polynomialmani.pl/polymani.pl

%% -*- mode: prolog-*-
%% vim: set softtabstop=4 shiftwidth=4 tabstop=4 expandtab:

/**
 *
 * polimani.pl
 *
 * Assignment 1 - Polynomial Manipulator
 * Programming in Logic - DCC-FCUP
 *
 * Diogo Peralta Cordeiro
 * up201705417@fc.up.pt
 *
 * Hugo David Cordeiro Sales
 * up201704178@fc.up.pt
 *
 *********************************************
 *   Follows 'Coding guidelines for Prolog'  *
 * https://doi.org/10.1017/S1471068411000391 *
 *********************************************/

/*
 * Import the Constraint Logic Programming over Finite Domains library
 * Essentially, this library improves the way Prolog deals with integers,
 * allowing more predicates to be reversible.
 * For instance, number(N) is always false, which prevents the
 * reversing of a predicate.
 */
:- use_module(library(clpfd)).
/*
 * Import Constraint Logic Programming for Reals library, which is somewhat
 * similar to clpfd, but for real numbers
 */
:- use_module(library(clpr)).


                 /*******************************
                 *        USER INTERFACE        *
                 *******************************/

/*
    poly2list/2 transforms a list representing a polynomial (second
    argument) into a polynomial represented as an expression (first
    argument) and vice-versa.
*/
poly2list(P, L) :-
    is_polynomial_valid_in_predicate(P, "poly2list"),
    polynomial_to_list(P, L),
    !.

/*
    simpolylist/2 simplifies a polynomial represented as a list into
    another polynomial as a list.
*/
simpoly_list(L, S) :-
    is_polynomial_as_list_valid_in_predicate(L, "simpoly_list"),
    simplify_polynomial_as_list(L, S),
    !.

/*
    simpoly/2 simplifies a polynomial represented as an expression
    as another polynomial as an expression.
*/
simpoly(P, S) :-
    is_polynomial_valid_in_predicate(P, "simpoly"),
    simplify_polynomial(P, S),
    !.

/*
    scalepoly/3 multiplies a polynomial represented as an expression by a scalar
    resulting in a second polynomial. The two first arguments are assumed to
    be ground. The polynomial resulting from the sum is in simplified form.
*/
scalepoly(P1, C, S) :-
    is_polynomial_valid_in_predicate(P1, "scalepoly"),
    is_number_in_predicate(C, "scalepoly"),
    scale_polynomial(P1, C, S),
    !.
%% Tests:
%% ?- scalepoly(3*x*z+2*z, 4, S).
%@ S = 12*x*z+8*z.
%% ?- scalepoly(3*x*z+2*z, 2, S).
%@ S = 6*x*z+4*z.

/*
    addpoly/3 adds two polynomials as expressions resulting in a
    third one. The two first arguments are assumed to be ground.
    The polynomial resulting from the sum is in simplified form.
*/
addpoly(P1, P2, S) :-
    is_polynomial_valid_in_predicate(P1, "addpoly"),
    is_polynomial_valid_in_predicate(P2, "addpoly"),
    add_polynomial(P1, P2, S),
    !.
%% Tests:
%% ?- addpoly(3 + x, 3 - x, S).
%@ S = 6.

%% is_polynomial_valid_in_predicate(+T, +F) is det
%
%  Returns true if valid polynomial, fails with UI message otherwise.
%  The failure message reports which polynomial is invalid and in which
%  predicate the problem ocurred.
%
is_polynomial_valid_in_predicate(P, _) :-
    %% If P is a valid polynomial, return true
    polynomial(P),
    !.
is_polynomial_valid_in_predicate(P, F) :-
    %% Otherwise, write the polynomial and fails
    write("Invalid polynomial in "),
    write(F),
    write(": "),
    write(P),
    fail.
%% Tests:
%% ?- is_polynomial_valid_in_predicate(1-x, "Test").
%@ true.
%% ?- is_polynomial_valid_in_predicate(a*4-0*x, "Test").
%@ Invalid polynomial in Test: a*4-0*x
%@ false.

%% is_polynomial_as_list_valid_in_predicate(+L, +F) is det
%
%  Returns true if the polynomial represented as list is valid,
%  fails with UI message otherwise.
%  The failure message reports which polynomial is invalid and
%  in which predicate the problem ocurred.
%
is_polynomial_as_list_valid_in_predicate(L, F) :-
    %% If L is a valid polynomial, return true
    list_to_polynomial(L, P),
    is_polynomial_valid_in_predicate(P, F).
%% Tests:
%% ?- is_polynomial_as_list_valid_in_predicate([1], "Test").
%@ true.
%% ?- is_polynomial_as_list_valid_in_predicate([0*x, a*4], "Test").
%@ Invalid polynomial in Test: a*4+0*x
%@ false.

%% is_number_in_predicate(+C:number, +F:string) is det
%
%  Validates that C is a number or prints F and it then it
%
is_number_in_predicate(C, _) :-
    number(C),
    !.
is_number_in_predicate(C, F) :-
    %% Writes the argument and fails
    write("Invalid number in "),
    write(F),
    write(": "),
    write(C),
    fail.

                 /*******************************
                 *            BACKEND           *
                 *******************************/

%% polynomial_variable_list(-List) is det
%
%  List of possible polynomial variables
%
polynomial_variable_list([x, y, z]).

%% polynomial_variable(?X:atom) is semidet
%
%  Returns true if X is a polynomial variable, false otherwise.
%
polynomial_variable(X) :-
    polynomial_variable_list(V),
    member(X, V).
%% Tests:
%% ?- polynomial_variable(x).
%@ true .
%% ?- polynomial_variable(a).
%@ false.

%% power(+X:atom) is semidet
%
%  Returns true if X is a power term, false otherwise.
%
power(P^N) :-
    %% CLPFD comparison. Reversible
    N #>= 1,
    polynomial_variable(P).
power(X) :-
    polynomial_variable(X).
%% Tests:
%% ?- power(x).
%@ true .
%% ?- power(a).
%@ false.
%% ?- power(x^1).
%@ true .
%% ?- power(x^3).
%@ true .
%% ?- power(x^(-3)).
%@ false.
%% ?- power(-x).
%@ false.
%% ?- power(X).
%@ X = x^_462546,
%@ _462546 in 1..sup ;
%@ X = y^_462546,
%@ _462546 in 1..sup ;
%@ X = z^_462546,
%@ _462546 in 1..sup ;
%@ X = x ;
%@ X = y ;
%@ X = z.

%% term(+N:atom) is semidet
%
%  Returns true if N is a term, false otherwise.
%
term(N) :-
    % If N is not a free variable
    nonvar(N),
    % Assert it as a number
    number(N).
term(N) :-
    % If N is a free variable and not compound
    not(compound(N)),
    var(N),
    % Assert it must be between negative and positive infinity
    % This uses the CLPR library, which makes this reversible,
    % whereas `number(N)` is always false, since it only succeeds
    % if the argument is bound (to a integer or float)
    {N >= 0; N < 0}.
term(X) :-
    power(X).
term(-X) :-
    power(X).
term(L * R) :-
    term(L),
    term(R).
%% Tests:
%% ?- term(2*x^3).
%@ true .
%% ?- term(x^(-3)).
%@ false.
%% ?- term(a).
%@ false.
%% ?- term(-1*x).
%@ true .
%% ?- term(-x).
%@ true .
%% ?- term((-3)*x^2).
%@ true .
%% ?- term(3.2*x).
%@ true .
%% ?- term(-x*(-z)).
%@ true .
%% ?- term(X).
%@ {X>=0.0} ;
%@ {X<0.0} ;
%@ X = x^_111514,
%@ _111514 in 1..sup ;
%@ X = y^_111514,
%@ _111514 in 1..sup ;
%@ X = z^_111514,
%@ _111514 in 1..sup ;
%@ X = x ;
%@ X = y ;
%@ X = z ;
%@ X = -x^_111522,
%@ _111522 in 1..sup ;
%@ X = -y^_111522,
%@ _111522 in 1..sup ;
%@ X = -z^_111522,
%@ _111522 in 1..sup ;
%@ X = -x ;
%@ X = -y ;
%@ X = -z ;

%% polynomial(+M:atom) is semidet
%
%  Returns true if polynomial, false otherwise.
%
polynomial(M) :-
    %% A polynomial is either a term
    term(M).
polynomial(L + R) :-
    %% Or a sum of terms
    polynomial(L),
    term(R).
polynomial(L - R) :-
    %% Or a subtraction of terms
    polynomial(L),
    term(R).
%% Tests:
%% ?- polynomial(x).
%@ true .
%% ?- polynomial(x^3).
%@ true .
%% ?- polynomial(3*x^7).
%@ true .
%% ?- polynomial(2 + 3*x + 4*x*y^3).
%@ true .
%% ?- polynomial(2 - 3*x + 4*x*y^3).
%@ true .
%% ?- polynomial(a).
%@ false.
%% ?- polynomial(x^(-3)).
%@ false.
%% ?- polynomial(-x + 3).
%@ true .
%% ?- polynomial(-x - -z).
%@ true .

%% power_to_canon(+T:atom, -T^N:atom) is semidet
%
%  Returns a canon power term.
%
power_to_canon(T^N, T^N) :-
    polynomial_variable(T),
    % CLP(FD) operator to ensure N is different from 1,
    % in a reversible way
    N #\= 1.
power_to_canon(T, T^1) :-
    polynomial_variable(T).
%% Tests:
%% ?- power_to_canon(x, X).
%@ X = x^1 .
%% ?- power_to_canon(-x, X).
%@ false.
%@ X = -1*x^1 .
%% ?- power_to_canon(X, x^1).
%@ X = x .
%% ?- power_to_canon(X, x^4).
%@ X = x^4 .
%% ?- power_to_canon(X, a^1).
%@ false.
%% ?- power_to_canon(X, x^(-3)).
%@ X = x^ -3 .
%% ?- power_to_canon(X, -1*x^1).
%@ X = -x .

%% term_to_list(?T, ?List) is semidet
%
%  Converts a term to a list of its monomials and vice versa.
%  Can verify if term and monomials list are compatible.
%
term_to_list(L * N, [N | TS]) :-
    number(N),
    term_to_list(L, TS).
term_to_list(L * P, [P2 | TS]) :-
    power(P),
    power_to_canon(P, P2),
    term_to_list(L, TS).
term_to_list(L * -P, [-P2 | TS]) :-
    power(P),
    power_to_canon(P, P2),
    term_to_list(L, TS).
term_to_list(N, [N]) :-
    number(N).
term_to_list(P, [P2]) :-
    power(P),
    power_to_canon(P, P2).
term_to_list(-P, [-P2]) :-
    power(P),
    power_to_canon(P, P2).
%% Tests:
%% ?- term_to_list(1, X).
%@ X = [1] .
%% ?- term_to_list(-1, X).
%@ X = [-1] .
%% ?- term_to_list(x, X).
%@ X = [x^1] .
%% ?- term_to_list(-x, X).
%@ X = [-x^1] .
%% ?- term_to_list(2 * 3, X).
%@ X = [3, 2] .
%% ?- term_to_list(1*2*y*z*23*x*y*x^3*x, X).
%@ X = [x^1, x^3, y^1, x^1, 23, z^1, y^1, 2, 1] .
%% ?- term_to_list(1*2*y*z*23*x*y*(-1), X).
%@ X = [-1, y^1, x^1, 23, z^1, y^1, 2, 1] .
%% ?- term_to_list(X, [-1]).
%@ X = -1 .
%% ?- term_to_list(X, [x^1, -1]).
%@ X = -1*x .
%% ?- term_to_list(X, [-x^1]).
%@ X = -x .
%% ?- term_to_list(X, [y^1, x^1]).
%@ X = x*y .
%% ?- term_to_list(X, [x^4]).
%@ X = x^4 .
%% ?- term_to_list(X, [y^6, z^2, x^4]).
%@ X = x^4*z^2*y^6 .
%% ?- term_to_list(X, [y^6, z^2, x^4, -2]).
%@ X = -2*x^4*z^2*y^6 .
%% ?- term_to_list(X, [x^1, 0]).
%@ X = 0*x .
%% ?- term_to_list(X, [y^1, -2]).
%@ X = -2*y .

%% simplify_term(+Term_In:term, ?Term_Out:term) is det
%
%  Simplifies a given term.
%  This function can also be be used to verify if
%  a term is simplified.
%
simplify_term(Term_In, Term_Out) :-
    term_to_list(Term_In, L),
    %% Sort the list of numbers and power to group them,
    %% simplifying the job of `join_similar_parts_of_term`
    sort(0, @=<, L, L2),
    (
        %% If there's a 0 in the list, then the whole term is 0
        member(0, L2),
        Term_Out = 0
    ;
        %% Otherwise
        (
            %% If there's only one element, then the term was already simplified
            %% This is done so that the `exclude` following doesn't remove all ones
            length(L2, 1),
            Term_Out = Term_In
        ;
            %% Remove all remaining ones
            exclude(==(1), L2, L3),
            join_similar_parts_of_term(L3, L4),
            %% Reverse the list, since the following call gives the result in the
            %% reverse order otherwise
            reverse(L4, L5),
            term_to_list(Term_Out, L5)
        )
    ),
    % First result is always the most simplified form.
    !.
%% Tests:
%% ?- simplify_term(1, X).
%@ X = 1.
%% ?- simplify_term(x, X).
%@ X = x.
%% ?- simplify_term(2*y*z*x^3*x, X).
%@ X = 2*x^4*y*z.
%% ?- simplify_term(1*y*z*x^3*x, X).
%@ X = x^4*y*z.
%% ?- simplify_term(0*y*z*x^3*x, X).
%@ X = 0.
%% ?- simplify_term(6*y*z*7*x*y*x^3*x, X).
%@ X = 42*x^5*y^2*z.
%% ?- simplify_term(-x, X).
%@ X = -x.
%% ?- simplify_term(-x*y*(-z)*3, X).
%@ X = 3* -x* -z*y.
%% ?- simplify_term(a, X).
%@ false.
%% ?- simplify_term(x^(-3), X).
%@ false.

%% join_similar_parts_of_term(+List, -List) is det
%
%  Combine powers of the same variable in the given list.
%  Requires that the list be sorted.
%
join_similar_parts_of_term([P1, P2 | L], L2) :-
    %% If both symbols are powers
    power(P1),
    power(P2),
    %% Decompose them into their parts
    B^N1 = P1,
    B^N2 = P2,
    %% Sum the exponent
    N is N1 + N2,
    join_similar_parts_of_term([B^N | L], L2),
    % First result is always the most simplified form.
    !.
join_similar_parts_of_term([N1, N2 | L], L2) :-
    %% If they are both numbers
    number(N1),
    number(N2),
    %% Multiply them
    N is N1 * N2,
    join_similar_parts_of_term([N | L], L2),
    % First result is always the most simplified form.
    !.
join_similar_parts_of_term([X | L], [X | L2]) :-
    %% Otherwise consume one element and recurse
    join_similar_parts_of_term(L, L2),
    % First result is always the most simplified form.
    !.
join_similar_parts_of_term([], []).
%% Tests:
%% ?- join_similar_parts_of_term([3], T).
%@ T = [3].
%% ?- join_similar_parts_of_term([x^2], T).
%@ T = [x^2].
%% ?- join_similar_parts_of_term([x^1, x^1, x^1, x^1], T).
%@ T = [x^4].
%% ?- join_similar_parts_of_term([2, 3, x^1, x^2], T).
%@ T = [6, x^3].
%% ?- join_similar_parts_of_term([2, 3, x^1, x^2, y^1, y^6], T).
%@ T = [6, x^3, y^7].
%% ?- join_similar_parts_of_term([2, 3, -x^1, -x^2], T).
%@ T = [6, -x^1, -x^2].

%% simplify_polynomial(+P:atom, -P2:atom) is det
%
%  Simplifies a polynomial.
%
simplify_polynomial(0, 0) :-
    % 0 is already fully simplified. This is an
    % exception to the following algorithm
    !.
simplify_polynomial(P, P2) :-
    polynomial_to_list(P, L),
    simplify_polynomial_as_list(L, L2),
    list_to_polynomial(L2, P2),
    %% The first result is the most simplified one
    !.
%% Tests:
%% ?- simplify_polynomial(1, X).
%@ X = 1.
%% ?- simplify_polynomial(0, X).
%@ X = 0.
%% ?- simplify_polynomial(x, X).
%@ X = x.
%% ?- simplify_polynomial(x*x, X).
%@ X = x^2.
%% ?- simplify_polynomial(2 + 2, X).
%@ X = 2*2.
%% ?- simplify_polynomial(x + x, X).
%@ X = 2*x.
%% ?- simplify_polynomial(0 + x*x, X).
%@ X = x^2.
%% ?- simplify_polynomial(x^2*x + 3*x^3, X).
%@ X = 4*x^3.
%% ?- simplify_polynomial(x^2*x + 3*x^3 + x^3 + x*x*x, X).
%@ X = 6*x^3.
%% ?- simplify_polynomial(x^2*x + 3*x^3 + x^3 + x*x*4 + z, X).
%@ X = 5*x^3+4*x^2+z.
%% ?- simplify_polynomial(x^2*x + 3*x^3 - x^3 - x*x*4 + z, X).
%@ X = 3*x^3-4*x^2+z.
%% ?- simplify_polynomial(x + 1 + x, X).
%@ X = 2*x+1.
%% ?- simplify_polynomial(x + 1 + x + 1 + x + 1 + x, X).
%@ X = 4*x+3.

%% simplify_polynomial_as_list(+L1:List,-L3:List) is det
%
%  Simplifies a polynomial represented as a list.
%
simplify_polynomial_as_list(L, L13) :-
    %% Convert each term to a list
    maplist(term_to_list, L, L2),
    %% Sort each sublist; done so the next
    %% sort gives the correct results
    maplist(sort(0, @>=), L2, L3),
    %% Sort the outer list
    sort(0, @>=, L3, L4),
    %% For each of the parts of the terms, join them
    maplist(join_similar_parts_of_term, L4, L5),
    %% Sort each of the sublists
    %% Done so the next call simplifies has less work
    maplist(sort(0, @=<), L5, L6),
    join_similar_terms(L6, L7),
    %% Exclude any sublist that includes a 0 (such as the
    %% equivalent to the term 0*x)
    exclude(member(0), L7, L8),
    %% Reverse each sublist, because the next call
    %% reverses the result
    maplist(reverse, L8, L9),
    maplist(term_to_list, L10, L9),
    %% Delete any 0 from the list
    delete(L10, 0, L11),
    %% Sort list converting back gives the result in the correct order
    sort(0, @=<, L11, L12),
    (
	%% If the list is empty, the result is a list with 0
	L12 = [], L13 = [0]
    ;
        %% Otherwise, this is the result
        L13 = L12
    ).
%% Tests:
%% ?- simplify_polynomial_as_list([x, 1, x^2, x*y, 3*x^2, 4*x], L).
%@ L = [1, 4*x^2, 5*x, x*y] .
%% ?- simplify_polynomial_as_list([1, x^2, x*y, 3*x^2, -4, -1*x], L).
%@ L = [-3, -1*x, 4*x^2, x*y] .
%% ?- simplify_polynomial_as_list([0*x, 0], L).
%@ L = [0] .

%% join_similar_terms(+P:ListList, -P2:ListList) is det
%
%  Joins similar sublists representing terms by using
%  `add_terms` to check if they can be merged and perform
%  the addition. Requires the list of list be sorted with
%    `maplist(sort(0, @>=), L, L2),
%     sort(0, @>=, L2, L3)`
%  and that the sublists to be sorted with
%  `sort(0, @=<)` since that is inherited from `add_terms`.
%
join_similar_terms([TL, TR | L], L2) :-
    %% Check if terms can be added and add them
    add_terms(TL, TR, T2),
    %% Recurse, accumulation on the first element
    join_similar_terms([T2 | L], L2),
    %% Give only first result. Red cut
    !.
join_similar_terms([X | L], [X | L2]) :-
    %% If a pair of elements can't be added, skip one
    %% and recurse
    join_similar_terms(L, L2),
    %% Give only first result. Red cut
    !.
join_similar_terms([], []).
%% Tests:
%% ?- join_similar_terms([[2, x^3], [3, x^3], [x^3]], L).
%@ L = [[6, x^3]].

%% term_to_canon(+T:List, -T2:List) is det
%
%  Adds the coefficient of the term as the first element of the list
%
%% Special cases to make this predicate reversible
term_to_canon([1], [1]) :-
    !.
term_to_canon(L2, [1 | L]) :-
    nonvar(L),
    L2 = L,
    !.
term_to_canon([-1], [-1]) :-
    !.
term_to_canon([-P | L2], [-1, P | L]) :-
    nonvar(L),
    L2 = L,
    !.
term_to_canon([N2 | L], [N | L]) :-
    number(N),
    N2 = N,
    !.
%% Normal case
term_to_canon(L, [N | L2]) :-
    term_to_canon_with_coefficient(N, L, L2),
    !.
%% Tests:
%% ?- term_to_canon([2], T).
%@ T = [2].
%% ?- term_to_canon([-x], T).
%@ T = [-1, x].
%% ?- term_to_canon([-x^3], T).
%@ T = [-1, x^3].
%% ?- term_to_canon([x^1], T).
%@ T = [1, x^1].
%% ?- term_to_canon([x^3], T).
%@ T = [1, x^3].
%% ?- term_to_canon([x^3, z], T).
%@ T = [1, x^3, z].
%% ?- term_to_canon([2, x^3], T).
%@ T = [2, x^3].
%% ?- term_to_canon([2, -x^3], T).
%@ T = [-2, x^3].
%% ?- term_to_canon([2, -x^3, -z], T).
%@ T = [2, x^3, z].
%% ?- term_to_canon(L, [-1]).
%@ L = [-1].
%% ?- term_to_canon(L, [1]).
%@ L = [1].
%% ?- term_to_canon(L, [-2]).
%@ L = [-2].
%% ?- term_to_canon(L, [-2, x]).
%@ L = [-2, x].
%% ?- term_to_canon(L, [1, x]).
%@ L = [x].
%% ?- term_to_canon(L, [-1, x]).
%@ L = [-x].
%% ?- term_to_canon(L, [1, x, z, y]).
%@ L = [x, z, y].
%% ?- term_to_canon(L, [-1, x, z, y]).
%@ L = [-x, z, y].

%% term_to_canon_with_coefficient(-N:number, +L:List, -L2:List) is semidet
%
%  Calculates the coefficient of the term and removes negations of powers,
%  accumulating the results in N
%
term_to_canon_with_coefficient(N, [N2 | TS], TS2) :-
    number(N2),
    term_to_canon_with_coefficient(N3, TS, TS2),
    N is N2 * N3,
    !.
term_to_canon_with_coefficient(N, [P | TS], [P2 | TS2]) :-
    sign_of_power(P, N2 * P2),
    term_to_canon_with_coefficient(N3, TS, TS2),
    N is N2 * N3,
    !.
term_to_canon_with_coefficient(N, [], []) :-
    nonvar(N);
    N = 1.
%% Tests:
%% ?- term_to_canon_with_coefficient(N, [x], L).
%@ N = 1,
%@ L = [x].
%% ?- term_to_canon_with_coefficient(N, [x, x^2, 2], L).
%@ N = 2,
%@ L = [x^1, x^2].
%% ?- term_to_canon_with_coefficient(N, [x, x^2, 2, 4, z], L).
%@ N = 8,
%@ L = [x, x^2, z].
%% ?- term_to_canon_with_coefficient(N, [x, x^2, 2, 4, -z], L).
%@ N = -8,
%@ L = [x, x^2, z].
%% ?- term_to_canon_with_coefficient(N, [x, -x^2, 2, 4, -z], L).
%@ N = 8,
%@ L = [x, x^2, z].
%% ?- term_to_canon_with_coefficient(N, L, [x]).
%@ N = 1,
%@ L = [x].
%% ?- term_to_canon_with_coefficient(N, L, [1]).
%@ N = 1,
%@ L = [1].
%% ?- term_to_canon_with_coefficient(N, L, [2]).
%@ N = 1,
%@ L = [2].

%% sign_of_power(P:power, P:term) is det
%
%  If there isn't a leading minus, multiplies the power by 1,
%  otherwise by a -1. This way it prefers the positive version.
%  Not idempotent
%
sign_of_power(P, 1*P) :-
    %% If P can't unify with a minus followed by an unnamed variable
    P \= -_,
    !.
sign_of_power(-P, -1*P).
%% Tests:
%% ?- sign_of_power(x, X).
%@ X = 1*x.
%% ?- sign_of_power(-x, X).
%@ X = -1*x.
%% ?- sign_of_power(X, 1*x).
%@ X = x.
%% ?- sign_of_power(X, -1*x).
%@ X = -x.

%% add_terms(+L:List, +R:List, -Result:List) is det
%
%  Adds two terms represented as list by adding
%  the coeficients if the power is the same.
%  Returns false if they can't be added
%  Requires the list of terms to be simplified.
%
add_terms([NL | TL], [NR | TR], [N2 | TL2]) :-
    %% Convert each term to a canon form. This ensures they
    %% have a number in front, so it can be added
    term_to_canon([NL | TL], [NL2 | TL2]),
    term_to_canon([NR | TR], [NR2 | TR2]),
    %% If the rest of the term is the same
    TL2 == TR2,
    %% Add the coeficients
    N2 is NL2 + NR2.
%% Tests
%% ?- add_terms([1], [1], R).
%@ R = [2].
%% ?- add_terms([x], [x], R).
%@ R = [2, x].
%% ?- add_terms([2, x^3], [x^3], R).
%@ R = [3, x^3].
%% ?- add_terms([2, x^3], [3, x^3], R).
%@ R = [5, x^3].
%% ?- add_terms([2, x^3], [3, x^2], R).
%@ false.

%% polynomial_to_list(+P:polynomial, -L:List) is det
%
%  Converts a polynomial in a list.
%
polynomial_to_list(L - T, [T2 | LS]) :-
    term(T),
    negate_term(T, T2),
    polynomial_to_list(L, LS),
    !.
polynomial_to_list(L + T, [T | LS]) :-
    term(T),
    polynomial_to_list(L, LS),
    !.
polynomial_to_list(T, [T]) :-
    term(T),
    !.
%% Tests:
%% ?- polynomial_to_list(2, S).
%@ S = [2].
%% ?- polynomial_to_list(x^2, S).
%@ S = [x^2].
%% ?- polynomial_to_list(x^2 + x^2, S).
%@ S = [x^2, x^2].
%% ?- polynomial_to_list(2*x^2+5+y*2, S).
%@ S = [y*2, 5, 2*x^2].
%% ?- polynomial_to_list(2*x^2+5-y*2, S).
%@ S = [-2*y, 5, 2*x^2].
%% ?- polynomial_to_list(2*x^2-5-y*2, S).
%@ S = [-2*y, -5, 2*x^2].

%% list_to_polynomial(+L:List, -P:Polynomial) is det
%
%  Converts a list in a polynomial.
%  An empty list will return false.
%
list_to_polynomial([T1|T2], P) :-
    % Start recursive calls until we are in the
    % end of the list. We know that the `-` will
    % always be at the left of a term.
    list_to_polynomial(T2, L1),
    (
        % If this is a negative term
        term_string(T1, S1),
        string_chars(S1, [First|_]),
        First = -,
        % Concat them
        term_string(L1, S2),
        string_concat(S2,S1,S3),
        term_string(P, S3)
    ;
        % Otherwise sum them
        P = L1+T1
    ),
    % The others computations are semantically meaningless
    !.
list_to_polynomial([T], T).
%% Tests:
%% ?- list_to_polynomial([1, x, x^2], P).
%@ P = x^2+x+1.
%% ?- list_to_polynomial([-1, -x, -x^2], P).
%@ P = -x^2-x-1.
%% ?- list_to_polynomial([1, -x, x^2], P).
%@ P = x^2-x+1.
%% ?- list_to_polynomial([x^2, x, 1], P).
%@ P = 1+x+x^2.
%% ?- list_to_polynomial([a,-e], P).
%@ P = -e+a.
%% ?- list_to_polynomial([], P).
%@ false.
%% ?- list_to_polynomial([a], P).
%@ P = a.

%% negate_term(T, T2) is det
%
%  Negate the coeficient of a term and return the negated term.
%
negate_term(T, T2) :-
    term_to_list(T, L),
    %% Ensure there is a coeficient
    term_to_canon(L, L2),
    [N | R] = L2,
    %% (-)/1 is an operator, needs to be evaluated, otherwise
    %% it gives a symbolic result, which messes with further processing
    N2 is -N,
    %% Convert the term back from canonic form
    term_to_canon(L3, [N2 | R]),
    %% Reverse the order of the list, because converting
    %% implicitly reverses it
    reverse(L3, L4),
    term_to_list(T2, L4),
    !.
%% Tests:
%% ?- negate_term(1, R).
%@ R = -1.
%% ?- negate_term(x, R).
%@ R = -x.
%% ?- negate_term(-x, R).
%@ R = x.
%% ?- negate_term(x^2, R).
%@ R = -x^2.
%% ?- negate_term(3*x*y^2, R).
%@ R = -3*y^2*x.

%% scale_polynomial(+P:Polynomial,+C:Constant,-S:Polynomial) is det
%
%  Multiplies a polynomial by a scalar.
%
scale_polynomial(P, C, S) :-
    polynomial_to_list(P, L),
    %% Convert each term to a list
    maplist(term_to_list, L, L2),
    %% Canonize terms
    maplist(term_to_canon, L2, L3),
    %% Append C to the start of each sublist
    maplist(cons(C), L3, L4),
    %% Convert to a list of terms
    maplist(term_to_list, L5, L4),
    %% Simplify the resulting polynomial
    simplify_polynomial_as_list(L5, L6),
    %% Return as a simplified polynomial
    list_to_polynomial(L6, S),
    !.
%% Tests:
%% ?- scale_polynomial(3*x^2, 2, S).
%@ S = 6*x^2.

%% cons(+C:atom, +L:List, -L2:List) is det
%
%  Add an atom C to the head of a list L.
%
cons(C, L, [C | L]).
%% Tests:
%% ?- cons(C, L, L2).
%@ L2 = [C|L].

%% add_polynomial(+P1:polynomial,+P2:polynomial,-S:polynomial) is det
%
%  S = P1 + P2.
%
add_polynomial(P1, P2, S) :-
    %% Convert both polynomials to lists
    polynomial_to_list(P1, L1),
    polynomial_to_list(P2, L2),
    %% Join them
    append(L1, L2, L3),
    %% Simplify the resulting polynomial
    simplify_polynomial_as_list(L3, L4),
    %% Convert back
    list_to_polynomial(L4, S),
    !.
%% Tests:
%% ?- add_polynomial(2, 2, S).
%@ S = 4.
%% ?- add_polynomial(x, x, S).
%@ S = 2*x.
%% ?- add_polynomial(2*x+5*z, 2*z+6*x, S).
%@ S = 8*x+7*z.