;;; -*-  Mode: Lisp -*- 
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;                                                                              
;;;  Copyright (C) 1999, 2002, 2009, 2015 Marek Rychlik <rychlik@u.arizona.edu>		 
;;;  		       								 
;;;  This program is free software; you can redistribute it and/or modify	 
;;;  it under the terms of the GNU General Public License as published by	 
;;;  the Free Software Foundation; either version 2 of the License, or		 
;;;  (at your option) any later version.					 
;;; 		       								 
;;;  This program is distributed in the hope that it will be useful,		 
;;;  but WITHOUT ANY WARRANTY; without even the implied warranty of		 
;;;  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the		 
;;;  GNU General Public License for more details.				 
;;; 		       								 
;;;  You should have received a copy of the GNU General Public License		 
;;;  along with this program; if not, write to the Free Software 		 
;;;  Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.	 
;;;										 
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; Load this file into Maxima to bootstrap the Grobner package. 
;; NOTE: This file does use symbols defined by Maxima, so it
;; will not work when loaded in Common Lisp.
;;
;; DETAILS: This file implements an interface between the Grobner
;; basis package NGROBNER, which is a pure Common Lisp package, and
;; Maxima. NGROBNER for efficiency uses its own representation of
;; polynomials. Thus, it is necessary to convert Maxima representation
;; to the internal representation and back. The facilities to do so
;; are implemented in this file.
;;
;; Also, since the NGROBNER package consists of many Lisp files, it is
;; necessary to load the files. It is possible and preferrable to use
;; ASDF for this purpose. The default is ASDF. It is also possible to
;; simply used LOAD and COMPILE-FILE to accomplish this task.
;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(in-package :maxima)

(macsyma-module cgb-maxima)


(eval-when
    #+gcl (load eval)
    #-gcl (:load-toplevel :execute)
    (format t "~&Loading maxima-grobner ~a ~a~%"
	    "$Revision: 2.0 $" "$Date: 2015/06/02 0:34:17 $"))

;;FUNCTS is loaded because it contains the definition of LCM
($load "functs")
#+sbcl(progn (require 'asdf) (load "ngrobner.asd")(asdf:load-system :ngrobner))

(use-package :ngrobner)


;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; Maxima expression ring
;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; This is how we perform operations on coefficients
;; using Maxima functions. 
;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(defparameter *maxima-ring*
    (make-ring 
     ;;(defun coeff-zerop (expr) (meval1 `(($is) (($equal) ,expr 0))))
     :parse #'(lambda (expr)
		(when modulus (setf expr ($rat expr)))
		expr)
     :unit #'(lambda () (if modulus ($rat 1) 1))
     :zerop #'(lambda (expr)
		;;When is exactly a maxima expression equal to 0?
		(cond ((numberp expr)
		       (= expr 0))
		      ((atom expr) nil)
		      (t
		       (case (caar expr)
			 (mrat (eql ($ratdisrep expr) 0))
			 (otherwise (eql ($totaldisrep expr) 0))))))
     :add #'(lambda (x y) (m+ x y))
     :sub #'(lambda (x y) (m- x y))
     :uminus #'(lambda (x) (m- x))
     :mul #'(lambda (x y) (m* x y))
     ;;(defun coeff-div (x y) (cadr ($divide x y)))
     :div #'(lambda (x y) (m// x y))
     :lcm #'(lambda (x y) (meval1 `((|$LCM|) ,x ,y)))
     :ezgcd #'(lambda (x y) (apply #'values (cdr ($ezgcd ($totaldisrep x) ($totaldisrep y)))))
     ;; :gcd #'(lambda (x y) (second ($ezgcd x y)))))
     :gcd #'(lambda (x y) ($gcd x y))))

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; Maxima expression parsing
;;
;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; Functions and macros dealing with internal representation
;; structure.
;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(defun equal-test-p (expr1 expr2)
  (alike1 expr1 expr2))

(defun coerce-maxima-list (expr)
  "Convert a Maxima list to Lisp list."
  (cond
   ((and (consp (car expr)) (eql (caar expr) 'mlist)) (cdr expr))
   (t expr)))

(defun free-of-vars (expr vars) (apply #'$freeof `(,@vars ,expr)))

(defun parse-poly (expr vars &aux (vars (coerce-maxima-list vars)))
  "Convert a maxima polynomial expression EXPR in variables VARS to internal form."
  (labels ((parse (arg) (parse-poly arg vars))
	   (parse-list (args) (mapcar #'parse args)))
    (cond
     ((eql expr 0) (make-poly-zero))
     ((member expr vars :test #'equal-test-p)
      (let ((pos (position expr vars :test #'equal-test-p)))
	(make-variable *maxima-ring* (length vars) pos)))
     ((free-of-vars expr vars)
      ;;This means that variable-free CRE and Poisson forms will be converted
      ;;to coefficients intact
      (coerce-coeff *maxima-ring* expr vars))
     (t
      (case (caar expr)
	(mplus (reduce #'(lambda (x y) (poly-add *maxima-ring* x y)) (parse-list (cdr expr))))
	(mminus (poly-uminus *maxima-ring* (parse (cadr expr))))
	(mtimes
	 (if (endp (cddr expr))		;unary
	     (parse (cdr expr))
	   (reduce #'(lambda (p q) (poly-mul *maxima-ring* p q)) (parse-list (cdr expr)))))
	(mexpt
	 (cond
	  ((member (cadr expr) vars :test #'equal-test-p)
	   ;;Special handling of (expt var pow)
	   (let ((pos (position (cadr expr) vars :test #'equal-test-p)))
	     (make-variable *maxima-ring* (length vars) pos (caddr expr))))
	  ((not (and (integerp (caddr expr)) (plusp (caddr expr))))
	   ;; Negative power means division in coefficient ring
	   ;; Non-integer power means non-polynomial coefficient
	   (mtell "~%Warning: Expression ~%~M~%contains power which is not a positive integer. Parsing as coefficient.~%"
		  expr)
	   (coerce-coeff *maxima-ring* expr vars))
	  (t (poly-expt *maxima-ring* (parse (cadr expr)) (caddr expr)))))
	(mrat (parse ($ratdisrep expr)))
	(mpois (parse ($outofpois expr)))
	(otherwise
	 (coerce-coeff *maxima-ring* expr vars)))))))

(defun parse-poly-list (expr vars)
  "Parse a Maxima representation of a list of polynomials."
  (case (caar expr)
    (mlist (mapcar #'(lambda (p) (parse-poly p vars)) (cdr expr)))
    (t (merror "Expression ~M is not a list of polynomials in variables ~M."
	       expr vars))))

(defun parse-poly-list-list (poly-list-list vars)
  "Parse a Maxima representation of a list of lists of polynomials."
  (mapcar #'(lambda (g) (parse-poly-list g vars)) (coerce-maxima-list poly-list-list)))


;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; Conversion from internal form to Maxima general form
;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(defun maxima-head ()
  (if $poly_return_term_list
      '(mlist)
    '(mplus)))

(defun coerce-to-maxima (poly-type object vars)
  (case poly-type
    (:polynomial 
     `(,(maxima-head) ,@(mapcar #'(lambda (term) (coerce-to-maxima :term term vars)) (poly-termlist object))))
    (:poly-list
     `((mlist) ,@(mapcar #'(lambda (p) (funcall *ratdisrep-fun* (coerce-to-maxima :polynomial p vars))) object)))
    (:term
     `((mtimes) ,(funcall *ratdisrep-fun* (term-coeff object))
       ,@(mapcar #'(lambda (var power) `((mexpt) ,var ,power))
		 vars (coerce (term-monom object) 'list))))
    ;; Assumes that Lisp and Maxima logicals coincide
    (:logical object)
    (otherwise
     object)))


;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; Unary and binary operation definition facility
;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(defmacro define-unop (maxima-name fun-name
		       &optional (documentation nil documentation-supplied-p))
  "Define a MAXIMA-level unary operator MAXIMA-NAME corresponding to unary function FUN-NAME."
  `(defun ,maxima-name (p vars
			     &aux
			     (vars (coerce-maxima-list vars))
			     (p (parse-poly p vars)))
     ,@(when documentation-supplied-p (list documentation))
     (coerce-to-maxima :polynomial (,fun-name *maxima-ring* p) vars)))

(defmacro define-binop (maxima-name fun-name
			&optional (documentation nil documentation-supplied-p))
  "Define a MAXIMA-level binary operator MAXIMA-NAME corresponding to binary function FUN-NAME."
  `(defmfun ,maxima-name (p q vars
			     &aux
			     (vars (coerce-maxima-list vars))
			     (p (parse-poly p vars))
			     (q (parse-poly q vars)))
     ,@(when documentation-supplied-p (list documentation))
     (coerce-to-maxima :polynomial (,fun-name *maxima-ring* p q) vars)))


;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; Order utilities
;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(defun find-order (order)
  "This function returns the order function bases on its name."
  (cond
   ((null order) nil)
   ((symbolp order)
    (case order
      ((lex :lex $lex) #'lex>) 
      ((grlex :grlex $grlex) #'grlex>)
      ((grevlex :grevlex $grevlex) #'grevlex>)
      ((invlex :invlex $invlex) #'invlex>)
      ((elimination-order-1 :elimination-order-1 elimination_order_1) #'elimination-order-1)
      (otherwise
       (mtell "~%Warning: Order ~M not found. Using default.~%" order))))
   (t
    (mtell "~%Order specification ~M is not recognized. Using default.~%" order)
    nil)))

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; Ring utilities
;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(defun find-ring (ring)
  "This function returns the ring structure bases on input symbol."
  (cond
   ((null ring) nil)
   ((symbolp ring)
    (case ring
      ((maxima-ring  :maxima-ring $expression_ring) *maxima-ring*) 
      ((ring-of-integers :ring-of-integers $ring_of_integers) *ring-of-integers*) 
      (otherwise
       (mtell "~%Warning: Ring ~M not found. Using default.~%" ring))))
   (t
    (mtell "~%Ring specification ~M is not recognized. Using default.~%" ring)
    nil)))


;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; Facilities for evaluating Grobner package expressions
;; within a prepared environment
;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(defmacro with-monomial-order ((order) &body body)
  "Evaluate BODY with monomial order set to ORDER."
  `(let ((*monomial-order* (or (find-order ,order) *monomial-order*)))
     . ,body))

(defmacro with-coefficient-ring ((ring) &body body)
  "Evaluate BODY with coefficient ring set to RING."
  `(let ((*maxima-ring* (or (find-ring ,ring) *maxima-ring*)))
     . ,body))

(defmacro with-ring-and-order ((ring order) &body body)
  "Evaluate BODY with monomial order set to ORDER and coefficient ring set to RING."
  `(let ((*monomial-order* (or (find-order ,order) *monomial-order*))
	 (*maxima-ring* (or (find-ring ,ring) *maxima-ring*)))
     . ,body))

(defmacro with-elimination-orders ((primary secondary elimination-order)
				   &body body)
  "Evaluate BODY with primary and secondary elimination orders set to PRIMARY and SECONDARY."
  `(let ((*primary-elimination-order* (or (find-order ,primary)  *primary-elimination-order*))
	 (*secondary-elimination-order* (or (find-order ,secondary) *secondary-elimination-order*))
	 (*elimination-order* (or (find-order ,elimination-order) *elimination-order*)))
     . ,body))


;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; Maxima-level interface functions
;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

;; Auxillary function for removing zero polynomial
(defun remzero (plist) (remove #'poly-zerop plist))

;;Simple operators

(define-binop $poly_add poly-add
  "Adds two polynomials P and Q")

(define-binop $poly_subtract poly-sub
  "Subtracts a polynomial Q from P.")

(define-binop $poly_multiply poly-mul
  "Returns the product of polynomials P and Q.")

(define-binop $poly_s_polynomial spoly
  "Returns the syzygy polynomial (S-polynomial) of two polynomials P and Q.")

(define-unop $poly_primitive_part poly-primitive-part
  "Returns the polynomial P divided by GCD of its coefficients.")

(define-unop $poly_normalize poly-normalize
  "Returns the polynomial P divided by the leading coefficient.")

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; Macro facility for writing Maxima-level wrappers for
;; functions operating on internal representation
;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

(defmacro with-parsed-polynomials (((maxima-vars &optional (maxima-new-vars nil new-vars-supplied-p))
				    &key (polynomials nil)
				    (poly-lists nil)
				    (poly-list-lists nil)
				    (value-type nil))
				   &body body
				   &aux (vars (gensym))
				     (new-vars (gensym)))
  `(let ((,vars (coerce-maxima-list ,maxima-vars))
	 ,@(when new-vars-supplied-p
		 (list `(,new-vars (coerce-maxima-list ,maxima-new-vars)))))
     (coerce-to-maxima
      ,value-type
      (with-coefficient-ring ($poly_coefficient_ring)
	(with-monomial-order ($poly_monomial_order)
	  (with-elimination-orders ($poly_primary_elimination_order
				    $poly_secondary_elimination_order
				    $poly_elimination_order)
	    (let ,(let ((args nil))
		       (dolist (p polynomials args)
			 (setf args (cons `(,p (parse-poly ,p ,vars)) args)))
		       (dolist (p poly-lists args)
			 (setf args (cons `(,p (parse-poly-list ,p ,vars)) args)))
		       (dolist (p poly-list-lists args)
			 (setf args (cons `(,p (parse-poly-list-list ,p ,vars)) args))))
	      . ,body))))
      ,(if new-vars-supplied-p
	   `(append ,vars ,new-vars)
	   vars))))


;;Functions

(defmfun $poly_expand (p vars)
  "This function is equivalent to EXPAND(P) if P parses correctly to a polynomial.
If the representation is not compatible with a polynomial in variables VARS,
the result is an error."
  (with-parsed-polynomials ((vars) :polynomials (p)
			    :value-type :polynomial)
			   p))

(defmfun $poly_expt (p n vars)
  (with-parsed-polynomials ((vars) :polynomials (p) :value-type :polynomial)
    (poly-expt *maxima-ring* p n)))

(defmfun $poly_content (p vars)
  (with-parsed-polynomials ((vars) :polynomials (p))
    (poly-content *maxima-ring* p)))

(defmfun $poly_pseudo_divide (f fl vars
			    &aux (vars (coerce-maxima-list vars))
				 (f (parse-poly f vars))
				 (fl (parse-poly-list fl vars)))
  (multiple-value-bind (quot rem c division-count)
      (poly-pseudo-divide *maxima-ring* f fl)
    `((mlist)
      ,(coerce-to-maxima :poly-list quot vars)
      ,(coerce-to-maxima :polynomial rem vars)
      ,c
      ,division-count)))

(defmfun $poly_exact_divide (f g vars)
  (with-parsed-polynomials ((vars) :polynomials (f g) :value-type :polynomial)
    (poly-exact-divide *maxima-ring* f g)))

(defmfun $poly_normal_form (f fl vars)
  (with-parsed-polynomials ((vars) :polynomials (f)
				   :poly-lists (fl)
				   :value-type :polynomial)
    (normal-form *maxima-ring* f (remzero fl) nil)))

(defmfun $poly_buchberger_criterion (g vars)
  (with-parsed-polynomials ((vars) :poly-lists (g) :value-type :logical)
    (buchberger-criterion *maxima-ring* g)))

(defmfun $poly_buchberger (fl vars)
  (with-parsed-polynomials ((vars) :poly-lists (fl) :value-type :poly-list)
    (buchberger *maxima-ring*  (remzero fl) 0 nil)))

(defmfun $poly_reduction (plist vars)
  (with-parsed-polynomials ((vars) :poly-lists (plist)
				   :value-type :poly-list)
    (reduction *maxima-ring* plist)))

(defmfun $poly_minimization (plist vars)
  (with-parsed-polynomials ((vars) :poly-lists (plist)
				   :value-type :poly-list)
    (minimization plist)))

(defmfun $poly_normalize_list (plist vars)
  (with-parsed-polynomials ((vars) :poly-lists (plist)
				   :value-type :poly-list)
    (poly-normalize-list *maxima-ring* plist)))

(defmfun $poly_grobner (f vars)
  (with-parsed-polynomials ((vars) :poly-lists (f)
				   :value-type :poly-list)
    (grobner *maxima-ring* (remzero f))))

(defmfun $poly_reduced_grobner (f vars)
  (with-parsed-polynomials ((vars) :poly-lists (f)
				   :value-type :poly-list)
    (reduced-grobner *maxima-ring* (remzero f))))

(defmfun $poly_depends_p (p var mvars
			&aux (vars (coerce-maxima-list mvars))
			     (pos (position var vars)))
  (if (null pos)
      (merror "~%Variable ~M not in the list of variables ~M." var mvars)
    (poly-depends-p (parse-poly p vars) pos)))

(defmfun $poly_elimination_ideal (flist k vars)
  (with-parsed-polynomials ((vars) :poly-lists (flist)
				   :value-type :poly-list)
    (elimination-ideal *maxima-ring* flist k nil 0)))

(defmfun $poly_colon_ideal (f g vars)
  (with-parsed-polynomials ((vars) :poly-lists (f g) :value-type :poly-list)
    (colon-ideal *maxima-ring* f g nil)))

(defmfun $poly_ideal_intersection (f g vars)
  (with-parsed-polynomials ((vars) :poly-lists (f g) :value-type :poly-list)  
    (ideal-intersection *maxima-ring* f g nil)))

(defmfun $poly_lcm (f g vars)
  (with-parsed-polynomials ((vars) :polynomials (f g) :value-type :polynomial)
    (poly-lcm *maxima-ring* f g)))

(defmfun $poly_gcd (f g vars)
  ($first ($divide (m* f g) ($poly_lcm f g vars))))

(defmfun $poly_grobner_equal (g1 g2 vars)
  (with-parsed-polynomials ((vars) :poly-lists (g1 g2))
    (grobner-equal *maxima-ring* g1 g2)))

(defmfun $poly_grobner_subsetp (g1 g2 vars)
  (with-parsed-polynomials ((vars) :poly-lists (g1 g2))
    (grobner-subsetp *maxima-ring* g1 g2)))

(defmfun $poly_grobner_member (p g vars)
  (with-parsed-polynomials ((vars) :polynomials (p) :poly-lists (g))
    (grobner-member *maxima-ring* p g)))

(defmfun $poly_ideal_saturation1 (f p vars)
  (with-parsed-polynomials ((vars) :poly-lists (f) :polynomials (p)
				   :value-type :poly-list)
    (ideal-saturation-1 *maxima-ring* f p 0)))

(defmfun $poly_saturation_extension (f plist vars new-vars)
  (with-parsed-polynomials ((vars new-vars)
			    :poly-lists (f plist)
			    :value-type :poly-list)
    (saturation-extension *maxima-ring* f plist)))

(defmfun $poly_polysaturation_extension (f plist vars new-vars)
  (with-parsed-polynomials ((vars new-vars)
			    :poly-lists (f plist)
			    :value-type :poly-list)
    (polysaturation-extension *maxima-ring* f plist)))

(defmfun $poly_ideal_polysaturation1 (f plist vars)
  (with-parsed-polynomials ((vars) :poly-lists (f plist)
				   :value-type :poly-list)
    (ideal-polysaturation-1 *maxima-ring* f plist 0 nil)))

(defmfun $poly_ideal_saturation (f g vars)
  (with-parsed-polynomials ((vars) :poly-lists (f g)
				   :value-type  :poly-list)
    (ideal-saturation *maxima-ring* f g 0 nil)))

(defmfun $poly_ideal_polysaturation (f ideal-list vars)
  (with-parsed-polynomials ((vars) :poly-lists (f)
				   :poly-list-lists (ideal-list)
				   :value-type :poly-list)
    (ideal-polysaturation *maxima-ring* f ideal-list 0 nil)))

(defmfun $poly_lt (f vars)
  (with-parsed-polynomials ((vars) :polynomials (f) :value-type :polynomial)
    (make-poly-from-termlist (list (poly-lt f)))))

(defmfun $poly_lm (f vars)
  (with-parsed-polynomials ((vars) :polynomials (f) :value-type :polynomial)
    (make-poly-from-termlist (list (make-term (poly-lm f) (funcall (ring-unit *maxima-ring*)))))))