maude: Phase 3 — equational matching modulo assoc/comm/id (28 tests, 119 total)

The chisel. lib/maude/matching.sx: multi-valued matcher mau/mm returning
ALL substitutions, dispatching on op theory (free/comm/assoc/AC). Identity
lets variables grab empty blocks. AC-canonical form (mau/canon) powers
ac-equal? and deterministic printout. AC rewriting extends f-AC equations
with rest vars so a rule fires on any sub-multiset/subword; mau/first-change
only commits rewrites that change the canonical form (idempotency/identity
terminate). Verified on multiset rewriting, set theory, group equations.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

plans/maude-on-sx.md

@@ -73,12 +73,12 @@ The novel substrate stress: equational matching. Pattern `X + Y` against `1 + 2
 - [x] Tests: peano arithmetic, list manipulation, propositional logic simplifier.
 
 ### Phase 3 — Equational matching (assoc / comm / id)
-- [ ] Extend matching to handle `assoc` operators (flatten then match across permutations of subterm groups).
-- [ ] Handle `comm` (try both argument orderings).
-- [ ] Handle `id: e` (X * e ≡ X).
-- [ ] Combinations: `assoc comm id` together.
-- [ ] Returns *all* matches, not just first — caller drives.
-- [ ] Tests: classic AC-matching examples (multiset rewriting, set theory, group equations).
+- [x] Extend matching to handle `assoc` operators (flatten then match across permutations of subterm groups).
+- [x] Handle `comm` (try both argument orderings).
+- [x] Handle `id: e` (X * e ≡ X).
+- [x] Combinations: `assoc comm id` together.
+- [x] Returns *all* matches, not just first — caller drives.
+- [x] Tests: classic AC-matching examples (multiset rewriting, set theory, group equations).
 
 ### Phase 4 — Conditional equations
 - [ ] `ceq L = R if Cond` — apply only when `Cond` reduces to true.
@@ -153,5 +153,29 @@ The novel substrate stress: equational matching. Pattern `X + Y` against `1 + 2
   non-linear `same(X,X)`. Innermost is fine for confluent terminating eq sets;
   Phase 3 will replace the matcher with AC-aware matching (multi-valued).
 
+- **Phase 3 (matching modulo assoc/comm/id) — DONE, 119/119 total. THE CHISEL.**
+  `lib/maude/matching.sx`. `mau/mm` is the multi-valued matcher (returns the
+  full list of substitutions): free=positional, comm=both orderings,
+  assoc=flatten f-spine + ordered sequence match (vars grab contiguous blocks),
+  assoc+comm=multiset match (vars grab sub-multisets via `mau/all-splits` =
+  2^n subset/complement pairs). `id: e` lets a var grab the empty block
+  (contributing e); `mau/var-kmin` gives kmin 0 under id. `mau/canon` is the
+  AC-canonical printout (flatten, drop identities, sort comm args) and powers
+  `mau/ac-equal?` (used for bound-var checks too). AC *rewriting* extends each
+  f-AC equation l=r with rest vars — comm: `f(l,$R)`; assoc: `f($L,l,$R)` —
+  so a rule fires on any sub-multiset/subword (`$`-prefixed rest vars allowed
+  empty). `mau/first-change` walks candidate matches and only commits a rewrite
+  that changes the canonical form — this is what makes idempotency (`X U X = X`)
+  and identity-absorbing matches terminate. API: `mau/ac-reduce` /
+  `mau/ac-reduce->str` / `mau/ac-canon` / `mau/match-all`. Verified: AC match
+  counts (X+Y vs a+b+c = 6), bag collapse, set dedup with empty, group
+  cancellation (assoc non-comm + inverse).
+  - Notes for next phases: AC matching is multi-valued — Phase 5 rule
+    application should iterate ALL of `mau/mm`'s results, not just first. The
+    `mau/ac-rewrite-eq` extension trick (rest vars) is the reusable core for
+    a future `lib/guest/rewriting/` (Phase 8). Keep `mau/canon` as the equality
+    oracle. `$EMPTY` is a transient marker for empty rest blocks w/o id; never
+    leaks past `mau/restv`.
+
 ## Blockers
 _(none)_