%
% DERIVATIVES + SIMPLIFICATION
%
% X' = 1
d( X, X, 1).

% Y' = 0 if Y is number or atom different from X
d( Y, X, 0) :-
    atomic(Y), Y \= X.

% subsumed by the more general rule below
% (C*Y)' = C*Y' if C is constant respect to X
d( C*Y, X, C*Z ) :-
    d( C, X, 0 ),    % C is constant wrt X <-> C'=0
    !,               % cut away all remaining choices to reduce bactrack
    d( Y, X, Z).

% (A+B)' = A' + B'
d( A+B, X, C+D ) :-
    d( A, X, B ),
    d( C, X, D).

% subsumed by A+B rule if we canonicalize the espression
% (A-B)' = A' - B'
d( A-B, X, C-D ) :-
    d( A, X, B ),
    d( C, X, D).

% (A*B)' = AB'+BA'
d( A*B, X, A*D+B*C ) :-
    d(A, X, C),
    d(B, X, D).

%
% TASK: add other derivation rules
%
% (X^C)' = CX^(C-1)    if C constant wrt X
%
% (F(G))'= F'(G)*G'
%
% (e^X)' = e^X
%
% ...

%
% TASK: apply simplification (see below) every time a derivation is applied?
%

%
% TASK: simplification (one single step)
%
% X*1 = X
% 1*X = X                       (**)
simp(X*1, X).
simp(1*X, X).

% 0+X = X
% X+0 = X                       (**)
simp(X+0, X).
simp(0+X, X).

% X^0 = 1
% X^1 = X
% X-X = 0                       (*)
% X+X = 2*X

% C*D = CD
simp( A*B, C ) :-
    number(A),
    number(B),
    C is A*B.

% AX+BX = (A+B)*X
% AX-BX = (A-B)*X               (*)
% X^2-Y^2 = (X-Y)*(X+Y)
%
% TASK: repeated simplification separated by single-step rules
%       one-step simplification should be repeated until there is no
%       more steps applicable, a new predicate could do it recursively
simplify(X, Z) :-
   simp(X, Y),
   !,
   simplify(Y,Z).
simplify(A+B, C+D) :-
   simplify(A, C),
   simplify(B, D).
simplify(A*B, C*D) :-
   simplify(A, C),
   simplify(B, D).

simplify(X, X).
%
% TASK: number canonicalization
% integers/rationals AND floats (WHY TWO TYPES?)
%       - choose one type and convert the other to it
%
% TASK: subtraction canonicalization to reduce operators (*)
%       -X = (-1)*X
%
% TASK: normalization of expressions to reduce the number of rules
% - commutative operations -> reorder the elements (or try both cases?)  (**)
%      A*B = B*A
% - distribution of * over +/-       (try both cases -> infinite loop)
%      A*(B+C) => A*B + A*C
% - associative operations -> reorder the operators or try all cases?
%      A*B*C = (A*B)*C = A*(B*C)
%
% TASK: monomial canonicalization   (order the literals)
%       numeric_constant * literal part
%
% TASK: polynomial canonicalization (order monomials and collect similar monomials)
%