Evaluating lambda calculus: if false false true

Question

The lambda calculus has the following expressions:

e ::=           Expressions
      x         Variables
      (λx.e)    Functions
      e e       Function application

From this base, we can define a number of additional constructs, such as Booleans and conditional statements:

let true = (λx.(λy.x))
false = (λx.(λy.y))
if = (λcond.(λthen.(λelse.cond then else)))

Showing your work, show the evaluation of the following program: if false false true.

Maybe if(false (then false ( else true then true)))?

But that would just mean exactly what it means. if false then false else true then true.

I don't know how to approach it.

Answer 1

Having the definitions

true = (λx.(λy.x))
false = (λx.(λy.y))
if = (λcond.(λthen.(λelse.cond then else)))

defined, means that

true x y = x
false x y = y
if cond then else = cond then else

Thus, e.g.,

   if true true false
-- if cond then else   = cond then else
                       = true true false
                     --  true x    y      = x
                                          = true

There's no more definitions to apply here, so we have our result.

Now you can try working out your example.