Assignment 16

Show, carefully and completely, how the following sentences are derived
according to our grammar (or if not, say why not, and propose a fix).
Discuss anything that needs to be discussed.


(1)  For there to have been a dead frog found in the soup would have 
     embarrassed the emperor.


(2)  We believe that for students to be given course credit for participation
     in protests would set a bad precedent.


(3)  My uncle's belief that there will be insects found on Mars is
     supported by the fact that there have been insects discovered inside
     volcanoes and under the polar ice.

In particular, what are 'for' and 'to'?