Not collapsing cardinals ≤ k in (< k)-support iterations

We deal with the problem of preserving various versions of completeness in (< k)-support iterations of forcing notions, generalizing the case "S-complete proper is preserved by CS iterations for a stationary costationary S ⊆ ω₁". We give applications to Uniformization and the Whitehead problem. In particular, for a strongly inaccessible cardinal k and a stationary set S ⊆ k, with fat complement we can have uniformization for every (A_δ : δ ∈ S′), Aδ ⊆ δ = sup A_δ, cf(δ) = otp(A_δ) and a stationary non-reflecting set S′ ⊆ S (see B.8.2).