Iterated forcing and changing cofinalities

We weaken the notion of proper to semi-proper, so that the important properties (e.g., being preserved by some interations) are preserved, and it includes some forcing which changes the confinality of a regular cardinal >א₁ to א₀. So, using the right iteractions, we can iterate such forcing without collapsing א₁. As a result, we solve the following problems of Friedman, Magidor and Avraham, by proving (modulo large cardinals) the consistency of the following with G.C.H.: (1) for every S {square image of or equal to} א₂, S or א₂-S contains a closed copy of ω₁ (2) there is a normal precipitous filter D on {Mathematical expression} (3) for every {Mathematical expression} is regular in L (δ ∩A)} is statonary. The results can be improved to equi-consistency; this will be discussed in a future paper.