host: relations-as-posts slice 4 — type ALGEBRA (intersection ∧ union)

An algebraic type is a post with operand edges: conj edges (intersection members),
disj edges (union members). host/blog-instances-of-expr computes its extent from the
operands' extents by set intersection/union, RECURSIVELY — operands can themselves be
algebraic (meta-circular; tested with (tag ∧ article) ∧ tag). host/blog-is-a-expr?
generalises is-a? to type expressions; make-and!/make-or! build them. Binary today
(nth 0/1, no fold over operands — robust on the serving JIT).

Operand edges are KV-only (host/blog--add-edge-kv!, read via host/blog-out), NOT in
lib/relations — feeding extra kinds into the Datalog graph blows up its per-query
re-saturation; load-edges! skips conj/disj on replay too.

conformance 295/295 (+4: intersection/union membership, extent = set op, nested expr).
(NB: host conformance can EXIT 124 purely from a sibling loop's CPU contention — ran
with timeout 1200.)

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

lib/host/blog.sx

@@ -177,10 +177,14 @@
                     (tail (substr body (+ p1 1))))
                 (let ((p2 (index-of tail "|")))
                   (when (>= p2 0)
-                    (relations/relate
-                      (host/blog--node src)
-                      (host/blog--node (substr tail (+ p2 1)))
-                      (string->symbol (substr tail 0 p2))))))))))
+                    (let ((ek (substr tail 0 p2)))
+                      ;; conj/disj are structural (type-algebra operands) — KV-only,
+                      ;; never replayed into the Datalog graph (it re-saturates per query).
+                      (when (not (or (= ek "conj") (= ek "disj")))
+                        (relations/relate
+                          (host/blog--node src)
+                          (host/blog--node (substr tail (+ p2 1)))
+                          (string->symbol ek)))))))))))
       (filter (fn (k) (starts-with? k "edge:"))
         (persist/backend-kv-keys host/blog-store)))))
 
@@ -277,6 +281,55 @@
       (host/blog--uniq
         (reduce (fn (acc t) (concat acc (host/blog-in t "is-a"))) (list) subtypes)))))
 
+;; ── Slice 4: type ALGEBRA — intersection (∧) and union (∨) types ─────
+;; An algebraic type is a post with operand edges: a `conj` edge per intersection
+;; member, a `disj` edge per union member. Its EXTENT is its operands' extents combined
+;; by set intersection / union, recursively — so types compose into an algebra in the
+;; same graph (meta-circular: an algebraic type is just another post). Binary today
+;; (nth 0/1, no fold over operands — robust on the serving JIT); n-ary is a follow-up.
+;; is-a-expr? generalises is-a? to type expressions.
+(define host/blog--set-intersect
+  (fn (xs ys) (filter (fn (x) (contains? ys x)) xs)))
+;; operand edges live in the KV ONLY (read back via host/blog-out), NOT in lib/relations:
+;; conj/disj are structural, and feeding extra kinds into the Datalog graph blows up its
+;; per-query re-saturation. host/blog-load-edges! skips them on replay for the same reason.
+(define host/blog--add-edge-kv!
+  (fn (src dst kind)
+    (persist/backend-kv-put host/blog-store (host/blog--edge-key src kind dst) 1)))
+(define host/blog-make-and!
+  (fn (t a b)
+    (begin
+      (host/blog-seed! t t
+        (str "(article (h1 \"" t "\") (p \"An intersection type (" a " ∧ " b ") — its instances are exactly those that are instances of BOTH.\"))")
+        "published")
+      (host/blog--add-edge-kv! t a "conj")
+      (host/blog--add-edge-kv! t b "conj"))))
+(define host/blog-make-or!
+  (fn (t a b)
+    (begin
+      (host/blog-seed! t t
+        (str "(article (h1 \"" t "\") (p \"A union type (" a " ∨ " b ") — its instances are those that are instances of EITHER.\"))")
+        "published")
+      (host/blog--add-edge-kv! t a "disj")
+      (host/blog--add-edge-kv! t b "disj"))))
+;; the EXTENT of a type expression: operands' extents combined by set ops (recursive).
+;; A plain type (no operands) falls through to its instances.
+(define host/blog-instances-of-expr
+  (fn (t)
+    (let ((conj (host/blog-out t "conj"))
+          (disj (host/blog-out t "disj")))
+      (cond
+        ((>= (len conj) 2)
+          (host/blog--set-intersect (host/blog-instances-of-expr (nth conj 0))
+                                    (host/blog-instances-of-expr (nth conj 1))))
+        ((>= (len disj) 2)
+          (host/blog--uniq (concat (host/blog-instances-of-expr (nth disj 0))
+                                   (host/blog-instances-of-expr (nth disj 1)))))
+        (else (host/blog-instances-of t))))))
+;; is `slug` a member of the type expression `t`? Generalises is-a? to the algebra.
+(define host/blog-is-a-expr?
+  (fn (slug t) (contains? (host/blog-instances-of-expr t) slug)))
+
 ;; ── tags (a tag is a post that is-a tag) ────────────────────────────
 (define host/blog-is-tag? (fn (slug) (host/blog-is-a? slug "tag")))
 (define host/blog-tags (fn (slug) (host/blog-out slug "tagged")))          ;; a post's tags

lib/host/tests/blog.sx

@@ -527,6 +527,29 @@
     (contains? (host/blog-out "alpha-post" "is-a") "beta-post"))
   false)
 
+;; -- Slice 4: type ALGEBRA — intersection (∧) and union (∨) types --
+;; ocaml is-a tag (seeded above); make it is-a article too, so it's in BOTH extents.
+(host/blog-relate! "ocaml" "article" "is-a")
+(host/blog-make-and! "taggy-article" "tag" "article")   ;; tag ∧ article
+(host/blog-make-or!  "tag-or-article" "tag" "article")   ;; tag ∨ article
+(host-bl-test "intersection (∧): a member iff it's an instance of BOTH operands"
+  (list (host/blog-is-a-expr? "ocaml" "taggy-article")   ;; is-a tag AND is-a article
+        (host/blog-is-a-expr? "ppost" "taggy-article"))  ;; neither
+  (list true false))
+(host-bl-test "union (∨): a member iff it's an instance of EITHER operand"
+  (list (host/blog-is-a-expr? "ocaml" "tag-or-article")  ;; is-a tag (and article)
+        (host/blog-is-a-expr? "ppost" "tag-or-article"))  ;; neither tag nor article
+  (list true false))
+(host-bl-test "the extent is the set intersection of the operands' extents"
+  (let ((ext (host/blog-instances-of-expr "taggy-article")))
+    (list (contains? ext "ocaml")                         ;; in tag ∩ article
+          (contains? ext "ppost")))                       ;; in neither
+  (list true false))
+;; algebra is META-CIRCULAR: an operand can itself be an algebraic type.
+(host/blog-make-and! "nested-and" "taggy-article" "tag")  ;; (tag ∧ article) ∧ tag
+(host-bl-test "nested type expression: (tag ∧ article) ∧ tag still admits ocaml"
+  (host/blog-is-a-expr? "ocaml" "nested-and") true)
+
 ;; -- Phase 3: tags as posts --  (ocaml is-a tag, from the seed-types test above)
 (host-bl-test "is-tag?: a post that is-a tag is a tag; others are not"
   (list (host/blog-is-tag? "ocaml") (host/blog-is-tag? "ppost"))

plans/relations-as-posts.md

@@ -85,16 +85,20 @@ relation-subtype closure when relations get subtyped; the boot title cache above
 - NOTE: `host/blog-relate!` (direct/seed) stays UNVALIDATED — the seed needs to write
   `X is-a relation` where `relation` isn't under `type`. Validation is a *handler* boundary.
 
-### Slice 4 — type algebra
-Types are posts + `subtype-of` is a partial order ⇒ a **lattice**, and `is-a?` is transitive
-set-membership ⇒ extents have set semantics. So algebra is expressible as posts:
-- **Intersection** `A ∧ B` — a type-post whose membership predicate is `is-a? A ∧ is-a? B`
-  (meet / GLB in the lattice). **Union** `A ∨ B` — `is-a? A ∨ is-a? B` (join / LUB).
-- **Refinement** `{x : T | φ(x)}` — a type-post with a `:constraint` predicate over a post
-  (generalises today's `article` schema "must have a heading"). Gradual: declaring the type
-  adds the obligation; the next save must satisfy it.
-- Algebraic types are *themselves posts* with edges to their operands — `is-a?` recurses on
-  the expression. Meta-circular: the algebra lives in the graph it describes.
+### Slice 4 — type algebra — DONE (intersection ∧ union)
+- An algebraic type is a post with operand edges: `conj` edges (intersection members),
+  `disj` edges (union members). `host/blog-instances-of-expr` computes its EXTENT from the
+  operands' extents by set intersection / union, RECURSIVELY — so operands can themselves be
+  algebraic (meta-circular; tested with `(tag ∧ article) ∧ tag`). `host/blog-is-a-expr?`
+  generalises `is-a?` to type expressions. `host/blog-make-and!` / `make-or!` build them.
+- Binary today (`nth 0/1`, no fold over operands — robust on the serving JIT); n-ary fold is
+  a follow-up once iteration-with-perform is JIT-reliable.
+- **Operand edges are KV-only** (`host/blog--add-edge-kv!`, read via `host/blog-out`), NOT in
+  lib/relations — feeding extra kinds into the Datalog graph blows up its per-query
+  re-saturation; `load-edges!` skips `conj`/`disj` on replay for the same reason.
+- **Refinement** `{x : T | φ(x)}` (a type-post with a `:constraint` predicate) → Slice 5,
+  with constraints-as-posts. (Process note: a sibling loop running heavy conformance saturates
+  the box; host conformance can EXIT 124 purely from CPU contention — use `timeout 1200`.)
 
 ### Slice 5 — constraints as posts + validation
 - Promote the schema/`:constraint` slot to **constraint-posts** (a predicate expr +