The strength of choiceless patterns of singular and weakly compact cardinals

We extend the core model induction technique to a choiceless context, and we exploit it to show that each one of the following two hypotheses individually implies that AD , the Axiom of Determinacy, holds in the L ( R ) of a generic extension of HOD : (a) ZF + every uncountable cardinal is singular, and (b) ZF + every infinite successor cardinal is weakly compact and every uncountable limit cardinal is singular.