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AKE principles in roughly deeply ramified henselian valued fields
We show that for any henselian valued field of mixed characteristic $(0,p)$, the (existential) $\mathcal{L}_{\mathrm{val}}$-theory of the valued field is determined by the (existential) theory of the value group in $\mathcal{L}_{\mathrm{oag}}$ with a constant for $v(p)$ and the (existential) theory
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