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Let T be an NIP
$\mathop T\limits^\~ $
be an enrichment. We give a sufficient condition on
for the underlying
-type of any definable (respectively invariant) type over a model of
to be definable (respectively invariant). These results are then applied to Scanlon’s model completion of valued differential fields.
We prove field quantifier elimination for valued fields endowed with both an analytic structure that is
-Henselian and an automorphism that is
-Henselian. From this result we can deduce various Ax–Kochen–Eršov type results with respect to completeness and the independence property. The main example we are interested in is the field of Witt vectors on the algebraic closure of
endowed with its natural analytic structure and the lifting of the Frobenius. It turns out we can give a (reasonable) axiomatization of its first-order theory and that this theory does not have the independence property.
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