% Natural numbers and operations: complete axiomatizations.

% nat(X) iff X is a natural number
nat(0).
nat(s(N)) :- nat(N).

% sum(X,Y,Z) iff Z = X + Y
sum(0,Y,Y).
sum(s(X),Y,s(Z)) :- sum(X,Y,Z).

% product(X,Y,P) iff P = X * Y
product(0,X,0).
product(s(X),Y,P) :- product(X,Y,Z), sum(Z,Y,P).

% lessthan(X,Y) iff X < Y.
lessthan(0,s(X)).
lessthan(s(X),s(Y)) :- lessthan(X,Y).


% Programming with Prolog's built- arithametics

% fact(N,M) will succeed with M = N!, if N is not a free variable initially
fact(N,1) :- N =< 1.
fact(N,M) :- N > 1, N2 is (N - 1), fact(N2,M2), M is (M2 * N).

sumto(1,1).
sumto(N,S) :- N > 1, N2 is (N - 1), sumto(N2,S2), S is (S2 + N).

% Tradeoffs:

% ?- sum(s(s(0)),Y,s(s(s(s(s0))))).   will return
% Y = s(s(s0))    , but:

% ?- 2 + Y = 5   will return
% no.

%  "=" in Prolog is (usualy) only syntactic pattern matching.


% The point in this demonstration is that, though all programming, and
% programming languages, are based in one way or another on symbolic 
% logic, various "trade-offs" has been made, especially in "imperative"
% languages such as C - to make programming more efficient and convenient 
% at the expense of logical clarity.  Even Prolog must make compromises
% to make itself friendlier to humans.  Reading "s(s(s(0)))" is fine
% for computers, but "3" is much easier for us.