-- The primitive recursive functions
--
-- Raul Rojas 
------------------------------------------

-- Zero function

clr [x] = 0

--Succesor function

suc [x] = x+1

--Projection functions. Tuples are represented by lists

p::Int->[Int]->Int

p 1 (a:b) = a
p n (a:b) = p (n-1) b

-- Operations
------------------------|

-- Composition

compose g fs xs = g (app fs xs)

-- Apply a list of functions to argument a

app   []   a = []
app (f:fs) a = (f a):(app fs a)

-- primitive recursion

pr::([Int]->Int)->([Int]->Int)->([Int]->Int)->[Int]->Int

pr phi psi chi (  0  :xs) = psi xs
pr phi psi chi ((y+1):xs) = chi ((phi (y:xs)):y:xs)


-- Convert an argument to a list

arg1 x = [x]

------------------------| some primitive recursive functions

--preddecessor
 
predd::[Int]->Int
predd  = pr  predd (const 0) (p 2) 

--Addition

add::[Int]->Int
add  = pr add (p 1) (compose suc [(p 1)])