;;;;;
;;;;;
;;;;; Arithmetic Operation
;;;;;
;;;;;

(define $to-math-expr (macro [$arg] (math-normalize b.+ 0 (apply to-math-expr' arg))))

(define $+' (cambda $xs (foldl b.+' (car xs) (cdr xs))))
(define $-' (cambda $xs (foldl b.-' (car xs) (cdr xs))))
(define $*' (cambda $xs (foldl b.*' (car xs) (cdr xs))))
(define $/' b./')
(define $.' (cambda $xs (foldl b..' (car xs) (cdr xs))))

(define $b.+
  (lambda [$x1 $x2]
    (match [x1 x2] [math-expr math-expr]
      {[[<div $p1 (& !,1 $p2)> <div $q1 ,p2>]
        (b./ (b.+ p1 q1) p2)]
       [[_ _] (reduce-fraction (math-normalize b.+' x1 x2))]})))

(define $b.-
  (lambda [$x1 $x2]
    (match [x1 x2] [math-expr math-expr]
      {[[<div $p1 (& !,1 $p2)> <div $q1 ,p2>]
        (b./ (b.- p1 q1) p2)]
       [[_ _] (reduce-fraction (math-normalize b.-' x1 x2))]})))

(define $b.*
  (lambda [$x1 $x2]
    (reduce-fraction (math-normalize b.*' x1 x2))))

(define $b./
  (lambda [$x1 $x2]
    (reduce-fraction (math-normalize b./' x1 x2))))

(define $+ (cambda $xs (foldl b.+ (car xs) (cdr xs))))
(define $- (cambda $xs (foldl b.- (car xs) (cdr xs))))
(define $* (cambda $xs (foldl b.* (car xs) (cdr xs))))
(define $/ b./)


(define $reduce-fraction
  (lambda [$mexpr]
    (match mexpr math-expr
      {[<div <poly $ts1>
             <poly $ts2>>
        (let {[$d (capply gcd {@ts2 @ts1})]}
          (/' (sum' (map (/' $ d) ts1)) (sum' (map (/' $ d) ts2))))]})))

(define $sum
  (lambda [$xs]
    (foldl + 0 xs)))

(define $sum'
  (lambda [$xs]
    (foldl +' 0 xs)))

(define $product
  (lambda [$xs]
    (foldl * 1 xs)))

(define $product'
  (lambda [$xs]
    (foldl *' 1 xs)))

(define $power
  (lambda [$x $n]
    (foldl * 1 (take n (repeat1 x)))))

(define $power'
  (lambda [$x $n]
    (foldl *' 1 (take n (repeat1 x)))))

(define $**
  (lambda [$x $n]
    (if (eq? x e)
      (exp n)
      (if (rational? n)
        (if (gte? n 0)
          (if (integer? n)
            (power x n)
            (`** x n))
          (/ 1 (** x (neg n))))
        (`** x n)))))

(define $**'
  (lambda [$x $n]
    (if (eq? x e)
      (exp n)
      (if (rational? n)
        (if (gte? n 0)
          (if (integer? n)
            (power' x n)
            (`** x n))
          (/' 1 (**' x (neg n))))
        (`** x n)))))

(define $gcd
  (cambda $xs
    (foldl b.gcd (car xs) (cdr xs))))

(define $gcd'
  (cambda $xs
    (foldl b.gcd' (car xs) (cdr xs))))

(define $b.gcd
  (lambda [$x $y]
    (match [x y] [term-expr term-expr]
      {[[_ ,0] x]
       [[,0 _] y]
       [[<term $a $xs> <term $b $ys>]
        (*' (b.gcd' (abs a) (abs b)) (foldl *' 1 (map 2#(**' %1 %2) (AC.intersect xs ys))))]})))

(define $b.gcd'
  (lambda [$x $y]
    (match [x y] [integer integer]
      {[[_ ,0] x]
       [[,0 _] y]
       [[_ ?(gte? $ x)] (b.gcd' (modulo y x) x)]
       [[_ _] (b.gcd' y x)]})))