Creating a new column by resetting cummin() of another column and a condition

This is my DataFrame:

import pandas as pd df = pd.DataFrame(    {'a': [98, 97, 100, 135, 103, 100, 105, 109, 130],'b': [100, 103, 101, 105, 110, 120, 101, 150, 160]    })

And this is the desired output. I want to create column c:

     a    b    c0   98  100  1001   97  103  1002  100  101  1003  135  105  1004  103  110  1105  100  120  1106  105  101  1017  109  150  1508  130  160  150

It is not so easy for me to describe the issue in pure English, since it is a little bit complicated.c is df.b.cummmin() but under certain conditions it changes. I describe it row by row:

The process starts with:

df['c'] = df.b.cummin()

The condition that changes c is :

cond = df.a.shift(1) > df.c.shift(1)

Now the rows that matter are the ones that cond == True. For these rows df.c = df.b And the cummin() of b RESETS.

For example, the first instance of cond is row 4. So c changes to 110 (in other words, whatever b is). And for row 5 it is the cummin() of b from row 4. The logic is the same to the end.

This is one of my attempts. But it does not work where the cond kicks in:

df['c'] = df.b.cummin()df.loc[df.a.shift(1) > df.c.shift(1), 'c'] = df.b