Lambda Calculus Problems

1. Define a Boolean OR function in lambda calculus. Show rigorously that for all four possible combinations of parameters (TRUE TRUE, TRUE FALSE, FALSE TRUE, FALSE FALSE), your OR function returns the right answer.
2. Show through a careful detailed derivation that (plus c4 c3) yields c7.
3. Show through a careful detailed derivation that (times c2 c3) yields c6.
4. Define a lambda calculus expression for raising one Church numeral to the power of another.
5. Find an alternative way to define the successor (plus1) function with Church numerals. Explain why your answer works, and how you came up with it.