  Monads can be interpreted as encoding formal expressions, or formal
operations in the sense of universal algebra. We give a construction which
formalizes the idea of "evaluating an expression partially": for example, "2+3"
can be obtained as a partial evaluation of "2+2+1". This construction can be
given for any monad, and it is linked to the famous bar construction, of which
it gives an operational interpretation: the bar construction induces a
simplicial set, and its 1-cells are partial evaluations.
  We study the properties of partial evaluations for general monads. We prove
that whenever the monad is weakly cartesian, partial evaluations can be
composed via the usual Kan filler property of simplicial sets, of which we give
an interpretation in terms of substitution of terms.
  In terms of rewritings, partial evaluations give an abstract reduction system
which is reflexive, confluent, and transitive whenever the monad is weakly
cartesian.
  For the case of probability monads, partial evaluations correspond to what
probabilists call conditional expectation of random variables.
  This manuscript is part of a work in progress on a general rewriting
interpretation of the bar construction.
