diff --git a/preserves-binary.md b/preserves-binary.md
index a2fe195..30054f0 100644
--- a/preserves-binary.md
+++ b/preserves-binary.md
@@ -21,34 +21,72 @@ syntax](preserves-text.html) also exists.
## Machine-Oriented Binary Syntax
A `Repr` is a binary-syntax encoding, or representation, of a `Value`.
-For a value `v`, we write `«v»` for the `Repr` of v.
### Type and Length representation.
Each `Repr` starts with a tag byte, describing the kind of information
-represented.
-
-However, inspired by [argdata][], a `Repr` does *not* describe its own
-length. Instead, the expected length of the `Repr` is always available
+represented. The expected length of the `Repr` is always available
from the surrounding context: either from a containing encoded value, or
from the overall container of the data, which could be a file, an HTTP
message, a UDP packet, etc.
-As a consequence, `Repr`s for `Compound` values store the lengths of
-their contained values. Each contained `Value` is represented as a
-length in bytes followed by its own `Repr`. Implementations use each
-stored length to decide when to stop reading the following `Repr`.
+### Atomic Values.
+
+**Booleans.** The "false" boolean's `Repr` is just tag `0xA0`; "true" is
+`0xA1`.
+
+**Floats and Doubles.** Both `Float` and `Double` values are represented
+as tag `0xA2` followed by big-endian 4- or 8-byte IEEE 754 binary
+representations of the values, respectively.
+
+**SignedIntegers.** A `SignedInteger` encodes as tag `0xA3` followed by
+a big-endian two's-complement binary representation of the value, taking
+at least as many whole bytes as needed to unambiguously identify the
+value and its sign. Zero may be represented as the tag alone, with no
+following bytes. The most-significant bit in the first byte after the
+tag is the sign bit.[^zero-intbytes] The shortest possible encoding
+*SHOULD* be used.[^overlong-signedinteger]
+
+ [^zero-intbytes]: The value 0 needs zero bytes to identify the value,
+ so `intbytes(0)` can be the empty byte string. Non-zero values need
+ at least one byte.
+
+ [^overlong-signedinteger]: **Implementation note.** The spec permits
+ overlong `SignedInteger` encodings to allow e.g. construction of
+ `Repr`s by filling in partially-completed templates, which can be
+ useful in resource-constrained situations.
+
+**Strings.** A `String` encodes as tag `0xA4` followed by the UTF-8
+encoding of the string, with an additional trailing `NUL` (0) byte. The
+`NUL` byte *MUST NOT* be treated as part of the `String`: it exists to
+permit zero-copy C interoperability.[^zero-copy-c-string-interop]
+
+ [^zero-copy-c-string-interop]: Some care must still be taken when
+ passing `String` `Repr`s directly to a C-style ABI, since `String`s
+ may contain the zero Unicode code point, which C library routines
+ will usually misinterpret as an end-of-string marker.
+
+**ByteStrings.** A `ByteString` encodes as tag `0xA5` followed by the
+bytes themselves.
+
+**Symbols.** A `Symbol` encodes as tag `0xA6` followed by the UTF-8
+encoding of the symbol's code points.
+
+### Compound Values.
+
+`Repr`s for `Compound` values store the lengths of their contained
+values. Each contained `Value` is converted to a `Repr` and stored as
+the length of the `Repr` in bytes followed by the `Repr` itself.
+Implementations use each stored length to decide when to stop reading
+the associated `Repr`. Similarly, no sentinel marks the end of a
+sequence of length-prefixed `Repr`s. Implementations use the length of
+the containing `Repr`, known from the surrounding context, to decide
+when to stop expecting more contained `Repr`s.
Each length is stored as an [argdata][]-compatible
big-endian base 128 *varint*.[^see-also-leb128] Each byte of a varint
stores seven bits of the length. All bytes have a clear upper bit,
-except the final byte, which has the upper bit set. We write
-`len(m)` for the varint-encoding of a non-negative integer `m`,
-defined recursively as follows:
-
- len(m) = e(m, 128)
- where e(v, d) = [v + d] if v < 128
- e(v / 128, 0) ++ [(v % 128) + d] if v ≥ 128
+except the final byte, which has the upper bit set.
[^see-also-leb128]: Argdata's length representation is very close to
[Variable-length quantity (VLQ)][VLQ] encoding, differing only in
@@ -56,10 +94,8 @@ defined recursively as follows:
big-endian, unlike [LEB128][] encoding ([as used by
Google][google-varint] in protobufs).
-We write `len(|r|)` for the varint-encoding of the length of `Repr` `r`.
-
-There is no requirement that a varint-encoded `m` in a `Repr` be the
-unique shortest encoding for that `m`.[^overlong-varint] However,
+There is no requirement that a varint-encoded length be the unique
+shortest encoding for the length.[^overlong-varint] However,
implementations *SHOULD* use the shortest encoding whereever possible
when writing, and *MAY* reject encodings with more than eight leading
`0` bytes when reading encoded values.
@@ -69,21 +105,24 @@ when writing, and *MAY* reject encodings with more than eight leading
anything other than a very low-level language, it is likely to be able to use
[IOList](./conventions.html#iolists)-style data structures to avoid unnecessary copying.
-### Records, Sequences, Sets and Dictionaries.
+**Records.** A `Record` is encoded as tag `0xA7` followed by the
+length-prefixed encodings of its label and fields.
- «» = [0xA7] ++ seq(«L», «F_1», ..., «F_m»)
- «[X_1...X_m]» = [0xA8] ++ seq(«X_1», ..., «X_m»)
- «#{E_1...E_m}» = [0xA9] ++ seq(«E_1», ..., «E_m»)
- «{K_1:V_1...K_m:V_m}» = [0xAA] ++ seq(«K_1», «V_1», ..., «K_m», «V_m»)
+**Sequences.** A `Sequence` is encoded as tag `0xA8` followed by the
+length-prefixed encodings of its members.
- seq(R_1, ..., R_m) = len(|R_1|) ++ R_1 ++...++ len(|R_m|) ++ R_m
+**Sets.** A `Set` is encoded like a `Sequence`, but with tag `0xA9`, and
+in some arbitrary order.
-There is *no* ordering requirement on the `E_i` elements or
-`K_i`/`V_i` pairs.[^no-sorting-rationale] They may appear in any
-order. However, the `E_i` and `K_i` *MUST* be pairwise distinct. In
-addition, implementations *SHOULD* default to writing set elements and
-dictionary key/value pairs in order sorted lexicographically by their
-`Repr`s[^not-sorted-semantically], and *MAY* offer the option of
+**Dictionaries.** A `Dictionary` encodes as tag `0xAA` followed by the
+length-prefixed keys and values, in an alternating key/value sequence.
+
+There is *no* ordering requirement on the elements of sets or the
+key/value pairs of dictionaries.[^no-sorting-rationale] However,
+elements of sets and keys in dictionaries *MUST* be pairwise distinct.
+In addition, implementations *SHOULD* default to writing set elements
+and dictionary key/value pairs in order sorted lexicographically by
+their `Repr`s[^not-sorted-semantically], and *MAY* offer the option of
serializing in some other implementation-defined order.
[^no-sorting-rationale]: In the BitTorrent encoding format,
@@ -109,93 +148,33 @@ serializing in some other implementation-defined order.
but encoding and then sorting byte strings is much more likely to
be within easy reach.
-No sentinel marks the end of a sequence of length-prefixed `Repr`s.
-During decoding, use the length of the containing `Repr` to decide when
-to stop expecting more contained `Repr`s.
+### Embedded Values.
-### SignedIntegers.
-
- «x» when x ∈ SignedInteger = [0xA3] ++ intbytes(x)
-
-The function `intbytes(x)` gives a big-endian two's-complement binary
-representation of `x`, taking at least as many whole bytes as needed to
-unambiguously identify the value and its sign; `intbytes(0)` may be the
-empty byte sequence.[^zero-intbytes] The most-significant bit in the
-first byte in `intbytes(x)` is the sign bit. While every `SignedInteger`
-*SHOULD* be represented with its shortest possible encoding (which will
-often include a necessary leading `0xFF` or `0x00`), redundant leading
-`0xFF` or `0x00` bytes *MAY* be used.[^overlong-signedinteger]
-
- [^zero-intbytes]: The value 0 needs zero bytes to identify the value,
- so `intbytes(0)` can be the empty byte string. Non-zero values need
- at least one byte.
-
- [^overlong-signedinteger]: **Implementation note.** The spec permits
- overlong `SignedInteger` encodings to allow e.g. construction of
- `Repr`s by filling in partially-completed templates, which can be
- useful in resource-constrained situations.
-
-### Strings, ByteStrings and Symbols.
-
- «S» = [0xA4] ++ utf8(S) ++ [0] if S ∈ String
- [0xA5] ++ S if S ∈ ByteString
- [0xA6] ++ utf8(S) if S ∈ Symbol
-
-For `String` and `Symbol`, the data following the tag is a UTF-8
-encoding of the `Value`'s code points, while for `ByteString` it is the
-raw data contained within the `Value` unmodified.
-
-Each `String` has a trailing zero byte appended. This extra byte *MUST
-NOT* be treated as part of the `Value`: it exists to permit zero-copy C
-interoperability.[^zero-copy-c-string-interop]
-
- [^zero-copy-c-string-interop]: Some care must still be taken when
- passing `String` `Repr`s directly to a C-style ABI, since `String`s
- may contain the zero Unicode code point, which C library routines
- will usually misinterpret as an end-of-string marker.
-
-### Booleans.
-
- «#f» = [0xA0]
- «#t» = [0xA1]
-
-### Floats and Doubles.
-
- «F» when F ∈ Float = [0xA2] ++ binary32(F)
- «D» when D ∈ Double = [0xA2] ++ binary64(D)
-
-The functions `binary32(F)` and `binary64(D)` yield big-endian 4- and
-8-byte IEEE 754 binary representations of `F` and `D`, respectively.
-
-### Embeddeds.
-
-The `Repr` of an `Embedded` is the `Repr` of a `Value` chosen to
-represent the denoted object, prefixed with `[0xAB]`.
-
- «#!V» = [0xAB] ++ «V»
+Embedded values are encoded as tag `0xAB` followed by the encoding of
+some `Value` chosen to represent the denoted embedded object.
### Annotations.
-To annotate a `Repr` `r` with some sequence of `Value`s `[v_1, ...,
-v_m]`, surround `r` as follows:
+The encoding of a sequence of annotations for a `Repr` uses tag `0xBF`,
+followed by the length-prefixed `Repr`, followed by the length-prefixed
+encoded annotations, in order. The `Repr` *MUST NOT* already have
+annotations (must not begin with `0xBF`), and there *MUST* be at least
+one `Value` in the sequence following the `Repr`.
- [0xBF] ++ len(|r|) ++ r ++ len(|«v_1»|) ++ «v_1» ++...++ len(|«v_m»|) ++ «v_m»
+## Examples (normative)
-The `Repr` `r` *MUST NOT* already have annotations; that is, it must not
-begin with `0xBF`. The sequence `[v_1, ..., v_m]` *MUST* contain at
-least one `Value`.
-
-## Examples
+We write `«v»` for the `Repr` of some `Value` `v`, and `varint(|«v»|)` for
+the varint-encoded length of the `Repr` of `v`.
### Varints (length representations).
The following table illustrates varint-encoding.
-| Number, `m` | `m` in binary, grouped into 7-bit chunks | `len(m)` bytes |
-|-------------|-------------------------------------------|-----------------|
-| 15 | `0001111` | 143 |
-| 300 | `0000010 0101100` | 2 172 |
-| 1000000000 | `0000011 1011100 1101011 0010100 0000000` | 3 92 107 20 128 |
+| Number, `m` | `m` in binary, grouped into 7-bit chunks | `varint(m)` bytes |
+|-------------|-------------------------------------------|-------------------|
+| 15 | `0001111` | 143 |
+| 300 | `0000010 0101100` | 2 172 |
+| 1000000000 | `0000011 1011100 1101011 0010100 0000000` | 3 92 107 20 128 |
### Atoms.
@@ -288,7 +267,9 @@ The `Repr` corresponding to textual syntax `@a@b[]`, i.e. an empty sequence anno
symbols, `a` and `b`, is
«@a @b []»
- = [0xBF] ++ len(|«[]»|) ++ «[]» ++ len(|«a»|) ++ «a» ++ len(|«b»|) ++ «b»
+ = [0xBF] ++ varint(|«[]»|) ++ «[]»
+ ++ varint(|«a»|) ++ «a»
+ ++ varint(|«b»|) ++ «b»
= [0xBF, 0x81, 0xA8, 0x82, 0xA6, 0x61, 0x82, 0xA6, 0x62]
## Security Considerations
@@ -346,7 +327,7 @@ undetermined number of `Value`s across, say, a TCP/IP connection:
- If the binary syntax is to be used for the connection, start the
connection with byte `0xA8` (sequence). After the initial byte, send
- each value `v` as `len(|«v»|) ++ «v»`. A side effect of this approach
+ each value `v` as `varint(|«v»|) ++ «v»`. A side effect of this approach
is that the entire stream, when complete, is a valid `Sequence`
`Repr`.