END_REF][START_REF] Ferrante | Yield determination, interplay between major components and yield stability in a traditional and a contemporary wheat across a wide range of environments[END_REF]Miralles and Slafer, 1995a). It is well established that, within spikelets, the grains in distal florets are smaller than in basal florets (e.g., [START_REF] Baillot | Different grain-filling rates explain grain-weight differences along the wheat ear[END_REF]Miralles and Slafer, 1995a;[START_REF] Rawson | The Pattern of Grain Growth within the Ear of Wheat[END_REF]. Taken together both relationships lead to a decrease in AGW spike [START_REF] Acreche | Grain weight response to increases in number of grains in wheat in a Mediterranean area[END_REF].

A problem is the estimation of the proportion of small grains within a population of grains is not straightforward, as the weight threshold criterion for 'small' is necessarily arbitrary. An alternative used by [START_REF] Acreche | Grain weight response to increases in number of grains in wheat in a Mediterranean area[END_REF] is to define the proportion of small grains as the ratio between the number of grains located in distal positions as a proportion of the total number of grains in the spike. However, this method does not account for actual individual grain weights and can be misleading as an additional grain in a distal position on a central spikelet is not necessarily smaller than a proximal grain in a more basal or apical spikelet, as suggested by the results of [START_REF] Feng | The Effect of Grain Position on Genetic Improvement of Grain Number and Thousand Grain Weight in Winter Wheat in North China[END_REF]Fig. 4).

Therefore, we used a method based on an analysis of individual grain weight deciles to compare cultivars. Within this analytical framework, the hypothesis of cultivar AGW differences driven by populational effects should result in higher weight differences between cultivars for the first deciles than for the last deciles. Our results clearly don't support this hypothesis as cultivar differences in individual grain weight tend to increase with deciles in a similar fashion under the two environmental conditions (Fig. 5, Table 4). In other words, the LCv cultivars do not have lower AGWs than the HCv cultivars due to an increased proportion of small grains but because all grains are smaller, in particular the larger ones.

In conclusion, we note that our data did not expose either of the two populational effects previously suggested as the main determinants of AGW difference between cultivars. This does not mean that such populational effects never occur (they would likely dominate the trade-off between GNM2 and AGW when large changes in GNM2 are produced) but that a component of the AGW differences observed between cultivars has a constitutive nature, as already suggested by [START_REF] Quintero | Trade-off between grain weight and grain number in wheat depends on GxE interaction: A case study of an elite CIMMYT panel (CIMCOG)[END_REF]. However, these authors' conclusions were based exclusively on the observation that both the proximal and distal grains (G2 and G4) of the two central spikelets were larger in the high AGW cultivars than in the low AGW cultivars. Thus, their analyses did not account for the diversity observed either in the spike population (variation in spike grain number distribution) or in the grain population caused by variation in spikelet number per spike. Our study explicitly evaluated the populational effects and showed that these were extremely weak and thus not sufficient to explain the cultivar differences in AGW. The constitutive nature of grain size difference has thus been highlighted for all grains (Fig. 5), independently of their weights and independently of the spike from which they originated. In agreement with [START_REF] Calderini | Overcoming the trade-off between grain weight and number in wheat by the ectopic expression of expansin in developing seeds leads to increased yield potential[END_REF], our conclusion implies that AGW can to some extent be manipulated independently of grain number.

Towards new targets for grain-weight breeding

It is widely accepted that the grain yield increases achieved in recent decades have come about mainly through increases in grain number per m². Meanwhile, AGW has been increased only marginally over the same period (e.g., [START_REF] Brancourt-Hulmel | Genetic Improvement of Agronomic Traits of Winter Wheat Cultivars Released in France from 1946 to 1992[END_REF][START_REF] Philipp | Grain number and grain yield distribution along the spike remain stable despite breeding for high yield in winter wheat[END_REF][START_REF] Shearman | Physiological Processes Associated with Wheat Yield Progress in the UK[END_REF] and has been selected principally for its effects on grain quality, rather than on grain yield [START_REF] Philipp | Grain number and grain yield distribution along the spike remain stable despite breeding for high yield in winter wheat[END_REF][START_REF] Xie | Pleiotropic effects of the wheat domestication gene Q on yield and grain morphology[END_REF]. The existence of a significant part of AGW genetic variation that can be attributed to constitutive cues (i.e., independent of variations in grain number) offers new avenues for manipulating yield. First, gene manipulation (in particular, expansin gene manipulation) has been proposed as a way to increase GW (Lizana et al., 2010), without having a negative effect on GN, and so increase GY [START_REF] Calderini | Overcoming the trade-off between grain weight and number in wheat by the ectopic expression of expansin in developing seeds leads to increased yield potential[END_REF]. Second, this approach could rely on the QTLs already identified for grain weight (e.g., Yan et al., 2017), and especially on those that have been reported to be independent of the trade-off between AGW and grain number [START_REF] Griffiths | Genetic Dissection of Grain Size and Grain Number Trade-Offs in CIMMYT Wheat Germplasm[END_REF]. Nevertheless, such QTLs should be used with care since the AGW quantification on which they rely were potentially compromised by populational effects that negatively influences the mean AGW. Using a methodology that provides access to single grain weights (as does the present study) should allow application of GWAS approaches, focusing specifically on the mean weights of the larger grains (defined, for example, as the 25% of larger grains). Our results suggest that this particular grain sub-population is unaffected by putative populational effects. This approach should be validated, independently of the target environment.

Fig. 1 .

 1 Fig.1. Dynamics of soil water reserve (A) and cumulative rainfall from sowing date (B) for the well-watered (blue line) and water-deficit (red line) conditions. Rectangles indicate the maximum range of observed anthesis dates for the four cultivars under well-watered (pale blue) and water-deficit (pink) conditions. Horizontal lines indicate the soil water reserves corresponding to field capacity, easily accessible water and permanent wilting point.

Fig. 2 :

 2 Fig. 2: Distributions of the grain number per spike (NGSspike) of the four cultivars under well-watered (A, B) and waterdeficit (C, D) conditions. Top panels (A, C) represent the density distribution of NGSspike per cultivar. Distribution parameters (mean, variance, skewness) for each condition and each cultivar: (A): Altigo (mean= 35.85, variance= 85.00, skewness= -0.02); Solehio (34.50, 105.36, 0.01); Cellule (40.53, 134.67, 0.06); SY Moisson (35.47, 92.11, -0.04); (D): Altigo (mean= 28.71, variance= 64.95, skewness= 0.24); Solehio (30.17, 81.29, -0.11); Cellule (33.11, 76.79, 0.02); SY Moisson (31.28, 115.18, 0.33). Bottom panels (B, D) show the decile values of NGSspike distributions for the four cultivars and the whole spike population pooling the four cultivars (All). Figures in the bottom panels stand for the values of the first decile, median and last decile.

Fig. 3 .

 3 Fig.3. Relationship between NGSspike (grains spike -1 ) and AGWspike (mg grain -1 ) for each cultivar under well-watered (A) and water-deficit conditions (B). For each condition, slope and intercept are calculated following the model described in the Material and Methods section: (A) Altigo (slope= -0.018; intercept= 52.65); Solehio (-0.018, 50.55); Cellule (0.051, 38.48); SY Moisson (0.051, 40.75); (B) Altigo (slope= 0.21, intercept= 42.82); Solehio (0.21, 42.34); Cellule (0.13, 34.13); SY Moisson (0.13, 36.25). The R² of the models are 0.67 (A) and 0.56 (B). Altigo and Solehio are High-AGW cultivars (HCv); Cellule and SY Moisson are Low-AGW cultivars (LCv).

Fig. 4 :

 4 Fig. 4: Distributions of the individual grain weight (mg) of the four cultivars under well-watered (A, B, C) and waterdeficit (D, E, F) conditions. Top panels (A, D) represent the density distribution of individual grain weight per cultivar. Distribution parameters (variance, skewness) for each condition and each cultivar: (A) Altigo (mean= 51.79, variance= 156.91, skewness= -0.69); Solehio (49.91, 178.58, -0.85); Cellule (40.60, 122.71, -0.73); SY Moisson (42.78, 83.04, -0.68); (D) Altigo (mean= 49.16, variance= 125.47, skewness= -0.56); Solehio (49.34, 136.68, -0.40); Cellule (38.96, 98.21, -0.65); SY Moisson (40.38, 70.07, -0.44). Middle panels (B, E) show the deciles values of individual grain weight distributions for the four cultivars and the whole grain population mixing the four cultivars (All). Numbers aregiven for the first decile, median and last decile. Bottom panels represent the shift function i.e., the differences between the deciles of each cultivar from the deciles of whole population.

Figure 5 :

 5 Figure 5: Mean individual grain weight per decile for two groups of cultivars HCv and LCv under well-watered (A, B) and water-deficit (C, D) conditions. HCv are Altigo and Solehio; LCv are Cellule and SY Moisson. Top panels (A, C) represent the mean individual grain weight per decile per group. Bottom panels represent the shift function i.e., the differences between the deciles of the HCv cultivars and the deciles of the LCv cultivars as a function of the HCv deciles. The numbers indicate the differences (mg) for each decile.

Table 1 .

 1 Average yield components for selected winter bread wheat cultivars in multi-environment field trials (mean ± Standard Deviation)

	Yield component	Altigo	Solehio	Cellule	SY Moisson
	Number of trials	56	135	126	57
	Years	2007 -2014	2009 -2017	2012 -2018	2012 -2018
	Spike number per m²	494 ± 92	487 ± 72	555 ± 83	560 ± 86
	Grain number per spike	37 ± 9	39 ± 5	43 ± 6	43 ± 6
	Grain number per m²	17607 ± 2729	18732 ± 2688	23473 ± 3269	23740 ± 2831
	Average grain weight (mg grain -1 ) Grain Yield (t ha -1 at 15% humidity)	52 ± 4 9.1 ± 1.4	50 ± 5 9.3 ± 1.3	41 ± 4 9.6 ± 1.4	42 ± 4 9.9 ± 1.0

Table 2 -

 2 Average yield and yield components for the four selected winter bread wheat cultivars under well-watered or water-deficit conditions.

	Condition	Group of cultivars 1	Cultivar	Flowering date (days from sowing)	SNM2 2 (spike per m 2 )	NGS plot 2 (grains spike -1 )	GNM2 2 (grains m -2 )	AGW plot 2 (mg grain -1 )	GY 2 (t ha -1 )
	Well-watered	HCv	Altigo	137	368 ± 24 a 3	35.9 ± 0.6 b	13182 ± 837a	51.96 ± 0.77 d	7.0 ± 4 a
			Solehio	137	430 ± 84 ab	34.5 ± 1.3 a	14821 ± 3007 ab	49.83 ± 0.79 c	7.5 ± 14a
		LCv	Cellule	133	378 ± 32 ab	40.5 ± 0.8 c	15336 ± 1209 ab	40.43 ± 0.86 a	6.3 ± 4a
			SY Moisson	136	485 ± 78 b	35.5 ± 1.0 ab	17146 ± 2478 b	42.54 ± 0.82 b	7.5 ± 10a
	Water-deficit	HCv	Altigo	134	362 ± 18 a	28.7 ± 1.2 a	10397 ± 648 a	48.77 ± 2.44 b	5.1 ± 5 a
			Solehio	134	344 ± 45 a	30.2 ± 1.8 a	10440 ± 1892 a	48.62 ± 1.03 b	5.1 ± 9 a
		LCv	Cellule	127	394 ± 50 a	33.1 ± 1.0 a	13042 ± 1645 b	38.43 ± 1.52 a	5.0 ± 7a
			SY Moisson	130	380± 21 a	31.2 ± 4.8 a	11793 ± 1141 ab	40.20 ± 1.34 a	4.8 ± 5a

1 : HCv: high-AGW cultivar; LCv low-AGW cultivar 2 : SNM2: Spike number per m 2 ; NGSplot: number of grains per spike determined on a plot basis; GNM2: grain number per m 2 ; AGWplot : Average Grain Weight determined on a plot basis; GY: Grain Yield. 3 : Per condition, different letters indicate statistical significance at level P=0.05 within the same column (according to a Tukey post-hoc test following ANOVA).

Table 3 .

 3 Analysis of variance (ANOVA) followed by the statistics of Wald test for AGW spike (mg grain -1 ) under well-watered and water-deficit conditions.

	Condition	Source of variation 1	Degrees of freedom	Percentage of variance explained	Wald statistic	P-value 2
	Well-watered	Number of grains per spike (NGS)	1	0.7	35.74	***
		Block (B)	3	0.8	40.31	***
		Group (G) of cultivars (Cv)	1	61.3	2992.91	***
		Cv nested in G	2	3.0	147.32	***
		NGS × G	1	0.4	17.44	***
		NGS × B	3	0.2	7.14	ns
		G × B	3	0.5	22.59	***
		G × B × NGS	3	0.1	2.79	ns
		Residual	1618	33.1
	Water-deficit	NGS	1	0.9	34.49	***
		B	3	4.2	155.77	***
		G	1	47.9	1758.29	***
		Cv nested in G	2	1.1	41.77	***
		NGS × G	1	0.2	6.89	**
		NGS × B	3	0.3	9.32	*
		G × B	3	0.8	28.07	***
		G × B × NGS	3	0.2	7.03	ns
		Residual	1631	44.4
	1 : G: HCv = High AGW cvs (Altigo and Solehio) ; LCv = Low AGW cvs (Cellule and SY Moisson)

2

: ns : P> 0.05; *: P<0.05; **: P<0.01 ; *** : P<0.001

Table 4 .

 4 Analysis of variance (ANOVA) followed by the statistics of Wald test for individual grain weight (mg) under well-watered and water-deficit conditions.

	Condition	Source of variation 1	Degrees of freedom	Percentage of variance explained	Wald statistic	P-value 2
	Well-watered	Block (B)	3	0.16	704.47	***
		Grain Decile (GrD)	9	70.69	316262.95	***
		Group (G) of Cultivars (Cv)	1	14.17	63398.26	***
		Cv nested in G	2	0.69	3101.77	***
		G × B	3	0.09	410.19	***
		G × GrD	9	0.86	3832.66	***
		B × GrD	27	0.01	56.03	***
		G × B × GrD	37	0.02	69.84	***
		Residual	59582	13.32
	Water-deficit	B	3	1.68	3903.87	***
		GrD	9	54.70	127044.86	***
		G	1	16.94	39343.56	***
		Cv nested in G	2	0.21	482.45	***
		G × B	3	0.35	819.85	***
		G × GrD	9	0.38	891.54	***
		B × GrD	27	0.06	133.76	***
		G × B × GrD	37	0.02	40.47	**
		Residual	59582	25.66

: Grain deciles (GrD) ; G: HCv = High AGW cvs (Altigo and Solehio) ; LCv = Low AGW cvs (Cellule and SY Moisson))

: **: P<0.01 ; *** : P<0.001.

Acknowledgements

This study was supported by the Agence Nationale de la Recherche (ANR) of the French government through the program ''Investissements d'Avenir'' (16-IDEX-0001 CAP 20-25), WOW! CAP 20-25 and INRAE DARESE (EIR-A 2019). A. Beral PhD Scholarship was funded by CAP 20-25 and INRAE AgroEcoSystem division. The authors also acknowledge the projects PHENOME (ANR-11-INBS-0012) for providing raw data. The authors also thank W. Ngo for technical assistance. D. Cormier and B. Adam are thanked for the management of the Pheno3C

Declaration of Competing Interest

The authors declare they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

platform and the development of the single grain weighting robot. The staff from UE 1074 PHACC is acknowledged for the implantation and the management of the trials. The authors thank Dr.

Sandy Lang (www.rescript.co.nz) for language editing and proofreading of the manuscript. Finally, the authors are grateful to the reviewers whose critical comments led to improve this manuscript.

Supplementary material