;; Translate output from parse to a pure list form
;; assuming variables are VARS

(defun alist-form (plist vars)
  "Translates an expression PLIST, which should be a list of polynomials
in variables VARS, to an alist representation of a polynomial.
It returns the alist. See also PARSE-TO-ALIST."
  (cond
   ((endp plist) nil)
   ((eql (first plist) '[)
    (cons '[ (mapcar #'(lambda (x) (alist-form x vars)) (rest plist))))
   (t
    (assert (eql (car plist) '+))
    (alist-form-1 (rest plist) vars))))

(defun alist-form-1 (p vars
			 &aux (ht (make-hash-table
				   :test #'equal :size 16))
			 stack)
  (dolist (term p)
    (assert (eql (car term) '*))
    (incf  (gethash (powers (cddr term) vars) ht 0) (second term)))
  (maphash #'(lambda (key value) (unless (zerop value)
				   (push (cons key value) stack))) ht)
  stack)
    
(defun powers (monom vars
		     &aux (tab (pairlis vars (make-list (length vars)
							:initial-element 0))))
  (dolist (e monom (mapcar #'(lambda (v) (cdr (assoc v tab))) vars))
    (assert (equal (first e) '^))
    (assert (integerp (third e)))
    (assert (= (length e) 3))
    (let ((x (assoc (second e) tab)))
      (if (null x) (error "Variable ~a not in the list of variables." 
			 (second e))
	(incf (cdr x) (third e))))))
  


(defun convert-number (number-or-poly n)
  "Returns NUMBER-OR-POLY, if it is a polynomial. If it is a number,
it converts it to the constant monomial in N variables. If the result
is a number then convert it to a polynomial in N variables."
  (if (numberp number-or-poly)
      (make-poly-from-termlist (list (make-term :monom (make-monom :dimension n) :coeff number-or-poly)))
      number-or-poly))

(defun $poly+ (ring-and-order p q n)
  "Add two polynomials P and Q, where each polynomial is either a
numeric constant or a polynomial in internal representation. If the
result is a number then convert it to a polynomial in N variables."
  (poly-add ring-and-order (convert-number p n) (convert-number q n)))

(defun $poly- (ring-and-order p q n)
  "Subtract two polynomials P and Q, where each polynomial is either a
numeric constant or a polynomial in internal representation. If the
result is a number then convert it to a polynomial in N variables."
  (poly-sub ring-and-order (convert-number p n) (convert-number q n)))

(defun $minus-poly (ring p n)
  "Negation of P is a polynomial is either a numeric constant or a
polynomial in internal representation. If the result is a number then
convert it to a polynomial in N variables."
  (poly-uminus ring (convert-number p n)))

(defun $poly* (ring-and-order p q n)
  "Multiply two polynomials P and Q, where each polynomial is either a
numeric constant or a polynomial in internal representation. If the
result is a number then convert it to a polynomial in N variables."
  (poly-mul ring-and-order (convert-number p n) (convert-number q n)))

(defun $poly/ (ring p q)
  "Divide a polynomials P which is either a numeric constant or a
polynomial in internal representation, by a number Q."
  (if (numberp p)
      (common-lisp:/ p q)
      (scalar-times-poly ring (common-lisp:/ q) p)))
    
(defun $poly-expt (ring-and-order p l n)
  "Raise polynomial P, which is a polynomial in internal
representation or a numeric constant, to power L. If P is a number,
convert the result to a polynomial in N variables."
  (poly-expt ring-and-order (convert-number p n) l))


(defun variable-basis (ring n &aux (basis (make-list n)))
  "Generate a list of polynomials X[i], i=0,1,...,N-1."
  (dotimes (i n basis)
    (setf (elt basis i) (make-poly-variable ring n i))))


(defun poly-eval-1 (expr vars &optional (ring *ring-of-integers*) (order #'lex>)
		    &aux
		      (ring-and-order (make-ring-and-order :ring ring :order order))
		      (n (length vars))
		      (basis (variable-basis ring (length vars))))
  "Evaluate an expression EXPR as polynomial by substituting operators
+ - * expt with corresponding polynomial operators and variables VARS
with the corresponding polynomials in internal form.  We use special
versions of binary operators $poly+, $poly-, $minus-poly, $poly* and
$poly-expt which work like the corresponding functions in the POLY
package, but accept scalars as arguments as well. The result is a
polynomial in internal form.  This operation is somewhat similar to
the function EXPAND in CAS."
  (cond
    ((numberp expr)
     (cond
       ((zerop expr) NIL)
       (t (make-poly-from-termlist (list (make-term :monom (make-monom :dimension n) :coeff expr))))))
    ((symbolp expr)
     (nth (position expr vars) basis))
    ((consp expr)
     (case (car expr)
       (expt
	(if (= (length expr) 3)
	    ($poly-expt ring-and-order
			(poly-eval-1 (cadr expr) vars ring order)
			(caddr expr)
			n)
	    (error "Too many arguments to EXPT")))
       (/
	(if (and (= (length expr) 3)
		 (numberp (caddr expr)))
	    ($poly/ ring (cadr expr) (caddr expr))
	    (error "The second argument to / must be a number")))
       (otherwise
	(let ((r (mapcar
		  #'(lambda (e) (poly-eval-1 e vars ring order))
		  (cdr expr))))
	  (ecase (car expr)
	    (+ (reduce #'(lambda (p q) ($poly+ ring-and-order p q n)) r))
	    (-
	     (if (endp (cdr r))
		 ($minus-poly ring (car r) n)
		 ($poly- ring-and-order
			 (car r)
			 (reduce #'(lambda (p q) ($poly+ ring-and-order p q n)) (cdr r))
			 n)))
	    (*
	     (reduce #'(lambda (p q) ($poly* ring-and-order p q n)) r))
	    )))))))


	     
(defun poly-eval (expr vars &optional (order #'lex>) (ring *ring-of-integers*))
  "Evaluate an expression EXPR, which should be a polynomial
expression or a list of polynomial expressions (a list of expressions
marked by prepending keyword :[ to it) given in Lisp prefix notation,
in variables VARS, which should be a list of symbols. The result of
the evaluation is a polynomial or a list of polynomials (marked by
prepending symbol '[) in the internal alist form. This evaluator is
used by the PARSE package to convert input from strings directly to
internal form."
  (cond
   ((numberp expr)
    (unless (zerop expr)
      (make-poly-from-termlist 
       (list (make-term :monom (make-monom :dimension (length vars)) :coeff expr)))))
   ((or (symbolp expr) (not (eq (car expr) :[)))
    (poly-eval-1 expr vars ring order))
   (t (cons '[ (mapcar #'(lambda (p) (poly-eval-1 p vars ring order)) (rest expr))))))


(defun parse-string-to-alist (str vars)
  "Parse string STR and return a polynomial as a sorted association
list of pairs (MONOM . COEFFICIENT). For example:
(parse-string-to-alist \"[x^2-y^2+(-4/3)*u^2*w^3-5,y]\" '(x y u w))
 ([ (((0 0 2 3) . -4/3) ((0 2 0 0) . -1) ((2 0 0 0) . 1)
     ((0 0 0 0) . -5))
    (((0 1 0 0) . 1)))
The functions PARSE-TO-SORTED-ALIST and PARSE-STRING-TO-SORTED-ALIST
sort terms by the predicate defined in the ORDER package."
  (with-input-from-string (stream str)
    (parse-to-alist vars stream)))


(defun parse-to-sorted-alist (vars &optional (order #'lex>) (stream t))
  "Parses streasm STREAM and returns a polynomial represented as 
a sorted alist. For example:
(WITH-INPUT-FROM-STRING (S \"X^2-Y^2+(-4/3)*U^2*W^3-5\")
  (PARSE-TO-SORTED-ALIST '(X Y U W) S))
returns
(((2 0 0 0) . 1) ((0 2 0 0) . -1) ((0 0 2 3) . -4/3) ((0 0 0 0) . -5))
and
(WITH-INPUT-FROM-STRING (S \"X^2-Y^2+(-4/3)*U^2*W^3-5\")
  (PARSE-TO-SORTED-ALIST '(X Y U W) T #'GRLEX>) S)
returns
(((0 0 2 3) . -4/3) ((2 0 0 0) . 1) ((0 2 0 0) . -1) ((0 0 0 0) . -5))"
  (sort-poly (parse-to-alist vars stream) order))

(defun parse-string-to-sorted-alist (str vars &optional (order #'lex>))
  "Parse a string to a sorted alist form, the internal representation
of polynomials used by our system."
  (with-input-from-string (stream str)
    (parse-to-sorted-alist vars order stream)))

(defun sort-poly-1 (p order)
  "Sort the terms of a single polynomial P using an admissible monomial order ORDER.
Returns the sorted polynomial. Destructively modifies P."
  (sort p order :key #'first))

;; Sort a polynomial or polynomial list
(defun sort-poly (poly-or-poly-list &optional (order #'lex>))
  "Sort POLY-OR-POLY-LIST, which could be either a single polynomial
or a list of polynomials in internal alist representation, using
admissible monomial order ORDER. Each polynomial is sorted using
SORT-POLY-1."
  (cond
   ((eql poly-or-poly-list :syntax-error) nil)
   ((null poly-or-poly-list) nil)
   ((eql (car poly-or-poly-list) '[)
    (cons '[ (mapcar #'(lambda (p) (sort-poly-1 p order)) 
		     (rest poly-or-poly-list))))
   (t (sort-poly-1 poly-or-poly-list order))))