Give the lambda term that needs to replace \(\alpha\) so that the reduction results as given.

1. \(\alpha \cnct a \cnct b = b \cnct a\)

2. \(\alpha \cnct a \cnct b \cnct c = a \cnct c \cnct b\)

3. \(\alpha \cnct a \cnct b \cnct c = a \cnct (c \cnct b)\)

4. \(\alpha \cnct a \cnct b = \lambda x.a \cnct x \cnct (b \cnct x)\)

5. \(\alpha \cnct (\lambda x.a \cnct x)c = ac\)

6. \(\alpha \cnct (\lambda x.a \cnct x)c = \lambda y.a(yc)\)

7. \(\alpha \cnct (\lambda x.sleeps'x) = sleeps'john'\)

8. \(\alpha \cnct sleeps' = sleeps'john'\)

9. \(\alpha (\lambda x. walks'x) john' \equiv_{\beta} slow'(walks'john')\)

10. \(\alpha (\lambda x. walks'x) john' \equiv_{\beta} slow'walks'john'\)

11. \(\alpha\cnct{mary'}john'(\lambda x.walks'x) \equiv_{\beta} walks'john' \land walks'mary'\)

12. \(\alpha(\lambda x. talks'x)(\lambda x.smiles'x)john'\equiv_{\beta} smiles'john' \land talks'john'\)