# Authoring surface — notation rationale whitepaper

## Candidate notations (same `map` encoded in each)

The reference target — the polymorphic `map` from `examples/list_map_poly.ail.json`:

```
data List a where Nil | Cons a (List a)
fn map : forall a b. ((a) -> b, List a) -> List b
       = \f xs. match xs of Nil -> Nil
                          | Cons h t -> Cons(f(h), map(f, t))
```

#### (A) S-expression with fully-tagged AST nodes

```
(module list_map_poly

  (data List (vars a)
    (ctor Nil)
    (ctor Cons a (con List a)))

  (fn inc
    (type (fn-type (params (con Int)) (ret (con Int))))
    (params x)
    (body (app + x 1)))

  (fn map
    (type
      (forall (vars a b)
        (fn-type
          (params (fn-type (params a) (ret b)) (con List a))
          (ret (con List b)))))
    (params f xs)
    (body
      (match xs
        (case (pat-ctor Nil) (term-ctor List Nil))
        (case (pat-ctor Cons h t)
          (term-ctor List Cons
            (app f h)
            (app map f t)))))))
```

Grammar core (3-rule lexical layer + ~25 named-form productions):
```
sexpr  ::= atom | "(" sexpr* ")"
atom   ::= integer | string | ident
ident  ::= any maximal non-whitespace, non-paren run that is not
           a recognised integer or string literal.
```

The lexer recognises one delimiter (`(` / `)`) and whitespace.
Every other maximal token is classified post-hoc:
- All-digit run with optional leading `-` → integer atom.
- `"`-delimited run → string atom.
- Otherwise → ident.

Consequence: arithmetic operators like `+` / `-` / `*` / `/` / `%`,
qualified names like `io/print_str`, and cross-module references
like `std_list.map` are all single ident tokens with no special
lex rule. The only reserved tokens are `(`, `)`, and whitespace.
Bool literals (`true`, `false`) and unit (`(lit-unit)`) are
disambiguated by parser context, not by lex. Comparison and
equality are class methods (`eq` / `compare`) and named fns
(`float_eq` / `float_lt` / etc.), not operators — see
[Prelude classes](../contracts/0017-prelude-classes.md).

Every AST node form has a unique head keyword (`module`, `data`,
`fn`, `forall`, `fn-type`, `con`, `var`, `app`, `lam`, `match`,
`case`, `pat-ctor`, `term-ctor`, `do`, `seq`, ...). A bare atom in
a positional slot (e.g. inside `(con List a)` second position) is
a name reference whose **sort** is determined by the parent slot:

- inside `(con NAME args...)` second-and-later positions → type
  expression. Bare atom there ⇒ `Type::Var { name }`.
- inside `(app HEAD args...)` first position ⇒ `Term::Var`.
- inside `(pat-ctor CTOR fields...)` field positions ⇒
  `Pattern::Var`.
- inside `(case PAT BODY)` second position ⇒ term.

There is **no lexical case rule**. To construct a value with a
ctor, write `(term-ctor TypeName CtorName args...)`. To match
against one, write `(pat-ctor CtorName fields...)`. Capitalised
identifiers carry no special meaning to the parser. This rules
out a class of silent errors ("I forgot to capitalise `Cons` and
it parsed as a function call").

**Pros:** smallest formal grammar of any candidate (the lexical
core is 3 rules; the named-form productions are uniform — every
node a tagged list). Foreign-LLM bar lowest. Round-trip with the
existing pretty-printer is a refactor of `pretty.rs` to emit this
tagged form, plus a new parser.

**Cons:** paren density is high. `(forall (vars a b) (fn-type
...))` has more visual nesting than the current pretty-printer's
`forall a. (...) -> ...`. Verbosity is ~2× JSON for the same node
when measured in characters, but ~8× shorter in lines (the
existing JSON `box.ail.json` of 160 lines becomes ~20 lines in
this form).

#### (B) Indented record-style with explicit terminators

```
module std_list

data List(a):
  Nil
  Cons(a, List(a))
end

fn map:
  type: forall a b. fn(fn(a) -> b, List(a)) -> List(b)
  params: f, xs
  body:
    match xs:
      Nil          => Nil
      Cons(h, t)   => Cons(f(h), map(f, t))
    end
end
```

Grammar core (~20–30 productions): module-level (def/data/end), type
sub-grammar (forall, fn, con, var), term sub-grammar (lam, match,
ctor, app, lit, var, seq), pattern sub-grammar.

**Pros:** higher information density per line, closer to mainstream
ML/Haskell shape. **Cons:** four sub-grammars instead of one.
`forall a b. fn(...)` keeps a pseudo-precedence (`->` binds tighter
than the outer `fn(...)` wrapper). Foreign-LLM bar higher.

#### (C) Pretty-printer-as-source

Use exactly the format `pretty::module` already emits, plus a parser
that accepts it. The existing pretty-printer's quirks (`::` for
type-of, `[params]` for fn-params, `<a>` for type-args, `forall a. ...`,
`!IO`, `()` ambiguous between unit-arg-list and empty-form) become
the spec.

**Pros:** zero churn — the existing pretty-printer is already the
spec; only the inverse is missing. Round-trip is the identity by
construction. **Cons:** the existing format mixes four mini-dialects
(s-expr at term level, ML-shape at type level, square brackets for
params, `<>` for type args). Formalising it crisply is harder than
designing a uniform form from scratch.

The chosen form is (A), shipped as the `ailang-surface` crate (see
[authoring surface](../contracts/0001-authoring-surface.md)); the
canonical JSON-AST it round-trips against is documented in
[Data model](../contracts/0002-data-model.md), and the round-trip
property itself in [Roundtrip Invariant](../contracts/0009-roundtrip-invariant.md).

For the prose projection used for human review (Form B), see
[prose projection](0006-prose-projection.md).