#lang pl

(define-type AE
  [R1 Prod AE]
  [R2 Prod])

(define-type Prod
  [R3 Atom Prod]
  [R4 Atom])

(define-type Atom
  [R5 Number]
  [R6 AE])

; Symbols:
; 'sum
; 'mul
; 'groups
; 'groupe

; parse-ae: s: Sexpr → AE
(: parse-ae : (Sexpr -> AE))
; Parse a Sexpr.
(define (parse-ae s)
  ; parseae: s: Sexpr, a: Sexpr → AE
  (: parseae : (Sexpr Sexpr -> AE))
  ; Parse a Sexpr and while accumulating in a.
  (define (parseae s a)
    (match s
      [(list prod ae) (parseae/group (rest s) '())]
      [(cons (number: n) r) (R2 (R4 (R5 n)))]))

  (parseae s '()))


(test (parse-ae '(3)) => (R2 (R4 (R5 3))))
(test (parse-ae '(groups 3 groupe)) => (R2 (R4 (R6 (R2 (R4 (R5 3)))))))
(test (parse-ae '(1 plus 1)) => (R1 (R4 (R5 1)) (R2 (R4 (R5 1)))))

; parse-ae/h: s: Sexpr, ae: AE → AE

#|
<AE> ::= <PROD> + <AE> (1)
         | <PROD> (2)

<PROD> ::= <ATOM> * <PROD> (3)
           | <ATOM> (4)

<ATOM> ::= <num> (5)
           | { <AE> } (6)

"5"
<atom> (5)
 <num> 5

"{5}"
<atom> (6)
 <ae> (2)
  <prod> (4)
   <atom> (5)
    <num> 5

"{2 + 5}"
<atom> (6)
 <ae> (1)
  <prod> (4)
   <atom> (5)
    <num> 2
  <ae> (2)
   <prod> (4)
    <atom> (5)
     <num> 5

"{{{2 + 5}} * {5 * {{3 + 6}}}}"
<atom> (6)
 <ae> (2)
  <prod> (3)
   <atom> (6)
    <atom> (6)
     <ae> (1)
      <prod> (4)
       <atom> (5)
        <num> 2
      <ae> (2)
       <prod> (4)
        <atom> (5)
         <num> 5
   <prod> (4)
    <atom> (6)
     <prod> (3)
      <atom> (5)
       <num> 5
      <prod> (4)
       <atom> (6)
        <atom> (6)
         <ae> (1)
          <prod> (4)
           <atom> (5)
            <num> 3
          <ae> (2)
           <prod> (4)
            <atom> (5)
             <num> 6

|#