equation_simplifier.pl

:- initialization(main).

main :- recursive_simplify_expr(((x+y+1)^2 = (x-y)^2),Output),writeln(["Output: ", Output]).
%mul([add([mul([add([mul([x,x]),mul([x,y2])])]),mul([add([mul([x,y1]),mul([y1,y2])])])])])=0

simplify_expr(mul([A|Rest]),mul([1|[A|Rest]])) :-
	atom(A).
simplify_expr(mul([A]),A).
simplify_expr(A/B,A1/B1) :-
	simplify_expr(A,A1),simplify_expr(B,B1).

%simplify_expr(add(A),add(B)) :-
%	replace_multiple(A,[add([A1|Rest]),add([A2|Rest])],[add([A1+A2|Rest])],B),integer(A1),integer(A2).
simplify_expr(A/B,A1/B1) :-
	integer(A),integer(B),gcd(A,B,C),A1 is A/C, B1 is B/C.

simplify_expr(mul([A|Rest])/mul([B|Rest1]),mul([A1|Rest])/mul([B1|Rest1])) :-
	integer(A),integer(B),gcd(A,B,C),A1 is A/C, B1 is B/C.

simplify_expr(A/1,A).
simplify_expr(A/(B/C),(A*C)/B).

simplify_expr(mul(A),mul(B)) :-
	replace_multiple(A,[A1,-A2],[A3,A2],B),number(A1),A3 is -A1,atom(A2).

simplify_expr(mul(A),mul(B)) :-
	replace_multiple(A,[add(A1),B1],[add(B2)],B),nth0(Index1,A,B1),nth0(Index2,A,add(A1)),dif(Index1,Index2),multiply_all(A1,B2,B1).

simplify_expr(add([X]),X) :-
	integer(X).
simplify_expr((A^Exp),A*(A^Exp2)) :-
	integer(Exp),Exp > 1, Exp2 is Exp - 1.
simplify_expr(A^1,A).
simplify_expr(A^0,0).
simplify_expr(add(A),add(B)) :-
	replace_multiple(A,[mul([sin(A1),sin(A1)]),mul([cos(A1),cos(A1)])],[1],B).
simplify_expr(add(A),add(B)) :-
	replace_multiple(A,[mul([Num,sin(A1),sin(A1)]),mul([Num,cos(A1),cos(A1)])],[1],B).

simplify_expr(add(A),add(B)) :-
	replace_multiple(A,[add(A1)],[],B1),append(B1,A1,B).

simplify_expr(add(A),add(B)) :-
	replace_multiple(A,[A1,mul([Num,A1])],[mul([A3,A1])],B),integer(Num),atom(A1),A3 is Num+1.
simplify_expr(-add(A),(-1)*add(A)).
simplify_expr(add([-add(A)]),-add(A)).
simplify_expr(add([0|X]),add(X)).
simplify_expr(X<Y,Y>X).
simplify_expr(0=0,true).
simplify_expr(X=0,false) :-
	number(X),dif(X,0).
simplify_expr(X=0,Y=0) :-
	simplify_expr(X,Y).
simplify_expr(X=Y,X-Y = 0) :-
	dif(Y,0).

simplify_expr(mul(A),0) :-
	member(0,A).

simplify_expr(-mul([A|B]),mul([-A|B])) :-
	integer(A).

simplify_expr(-A,B) :-
	integer(A), B is -A.

simplify_expr(A - B, A+(-B)).

simplify_expr(-A,-B) :-
	simplify_expr(A,B).

simplify_expr(mul(A),mul(B)) :-
	replace_multiple(A,[mul(A1)],A1,B).

simplify_expr(mul([A1|[A2|Rest]]),mul([A3|Rest])) :-
	integer(A1),integer(A2),A3 is A1*A2.

simplify_expr(add(A),add(B)) :-
	nth0(A1_index,A,mul([A1|Rest])),nth0(A2_index,A,mul([A2|Rest])),dif(A1_index,A2_index),replace_multiple(A,[mul([A1|Rest]),mul([A2|Rest])],[mul([A3|Rest])],B), integer(A1), integer(A2), A3 is A1 + A2.

simplify_expr(add([mul(X)]),mul(X)).
simplify_expr(X+Y,add([X,Y])).
simplify_expr(X*Y,mul([X,Y])).
simplify_expr(add([A,B]),C) :- integer(A),integer(B),C is A+B.

simplify_expr(add(A),add(B)) :-
	simplify_all(A,B).
simplify_expr(add(A),add(B)) :-
	msort(A,B).
simplify_expr(mul(A),mul(B)) :-
	msort(A,B).
simplify_expr(mul(A),mul(B)) :-
	simplify_all(A,B).

simplify_all([],[]).
simplify_all([A|Rest],[A1|Rest1]) :-
	recursive_simplify_expr(A,A1),simplify_all(Rest,Rest1).

recursive_simplify_expr(A,Output) :-
	simplify_expr(A,B),dif(A,B),recursive_simplify_expr(B,Output);A=Output.



replace_multiple(List,To_replace,Replacements,Output) :-
	members(To_replace,List),remove_all(List,To_replace,Output1),append(Output1,Replacements,Output).

remove_all(List,[],List).
remove_all(List,[To_remove|Rest],Result) :-
	members([To_remove|Rest],List),delete_one(To_remove,List,Result1),remove_all(Result1,Rest,Result).

delete_one(_, [], []).
delete_one(Term, [Term|Tail], Tail).
delete_one(Term, [Head|Tail], [Head|Result]) :-
  delete_one(Term, Tail, Result).

members([],_).
members([A|A1],B) :- member(A,B),members(A1,B).

multiply_all([],[],_).
multiply_all([First|Rest],[First1|Rest1],B) :-
	First1 = B*First, multiply_all(Rest,Rest1,B).

%from https://stackoverflow.com/a/22452766/975097
gcd(X, Y, Z) :-
    X < 0, !,
    gcd(-X, Y, Z).
gcd(X, Y, Z) :-
    Y < 0, !,
    gcd(X, -Y, Z).
gcd(X, 0, X) :- X > 0.
gcd(0, Y, Y) :- Y > 0.
gcd(X, Y, Z) :-
    X > Y, Y > 0,
    X1 is X - Y,
    gcd(Y, X1, Z).
gcd(X, Y, Z) :-
    X =< Y, X > 0,
    Y1 is Y - X,
    gcd(X, Y1, Z).