An Introduction to Functional Programming Through Lambda Calculus

- 115 -rec DOUBLE [] = []or DOUBLE (H::T) = (2*H)::(DOUBLE T)so:DOUBLE [1,2,3] => ... =>(2*1)::(DOUBLE [2,3]) -> ... ->2::(2*2)::(DOUBLE [3]) -> ... ->2::4::(2*3)::(DOUBLE []) -> ... ->2::4::6::[] ==[2,4,6]Now consider the function which turns all the words in a list into plurals:so:rec PLURAL [] = []or PLURAL (H::T) = (APPEND H "s")::(PLURAL T)PLURAL ["cat","dog","pig"] => ... =>(APPEND "cat" "s")::(PLURAL ["dog","pig"]) -> ... ->"cats"::(APPEND "dog" "s")::(PLURAL ["pig"]) -> ... ->"cats"::"dogs"::(APPEND "pig" "s")::(PLURAL []) -> ... ->"cats"::"dogs"::"pigs"::[] ==["cats","dogs","pigs"]The functions DOUBLE and PLURAL both apply a function repeatedly to the consecutive heads of their list arguments.In LISP this is known as a CAR mapping because the function is mapped onto the CARs of the list. We can abstract acommon structure from DOUBLE and PLURAL as:rec MAPCAR FUNC [] = []or MAPCAR FUNC (H::T) = (FUNC H)::(MAPCAR FUNC T)Thus, we can define DOUBLE as:def DOUBLE = MAPCAR λX.(2*X)so DOUBLEs definition expands as:rec DOUBLE [] = []or DOUBLE (H::T) =(λX.(2*X) H)::(MAPCAR λX.(2*X) T)Simplifying, we get:def DOUBLE [] = []or DOUBLE (H::T) = (2*H)::(MAPCAR λX.(2*X) T)