Native natural-number math

native.tri provides binary natural-number arithmetic and ASCII classifiers. The source lexer and executable-digest validator use its canonical-number fast path. Checked public arithmetic in base.tri validates inputs before delegating to these algorithms. The implementation uses ordinary Tree Calculus terms and stable direct recursion.

Input/output contract

All numeric arguments must be canonical natural-number trees:

• Zero is Leaf (t).
• A positive number is Fork bit rest, least-significant bit first; bit is exactly Leaf (0) or Stem Leaf (1), and the highest represented bit is 1.
• Outputs use the same canonical encoding.
• Booleans are exactly Leaf (false) or Stem Leaf (true).

This fast path requires canonical inputs. Validate external data at the boundary, then pass accepted values to these operations. Each demanded observation must terminate.

equal compares canonical natural numbers numerically. base.equal? provides structural equality for arbitrary trees.

Exported interface

Exports	Meaning
compare a b	Leaf = equal; Stem Leaf = less; Fork Leaf Leaf = greater. These are structural ordering tags.
equal, lt, lte, gt, gte	Two-argument numeric predicates.
inRange value low high	Inclusive bounds; false when low > high.
minimum a b, maximum a b	Canonical selected operand.
zero value, odd value, even value	Unary predicates.
successor value, predecessor value	Increment/decrement; predecessor of zero is zero.
add a b, subtract a b	Sum and subtraction saturating at zero.
multiply a b	Product.
divMod a b	Pair of quotient and remainder; a zero divisor yields (0, a).
quotient a b, remainder a b	Projections of divMod.
power a b	Exponentiation by squaring; 0^0 is 1.
decimal value digit	10 * value + digit; digit is a numeric value in 0..9.
ascii value	Membership in 0..127.
digit value	ASCII decimal digit, 48..57.
letter value	ASCII A..Z or a..z.
lowerHex value	ASCII 0..9 or lowercase a..f.
printable value	ASCII 32..126.

Use namespace-qualified calls. In a native flat workspace, put this source at native.tri, then declare sources before uses:

!import native as N
!export main
main input = N.add input 1

Select a retained compiler artifact explicitly, as described in the portable tool guide. Internal helpers are private. The test runner prepares such a workspace automatically.

Implementation and limits

Comparison walks binary tails, allowing the more significant differing bit to win. Equality stops at the first differing low bit. Addition uses carry bits; subtraction first checks ordering, then propagates borrow and removes high zero bits. Multiplication uses binary shift/add; decimal accumulation uses shifts and addition. digit, letter, and lowerHex classify fixed bit slices. Digit recognition checks a high part of 3 and a low nibble in 0..9. Letter recognition ignores the ASCII case bit and checks a low five-bit value in 1..26; lowercase hex letters have a high part of 12 and low three bits in 1..6. High-part equality rejects larger naturals, including Unicode characters outside ASCII. ascii and printable retain binary comparisons. These classifiers require the canonical-natural invariant.

Comparison, addition, and subtraction walk binary digits. Multiplication uses shift/add; division performs binary long division.

The raw zero-divisor policy keeps divMod total on canonical naturals. Checked public division rejects zero with the "non-zero" contract. Choose the appropriate boundary for the caller.

Validation

The numeric checks use independent integer, Boolean, and complete-tree expectations. Candidate preparation and fixture publication are separate from reference-artifact tests. Measure selected workloads with a recorded compiler; changing function trees can affect both performance and intensional observations.

Binary64 arithmetic

The binary64 source interface provides checked IEEE-word import, exact export and trusted working-value arithmetic for the four basic RNE operations, with explicit flags. Its guide describes the source workspace and calling conventions.