We establish natural criteria under which normally iterable premice are iterable for stacks of normal trees. Let ΩΩ be a regular uncountable cardinal. Let m<ωm<ω and MM be an mm-sound premouse and ΣΣ be an (m,Ω+1)(m,Ω+1)-iteration strategy for MM (roughly, a normal (Ω+1)(Ω+1)-strategy). We define a natural condensation property for iteration strategies, inflation condensation. We show that if ΣΣ has inflation condensation then MM is (m,Ω,Ω+1)∗(m,Ω,Ω+1)∗-iterable (roughly, MM is iterable for length ≤Ω≤Ω stacks of normal trees each of length <Ω<Ω), and moreover, we define a specific such strategy ΣstΣst and a reduction of stacks via ΣstΣst to normal trees via ΣΣ. If ΣΣ has the Dodd-Jensen property and card(M)<Ωcard(M)<Ω then ΣΣ has inflation condensation. We also apply some of the techniques developed to prove that if ΣΣ has strong hull condensation (introduced independently by John Steel), and GG is VV-generic for an ΩΩ-cc forcing, then ΣΣ extends to an (m,Ω+1)(m,Ω+1)-strategy Σ+Σ+ for MM with strong hull condensation, in the sense of V[G]V[G]. Moreover, this extension is unique. We deduce that if GG is VV-generic for a ccc forcing then VV and V[G]V[G] have the same ωω-sound, (ω,Ω+1)(ω,Ω+1)-iterable premice which project to ωω.
