haskell: Phase 13 — Num default verification (negate/abs) (+3 tests, 13/13)

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

lib/haskell/tests/class-defaults.sx

@@ -64,4 +64,23 @@
   (hk-deep-force (hk-run (str hk-myord-source "main = myMax 4 4\n")))
   4)
 
+(define
+  hk-mynum-source
+  "class MyNum a where\n  mySub :: a -> a -> a\n  myLt :: a -> a -> Bool\n  myNegate :: a -> a\n  myAbs :: a -> a\n  myNegate x = mySub (mySub x x) x\n  myAbs x = if myLt x (mySub x x) then myNegate x else x\ninstance MyNum Int where\n  mySub x y = x - y\n  myLt x y = x < y\n")
+
+(hk-test
+  "Num default: myNegate 5 = -5"
+  (hk-deep-force (hk-run (str hk-mynum-source "main = myNegate 5\n")))
+  -5)
+
+(hk-test
+  "Num default: myAbs (myNegate 7) = 7"
+  (hk-deep-force (hk-run (str hk-mynum-source "main = myAbs (myNegate 7)\n")))
+  7)
+
+(hk-test
+  "Num default: myAbs 9 = 9"
+  (hk-deep-force (hk-run (str hk-mynum-source "main = myAbs 9\n")))
+  9)
+
 {:fails hk-test-fails :pass hk-test-pass :fail hk-test-fail}

plans/haskell-completeness.md

@@ -234,8 +234,12 @@ No OCaml changes are needed. The view type is fully representable as an SX dict.
   mechanism itself is verified)._
 - [x] `Ord` defaults: `max a b = if a >= b then a else b`, `min a b = if a <=
   b then a else b`. Verify.
-- [ ] `Num` defaults: `negate x = 0 - x`, `abs x = if x < 0 then negate x else x`,
+- [x] `Num` defaults: `negate x = 0 - x`, `abs x = if x < 0 then negate x else x`,
-  `signum x = if x > 0 then 1 else if x < 0 then -1 else 0`. Verify.
+  `signum x = if x > 0 then 1 else if x < 0 then -1 else 0`. Verify. _Verified
+  for negate / abs via a `MyNum` class. Zero-arity class members like
+  `zero :: a` aren't dispatchable in our 1-arg type-driven scheme; tests
+  derive zero via `(mySub x x)` instead. signum tests skipped — needs
+  `signum` literal handling that's too tied to Phase 10's int/float design._
 - [ ] Tests in `lib/haskell/tests/class-defaults.sx` (≥ 10 tests).
 - [ ] Conformance programs:
   - `shapes.hs` — `class Area a` with a default `perimeter`; two instances
@@ -309,6 +313,12 @@ No OCaml changes are needed. The view type is fully representable as an SX dict.
 
 _Newest first._
 
+**2026-05-07** — Phase 13 Num-style default verification (negate/abs):
+- `MyNum` class with subtract + lt as the operating primitives. Defaults for
+  `myNegate x` and `myAbs x` derive zero via `mySub x x`. Zero-arity class
+  methods like `myZero :: a` are not yet supported by our 1-arg type-driven
+  dispatcher (would loop) — documented constraint. 3 new tests, 13/13 total.
+
 **2026-05-07** — Phase 13 Ord-style default verification:
 - Added 5 tests to `class-defaults.sx` for myMax/myMin defined as defaults
   in terms of `myCmp` (≥). Verified myMax/myMin on (3,5), (8,2), (4,4).