Add better support for negative numbers

polymani.pl

@@ -27,6 +27,11 @@
  * 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)).
 
 
                  /*******************************
@@ -82,6 +87,51 @@ addpoly(P1, P2, S) :-
     is_polynomial_valid_in_predicate(P2, "addpoly"),
     add_polynomial(P1, P2, S),
     !.
+%% Tests:
+%% ?- addpoly(3 + x, 3 - x, S).
+%@ S = 0*x+6.  TODO HERE
+
+%% 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) :-
+    %% Writes the polynomial and fails otherwise
+    write("Invalid polynomial in "),
+    write(F),
+    write(": "),
+    write(P),
+    fail.
+%% Tests:
+%% ?- is_polynomial_valid_in_predicate(1, "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.
 
 
                  /*******************************
@@ -112,12 +162,8 @@ polynomial_variable(X) :-
 %  Returns true if X is a power term, false otherwise.
 %
 power(P^N) :-
-    (
+    N #>= 1,
-        N #>= 1,
+    polynomial_variable(P).
-        polynomial_variable(P)
-    ;
-        fail
-    ).
 power(X) :-
     polynomial_variable(X).
 %% Tests:
@@ -131,13 +177,15 @@ power(X) :-
 %@ true .
 %% ?- power(x^(-3)).
 %@ false.
+%% ?- power(-x).
+%@ false.
 %% ?- power(X).
-%@ X = x^_2420,
+%@ X = x^_462546,
-%@ _2420 in 0..sup ;
+%@ _462546 in 1..sup ;
-%@ X = y^_2420,
+%@ X = y^_462546,
-%@ _2420 in 0..sup ;
+%@ _462546 in 1..sup ;
-%@ X = z^_2420,
+%@ X = z^_462546,
-%@ _2420 in 0..sup ;
+%@ _462546 in 1..sup ;
 %@ X = x ;
 %@ X = y ;
 %@ X = z.
@@ -148,22 +196,25 @@ power(X) :-
 %
 term(N) :-
     (
-        % If N is non a free variable
+        % If N is not a free variable
         nonvar(N),
         % Assert it as a number
         number(N)
     ;
         % If N is a free variable
-        not(compound(N)),
+        %% not(compound(N)),
         var(N),
         % Assert it must be between negative and positive infinity
         % This uses the CLP(FD) library, which makes this reversible,
         % whereas `number(N)` is always false, since it only succeeds
         % if the argument is bound to a intger or float
-        N in inf..sup
+        %% N in inf..sup
+	{N >= 0; N < 0}
     ).
 term(X) :-
     power(X).
+term(-X) :-
+    power(X).
 term(L * R) :-
     term(L),
     term(R).
@@ -176,78 +227,35 @@ term(L * R) :-
 %@ 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 in inf..sup ;
+%@ {X>=0.0} ;
-%@ X = x^_1242,
+%@ {X<0.0} ;
-%@ _1242 in 1..sup ;
+%@ X = x^_111514,
-%@ X = y^_1242,
+%@ _111514 in 1..sup ;
-%@ _1242 in 1..sup ;
+%@ X = y^_111514,
-%@ X = z^_1242,
+%@ _111514 in 1..sup ;
-%@ _1242 in 1..sup ;
+%@ X = z^_111514,
+%@ _111514 in 1..sup ;
 %@ X = x ;
 %@ X = y ;
 %@ X = z ;
-%@ X = _1330*_1332,
+%@ X = -x^_111522,
-%@ _1330 in inf..sup,
+%@ _111522 in 1..sup ;
-%@ _1332 in inf..sup ;
+%@ X = -y^_111522,
-%@ X = _1406*x^_1414,
+%@ _111522 in 1..sup ;
-%@ _1406 in inf..sup,
+%@ X = -z^_111522,
-%@ _1414 in 1..sup ;
+%@ _111522 in 1..sup ;
-%@ X = _1406*y^_1414,
+%@ X = -x ;
-%@ _1406 in inf..sup,
+%@ X = -y ;
-%@ _1414 in 1..sup ;
+%@ X = -z ;
-%@ X = _1406*z^_1414,
-%@ _1406 in inf..sup,
-%@ _1414 in 1..sup ;
-%@ X = _1188*x,
-%@ _1188 in inf..sup .
-%% Doesn't give all possible terms, because number(N) is not reversible
-
-%% 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) :-
-    %% Writes the polynomial and fails otherwise
-    write("Invalid polynomial in "),
-    write(F),
-    write(": "),
-    write(P),
-    fail.
-%% Tests:
-%% ?- is_polynomial_valid_in_predicate(1, "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.
 
 %% polynomial(+M:atom) is semidet
 %
@@ -279,6 +287,10 @@ polynomial(L - R) :-
 %@ false.
 %% ?- polynomial(x^(-3)).
 %@ false.
+%% ?- polynomial(-x + 3).
+%@ true .
+%% ?- polynomial(-x - -z).
+%@ true .
 
 %% power_to_canon(+T:atom, -T^N:atom) is semidet
 %
@@ -291,9 +303,14 @@ power_to_canon(T^N, T^N) :-
     N #\= 1.
 power_to_canon(T, T^1) :-
     polynomial_variable(T).
+%% power_to_canon(-P, -P2) :-
+%%     power_to_canon(P, P2).
 %% 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).
@@ -302,6 +319,8 @@ power_to_canon(T, T^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
 %
@@ -317,16 +336,28 @@ 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] .
+%@ 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).
@@ -337,6 +368,9 @@ term_to_list(P, [P2]) :-
 %@ X = -1 .
 %% ?- term_to_list(X, [x^1, -1]).
 %@ X = -1*x .
+%@ 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]).
@@ -393,6 +427,10 @@ simplify_term(Term_In, Term_Out) :-
 %@ 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).
@@ -439,6 +477,8 @@ join_similar_parts_of_term([], []).
 %@ 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
 %
@@ -511,8 +551,12 @@ simplify_polynomial_as_list(L, L11) :-
 %% 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] .
+%@ 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] .
+%@ L = [-3, -1*x, 4*x^2, x*y] .
+%% ?- simplify_polynomial_as_list([0*x], L).
+%@ L = [0*x] .
 
 %% join_similar_terms(+P:ListList, -P2:ListList) is det
 %
@@ -544,34 +588,130 @@ join_similar_terms([], []).
 
 %% term_to_canon(+T:List, -T2:List) is det
 %
-%  Adds a 1 if there's no number in the list.
+%  Adds the coefficient of the term as the first element of the list
-%  Requires the list to be sorted such that the
-%  numbers come first.
-%  For instance with `sort(0, @=<)`.
 %
-term_to_canon([T | TS], [1, T | TS]) :-
+term_to_canon([1], [1]) :-
-    %% Since the list is sorted, if the first element
-    %% is not a number, then we need to add the 1
-    not(number(T)),
-    %% Give only first result. Red cut
     !.
-term_to_canon([T | TS], [N, T | TS]) :-
+term_to_canon(L2, [1 | L]) :-
-    %% Since the list is sorted, if the first element
+    nonvar(L),
-    %% is not a number, then we need to add the 1
+    L2 = L,
-    not(number(T)),
+    !.
-    N is -1,
+term_to_canon([-1], [-1]) :-
-    %% Give only first result. Red cut
+    !.
+term_to_canon([-P | L2], [-1, P | L]) :-
+    nonvar(L),
+    L2 = L,
+    !.
+term_to_canon([N2 | L], [N | L]) :-
+    number(N),
+    N2 = N,
+    !.
+term_to_canon(L, [N | L2]) :-
+    %% N == 1 -> L = L2
+    %% ;
+    term_to_canon_with_coefficient(N, L, L2),
     !.
-term_to_canon(L, L).
 %% 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),
+    %% {N2 >= 0; N2 < 0},
+    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
 %
@@ -678,28 +818,29 @@ list_to_polynomial([T], T).
 %
 negate_term(T, T2) :-
     term_to_list(T, L),
-    %% Sort the list, so the coeficient is the first element
-    sort(0, @=<, L, L2),
     %% Ensure there is a coeficient
-    term_to_canon(L2, L3),
+    term_to_canon(L, L2),
-    [N | R] = L3,
+    [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,
     %% Reverse the order of the list, because converting
     %% implicitly reverses it
-    reverse([N2 | R], L4),
+    term_to_canon(L3, [N2 | R]),
+    reverse(L3, L4),
     term_to_list(T2, L4),
     !.
 %% Tests:
 %% ?- negate_term(1, R).
 %@ R = -1.
 %% ?- negate_term(x, R).
-%@ R = -1*x.
+%@ R = -x.
+%% ?- negate_term(-x, R).
+%@ R = x.
 %% ?- negate_term(x^2, R).
-%@ R = -1*x^2.
+%@ R = -x^2.
 %% ?- negate_term(3*x*y^2, R).
-%@ R = -3*x*y^2.
+%@ R = -3*y^2*x.
 
 %% scale_polynomial(+P:Polynomial,+C:Constant,-S:Polynomial) is det
 %