A minimal Magidor-type forcing (countable case)

In their paper from 2013, Koepke, Rasch and Schlicht defined a minimal Prikry-type forcing, which satisfies that any intermediate model is either the ground model or the generic extension.

In this talk, we try to generalize this result, and prove that for a countable limit ordinal delta, we can define a minimal Magidor-type forcing, which adds an increasing continuous sequence C_G of length delta such that any intermediate model between the ground model and the generic extension is of the form "V[C_G restricted to alpha]" for some limit ordinal alpha <= delta.