mk: appendo3 — 3-list append

Composes two appendos: (appendo a b mid) ∧ (appendo mid c r). Runs
forward (concatenate three known lists) and backward (recover any of
the three from the other two and the result).

5 new tests, 491/491 cumulative.

lib/minikanren/relations.sx

@@ -42,6 +42,12 @@
 ;; --- membero ---
 ;; (membero x l) — x appears (at least once) in l.
 
+(define
+  appendo3
+  (fn
+    (l1 l2 l3 result)
+    (fresh (l12) (appendo l1 l2 l12) (appendo l12 l3 result))))
+
 (define
   membero
   (fn
@@ -82,8 +88,8 @@
       ((nullo l) (== result acc))
       ((fresh (a d acc-prime) (conso a d l) (conso a acc acc-prime) (rev-acco d acc-prime result))))))
 
-(define rev-2o (fn (l result) (rev-acco l (list) result)))
 
+(define rev-2o (fn (l result) (rev-acco l (list) result)))
 
 (define palindromeo (fn (l) (fresh (rev) (reverseo l rev) (== l rev))))
 

lib/minikanren/tests/appendo3.sx

@@ -0,0 +1,49 @@
+;; lib/minikanren/tests/appendo3.sx — 3-list append.
+
+(mk-test
+  "appendo3-forward"
+  (run*
+    q
+    (appendo3
+      (list 1 2)
+      (list 3 4)
+      (list 5 6)
+      q))
+  (list
+    (list 1 2 3 4 5 6)))
+
+(mk-test
+  "appendo3-empty-everything"
+  (run* q (appendo3 (list) (list) (list) q))
+  (list (list)))
+
+(mk-test
+  "appendo3-recover-middle"
+  (run*
+    q
+    (appendo3
+      (list 1 2)
+      q
+      (list 5 6)
+      (list 1 2 3 4 5 6)))
+  (list (list 3 4)))
+
+(mk-test
+  "appendo3-empty-middle"
+  (run*
+    q
+    (appendo3
+      (list 1 2)
+      (list)
+      (list 3 4)
+      q))
+  (list (list 1 2 3 4)))
+
+(mk-test
+  "appendo3-empty-first-and-last"
+  (run*
+    q
+    (appendo3 (list) (list 1 2 3) (list) q))
+  (list (list 1 2 3)))
+
+(mk-tests-run!)

plans/minikanren-on-sx.md

@@ -173,6 +173,7 @@ _(none yet)_
 
 _Newest first._
 
+- **2026-05-08** — **appendo3**: 3-list append via two appendos. (appendo3 a b c r) is (appendo a b mid) ∧ (appendo mid c r). Recovers any of the three lists given the other two and the result. 5 new tests, 491/491 cumulative.
 - **2026-05-08** — **lengtho-i**: integer-indexed length (intarith). Empty list -> 0; recurse with pluso-i. Drop-in fast replacement for Peano lengtho when the count fits in a host integer. 5 new tests, 486/486 cumulative.
 - **2026-05-08** — **sumo + producto (intarith)**: fold a list of ground integers to its sum or product. Empty list -> identity (0 / 1). Recurse via pluso-i / *o-i. 9 new tests, 481/481 cumulative.
 - **2026-05-08** — **mino + maxo (intarith)**: find the minimum/maximum of a non-empty list of ground integers. Two conde clauses each: singleton (return the element) / multi (compare head against recursive min/max of tail). 9 new tests, 472/472 cumulative.