C-H has nothing to do with "property X is true about all things of type Y". It says "if you can find an object of type Y, you can prove global property Z."
For example, the property corresponding to the type "(Either a a) -> a" is "(A or A) implies A" which doesn't tell you any properties of actual objects of the type "(Either a a) -> a"
The property corresponding to the type "(Either a b) -> a" is "(A or B) implies A" which is not provable, so you can't find any object of this type.
For example, the property corresponding to the type "(Either a a) -> a" is "(A or A) implies A" which doesn't tell you any properties of actual objects of the type "(Either a a) -> a"
The property corresponding to the type "(Either a b) -> a" is "(A or B) implies A" which is not provable, so you can't find any object of this type.