  This paper is concerned by the concept of algebra up to homotopy for a
structure defined by two operations $.$ and [,]. An important example of such a
structure is the Gerstenhaber algebra (commutatitve and Lie). The notion of
Gerstenhaber algebra up to homotopy ($G_\infty$ algebra) is known. Here, we
give a definition of pre-Gerstenhaber algebra (pre-commutative and pre-Lie)
allowing the construction of $\hbox{pre}G_\infty$ algebra. Given a structure of
pre-commutative (Zinbiel) and pre-Lie algebra and working over the
corresponding dual operads, we will give an explicit construction of the
associated pre-Gerstenhaber algebra up to homotopy, this is a bicogebra
(Leibniz and permutative) equipped with a codifferential which is a
coderivation for the two coproducts.
