Editing FP (programming language) (section)

== Overview ==

The '''values''' that FP programs map into one another comprise a [[set (abstract data type)|set]] which is [[closure (mathematics)|closed]] under '''sequence formation''':

 if '''x'''<sub>1</sub>,...,'''x'''<sub>n</sub> are '''values''', then the '''sequence''' 〈'''x'''<sub>1</sub>,...,'''x'''<sub>n</sub>〉 is also a '''value'''

These values can be built from any set of atoms: booleans, integers, reals, characters, etc.:
 '''boolean'''   : {'''T''', '''F'''}
 '''integer'''   : {0,1,2,...,∞}
 '''character''' : {'a','b','c',...}
 '''symbol'''    : {'''x''','''y''',...}

'''⊥''' is the '''undefined''' value, or '''bottom'''. Sequences are ''bottom-preserving'':

 〈'''x'''<sub>1</sub>,...,'''⊥''',...,'''x'''<sub>n</sub>〉  =  '''⊥'''

FP programs are ''functions'' '''f''' that each map a single ''value'' '''x''' into another:

 '''f''':'''x''' represents the '''value''' that results from applying the '''function''' '''f''' 
     to the '''value''' '''x'''

Functions are either primitive (i.e., provided with the FP environment) or are built from the primitives by '''program-forming operations''' (also called '''functionals''').

An example of primitive function is '''constant''', which transforms a value '''x''' into the constant-valued function '''x̄'''. Functions are [[strict function|strict]]:
 '''f''':'''⊥''' = '''⊥'''

Another example of a primitive function is the '''selector''' function family, denoted by '''1''','''2''',... where:
 '''''i''''':〈'''x'''<sub>1</sub>,...,'''x'''<sub>n</sub>〉  =  '''x'''<sub>i</sub>  if  1 ≤ '''''i''''' ≤ n
               =  ⊥   otherwise