#lang pl

#|

"{{{2 + 5}} * {5 * {{3 + 6}}}}"
The parenthesis    ^here & ^here are necessary because otherwise it would be an AE, not a Prod as R3 requires. It requires this to maintain order of operations.

"{{{2 + 5}} * {5 * {{3 * 6}}}}"
The parenthesis are no longer needed, as {3 * 6} is itself a prod satisfying R3.

If you have a grammar language with easy precedence, it's nice. Otherwise, it's cursed.

In Yacc:
%left ADD SUB
%left MUL DIV
%precedence NEG

exp:
    ...
    | NUM {...}
    | GROUPSTART exp GROUPEND { ... }
    | SUB exp {...}
    | exp SUM exp {...}
    | exp SUB exp {...}
    | exp MUL exp {...}
    | exp DIV exp {...}

|#

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

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

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

#|
<AE> ::= <PROD> + <AE> (1)
       | <PROD> (2)
<PROD> ::= <ATOM> * <PROD> (3)
         | <ATOM> (4)
<ATOM> ::= <num> (5)
         | { <AE> } (6)
|#

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

(: parseae : (Sexpr -> AE))
(define (parseae s)
  (match s
    [(list prod '+ ae) (R1 (parseprod prod) (parseae ae))]
    [prod (R2 (parseprod prod))]))

(: parseprod : (Sexpr -> Prod))
(define (parseprod s)
  (match s
    [(list atom '* prod) (R3 (parseatom atom) (parseprod prod))]
    [ atom (R4 (parseatom atom))]))

(: parseatom : (Sexpr -> Atom))
(define (parseatom s)
  (match s
    [ (number: n) (R5 n)]
    [(list ae) (R6 (parseae ae))]))

(: parse : (String -> AE))
(define (parse s) (parseae (string->sexpr s)))

(test (parse "5") => (R2 (R4 (R5 5))))
(test (parse "{2 + 5}") => (R1 (R4 (R5 2)) (R2 (R4 (R5 5)))))
(test (parse "{{{2 + 5}} * {5 * {{3 + 6}}}}") => (R2 (R3 (R6 (R1 (R4 (R5 2)) (R2 (R4 (R5 5))))) (R3 (R5 5) (R4 (R6 (R1 (R4 (R5 3)) (R2 (R4 (R5 6)))))))))
      )

; 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)

These are not wrapped in the few rules at the top level to convert them to AEs

"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

|#