wcfs: xbtree: blib += PPTreeSubSet, ΔPPTreeSubSet

This data structures will be used in ΔBtail to maintain sef of tracked
BTree nodes, and to represent δ to such set.

Some preliminary history:

kirr/wendelin.core@78f2f88b    X wcfs/xbtree: Fix treediff(a, ø)
kirr/wendelin.core@5324547c    X wcfs/xbtree: root(a) must stay in trackSet even after treediff(a,ø)
kirr/wendelin.core@f65f775b    X wcfs/xbtree: treediff(ø, b)
kirr/wendelin.core@66bc41ce    X Fix bug in PPTreeSubSet.Difference  - it was always leaving root node alive
kirr/wendelin.core@ddb28043    X rebuild: Don't return nil for empty ΔPPTreeSubSet - that leads to SIGSEGV
kirr/wendelin.core@a87cc6de    X rebuild: tests: Don't recompute trackSet(keys1R2) several times

Quoting PPTreeSubSet and ΔPPTreeSubSet documentation:

---- 8< ----

PPTreeSubSet represents PP-connected subset of tree node objects.

It is

    PP(xleafs)

where PP(node) maps node to {node, node.parent, node.parent,parent, ...} up
to top root from where the node is reached.

The nodes in the set are represented by their Oid.

Usually PPTreeSubSet is built as PP(some-leafs), but in general the starting
nodes are arbitrary. PPTreeSubSet can also have many root nodes, thus not
necessarily representing a subset of a single tree.

Usual set operations are provided: Union, Difference and Intersection.

Nodes can be added into the set via AddPath. Path is reverse operation - it
returns path to tree node given its oid.

Every node in the set comes with .parent pointer.

~~~~

ΔPPTreeSubSet represents a change to PPTreeSubSet.

It can be applied via PPTreeSubSet.ApplyΔ .

The result B of applying δ to A is:

    B = A.xDifference(δ.Del).xUnion(δ.Add)		(*)

(*) NOTE δ.Del and δ.Add might have their leafs starting from non-leaf nodes in A/B.
    This situation arises when δ represents a change in path to particular
    node, but that node itself does not change, for example:

           c*             c
          / \            /
        41*  42         41
         |    |         | \
        22   43        46  43
              |         |   |
             44        22  44

    Here nodes {c, 41} are changed, node 42 is unlinked, and node 46 is added.
    Nodes 43 and 44 stay unchanged.

        δ.Del = c-42-43   | c-41-22
        δ.Add = c-41-43   | c-41-46-22

    The second component with "-22" builds from leaf, but the first
    component with "-43" builds from non-leaf node.

        ΔnchildNonLeafs = {43: +1}

    Only complete result of applying all

        - xfixup(-1, ΔnchildNonLeafs)
        - δ.Del,
        - δ.Add, and
        - xfixup(+1, ΔnchildNonLeafs)

    produces correctly PP-connected set.

wcfs/internal/xbtree/blib/blib.go

@@ -25,6 +25,8 @@ import (
 	"math"
 
 	"lab.nexedi.com/kirr/neo/go/zodb/btree"
+
+	"lab.nexedi.com/nexedi/wendelin.core/wcfs/internal/set"
 )
 
 // XXX instead of generics
@@ -39,6 +41,8 @@ type KeyRange    = btree.LKeyRange
 const KeyMax Key = math.MaxInt64
 const KeyMin Key = math.MinInt64
 
+type setOid = set.Oid
+
 
 // KStr formats key as string.
 func KStr(k Key) string {
@@ -50,3 +54,8 @@ func KStr(k Key) string {
 	}
 	return fmt.Sprintf("%d", k)
 }
+
+
+func panicf(format string, argv ...interface{}) {
+	panic(fmt.Sprintf(format, argv...))
+}

wcfs/internal/xbtree/blib/pptreesubset.go

+// Copyright (C) 2018-2021  Nexedi SA and Contributors.
+//                          Kirill Smelkov <kirr@nexedi.com>
+//
+// This program is free software: you can Use, Study, Modify and Redistribute
+// it under the terms of the GNU General Public License version 3, or (at your
+// option) any later version, as published by the Free Software Foundation.
+//
+// You can also Link and Combine this program with other software covered by
+// the terms of any of the Free Software licenses or any of the Open Source
+// Initiative approved licenses and Convey the resulting work. Corresponding
+// source of such a combination shall include the source code for all other
+// software used.
+//
+// This program is distributed WITHOUT ANY WARRANTY; without even the implied
+// warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
+//
+// See COPYING file for full licensing terms.
+// See https://www.nexedi.com/licensing for rationale and options.
+
+package blib
+// PP-connected subset of tree nodes.
+
+import (
+	"fmt"
+
+	"lab.nexedi.com/kirr/neo/go/zodb"
+)
+
+const tracePPSet = false
+const debugPPSet = false
+
+// PPTreeSubSet represents PP-connected subset of tree node objects.
+//
+// It is
+//
+//	PP(xleafs)
+//
+// where PP(node) maps node to {node, node.parent, node.parent,parent, ...} up
+// to top root from where the node is reached.
+//
+// The nodes in the set are represented by their Oid.
+//
+// Usually PPTreeSubSet is built as PP(some-leafs), but in general the starting
+// nodes are arbitrary. PPTreeSubSet can also have many root nodes, thus not
+// necessarily representing a subset of a single tree.
+//
+// Usual set operations are provided: Union, Difference and Intersection.
+//
+// Nodes can be added into the set via AddPath. Path is reverse operation - it
+// returns path to tree node given its oid.
+//
+// Every node in the set comes with .parent pointer.
+//
+// XXX we only allow single parent/root case and report "tree corrupt" otherwise.
+type PPTreeSubSet map[zodb.Oid]*nodeInTree
+
+// nodeInTree represents tracking information about a node.
+type nodeInTree struct {
+	parent  zodb.Oid // parent node | InvalidOid for root
+	nchild  int      // number of direct children in PPTreeSubSet referring to this node
+}
+
+// Parent returns parent of this node.
+func (n *nodeInTree) Parent() zodb.Oid {
+	return n.parent
+}
+
+// NChild returns number of children of this node in the tree subset.
+func (n *nodeInTree) NChild() int {
+	return n.nchild
+}
+
+// Has returns whether node is in the set.
+func (S PPTreeSubSet) Has(oid zodb.Oid) bool {
+	_, ok := S[oid]
+	return ok
+}
+
+// Path returns path leading to the node specified by oid.
+//
+// The node must be in the set.
+func (S PPTreeSubSet) Path(oid zodb.Oid) (path []zodb.Oid) {
+	for {
+		t, ok := S[oid]
+		if !ok {
+			panicf("node %s is not in the set  <- %v", oid, path)
+		}
+
+		path = append([]zodb.Oid{oid}, path...)
+		oid = t.parent
+
+		if oid == zodb.InvalidOid {
+			break
+		}
+	}
+
+	return path
+}
+
+// AddPath adds path to a node to the set.
+//
+// Note: embedded buckets (leaf node with InvalidOid) are removed from the path.
+func (S PPTreeSubSet) AddPath(path []zodb.Oid) {
+	S.verify()
+	defer S.verify()
+
+	l := len(path)
+	if l == 0 {
+		panic("empty path")
+	}
+
+	// normalize path: remove embedded bucket and check whether it was an
+	// artificial empty tree.
+	path = NormPath(path)
+
+	// go through path and add nodes to the set
+	parent := zodb.InvalidOid
+	var pt *nodeInTree = nil
+	for _, oid := range path {
+		if oid == zodb.InvalidOid {
+			panicf("path has node with invalid oid: %v", path)
+		}
+
+		t, already := S[oid]
+		if !already {
+			t = &nodeInTree{parent: parent, nchild: 0}
+			S[oid] = t
+		}
+		if t.parent != parent {
+			// XXX -> error (e.g. due to corrupt data in ZODB) ?
+			panicf("node %s is reachable from multiple parents: %s %s",
+				oid, t.parent, parent)
+		}
+
+		if pt != nil && !already {
+			pt.nchild++
+		}
+
+		parent = oid
+		pt = t
+	}
+}
+
+// NormPath normalizes path.
+//
+// It removes embedded buckets and artificial empty trees.
+// Returned slice is subslice of path and aliases its memory.
+func NormPath(path []zodb.Oid) []zodb.Oid {
+	l := len(path)
+
+	// don't keep track of artificial empty tree
+	if l == 1 && path[0] == zodb.InvalidOid {
+		return nil
+	}
+
+	// don't explicitly keep track of embedded buckets - they all have
+	// InvalidOid, and thus, if kept in S, e.g. T/B1:a and another
+	// T/B2:b would lead to InvalidOid having multiple parents.
+	if l == 2 && path[1] == zodb.InvalidOid {
+		return path[:1]
+	}
+
+	return path
+}
+
+// ---- Union/Difference/Intersection ----
+
+// Union returns U = PP(A.leafs | B.leafs)
+//
+// In other words it adds A and B nodes.
+func (A PPTreeSubSet) Union(B PPTreeSubSet) PPTreeSubSet {
+	U := A.Clone()
+	U.UnionInplace(B)
+	return U
+}
+
+// UnionInplace sets A = PP(A.leafs | B.leafs)
+//
+// In other words it adds B nodes to A.
+func (A PPTreeSubSet) UnionInplace(B PPTreeSubSet) {
+	if tracePPSet {
+		fmt.Printf("\n\nUnion:\n")
+		fmt.Printf("  A: %s\n", A)
+		fmt.Printf("  B: %s\n", B)
+		defer fmt.Printf("->U: %s\n", A)
+	}
+
+	A.verify()
+	B.verify()
+	defer A.verify()
+
+	A.xUnionInplace(B)
+}
+
+// Difference returns D = PP(A.leafs \ B.leafs)
+//
+// In other words it removes B nodes from A while still maintaining A as PP-connected.
+func (A PPTreeSubSet) Difference(B PPTreeSubSet) PPTreeSubSet {
+	D := A.Clone()
+	D.DifferenceInplace(B)
+	return D
+}
+
+
+// DifferenceInplace sets A = PP(A.leafs \ B.leafs)
+//
+// In other words it removes B nodes from A while still maintaining A as PP-connected.
+func (A PPTreeSubSet) DifferenceInplace(B PPTreeSubSet) {
+	if tracePPSet {
+		fmt.Printf("\n\nDifference:\n")
+		fmt.Printf("  A: %s\n", A)
+		fmt.Printf("  B: %s\n", B)
+		defer fmt.Printf("->D: %s\n", A)
+	}
+
+	A.verify()
+	B.verify()
+	defer A.verify()
+
+	A.xDifferenceInplace(B)
+}
+
+// TODO Intersection
+
+func (A PPTreeSubSet) xUnionInplace(B PPTreeSubSet) {
+	if tracePPSet {
+		fmt.Printf("\n\n  xUnion:\n")
+		fmt.Printf("    a: %s\n", A)
+		fmt.Printf("    b: %s\n", B)
+		defer fmt.Printf("  ->u: %s\n", A)
+	}
+
+	δnchild := map[zodb.Oid]int{}
+
+	for oid, t2 := range B {
+		t, already := A[oid]
+		if !already {
+			t = &nodeInTree{parent: t2.parent, nchild: 0}
+			A[oid] = t
+			// remember to nchild++ in parent
+			if t.parent != zodb.InvalidOid {
+				δnchild[t.parent] += 1
+			}
+		} else {
+			if t2.parent != t.parent {
+				// XXX or verify this at Track time and require
+				// that update is passed only entries with the
+				// same .parent? (then it would be ok to panic here)
+				// XXX -> error (e.g. due to corrupt data in ZODB)
+				panicf("node %s is reachable from multiple parents: %s %s",
+					oid, t.parent, t2.parent)
+			}
+		}
+	}
+
+	A.fixup(δnchild)
+}
+
+func (A PPTreeSubSet) xDifferenceInplace(B PPTreeSubSet) {
+	if tracePPSet {
+		fmt.Printf("\n\n  xDifference:\n")
+		fmt.Printf("    a: %s\n", A)
+		fmt.Printf("    b: %s\n", B)
+		defer fmt.Printf("  ->d: %s\n", A)
+	}
+
+	δnchild := map[zodb.Oid]int{}
+
+	// remove B.leafs and their parents
+	for oid, t2 := range B {
+		if t2.nchild != 0 {
+			continue // not a leaf
+		}
+
+		t, present := A[oid]
+		if !present {
+			continue // already not there
+		}
+
+		if t2.parent != t.parent {
+			// XXX or verify this at Track time and require
+			// that update is passed only entries with the
+			// same .parent? (then it would be ok to panic here)
+			// XXX -> error (e.g. due to corrupt data in ZODB)
+			panicf("node %s is reachable from multiple parents: %s %s",
+				oid, t.parent, t2.parent)
+		}
+
+		delete(A, oid)
+		if t.parent != zodb.InvalidOid {
+			δnchild[t.parent] -= 1
+		}
+	}
+
+	A.fixup(δnchild)
+}
+
+// fixup performs scheduled δnchild adjustment.
+func (A PPTreeSubSet) fixup(δnchild map[zodb.Oid]int) {
+	A.xfixup(+1, δnchild)
+}
+func (A PPTreeSubSet) xfixup(sign int, δnchild map[zodb.Oid]int) {
+	if debugPPSet {
+		ssign := "+"
+		if sign < 0 {
+			ssign = "-"
+		}
+		fmt.Printf("\n  fixup:\n")
+		fmt.Printf("      ·: %s\n", A)
+		fmt.Printf("     %sδ: %v\n", ssign, δnchild)
+		defer fmt.Printf("    ->·: %s\n\n", A)
+	}
+
+	gcq := []zodb.Oid{}
+	for oid, δnc := range δnchild {
+		t := A[oid] // t != nil as A is PP-connected
+		t.nchild += sign*δnc
+		if t.nchild == 0 {
+			gcq = append(gcq, oid)
+		}
+	}
+
+	// GC parents that became to have .nchild == 0
+	for _, oid := range gcq {
+		A.gc1(oid)
+	}
+}
+
+// gc1 garbage-collects oid and cleans up its parent down-up.
+func (S PPTreeSubSet) gc1(oid zodb.Oid) {
+	t, present := S[oid]
+	if !present {
+		return // already not there
+	}
+	if t.nchild != 0 {
+		panicf("gc %s %v  (nchild != 0)", oid, t)
+	}
+
+	delete(S, oid)
+	oid = t.parent
+	for oid != zodb.InvalidOid {
+		t := S[oid]
+		t.nchild--
+		if t.nchild > 0 {
+			break
+		}
+		delete(S, oid)
+		oid = t.parent
+	}
+}
+
+
+// ---- verify ----
+
+// verify checks internal consistency of S.
+func (S PPTreeSubSet) verify() {
+	// TODO !debug -> return
+
+	var badv []string
+	badf := func(format string, argv ...interface{}) {
+		badv = append(badv, fmt.Sprintf(format, argv...))
+	}
+	defer func() {
+		if badv != nil {
+			emsg := "S.verify: fail:\n\n"
+			for _, bad := range badv {
+				emsg += fmt.Sprintf("- %s\n", bad)
+			}
+			emsg += fmt.Sprintf("\nS: %s\n", S)
+			panic(emsg)
+		}
+	}()
+
+	// recompute {} oid -> children  and verify .nchild against it
+	children := make(map[zodb.Oid]setOid, len(S))
+	for oid, t := range S {
+		if t.parent != zodb.InvalidOid {
+			cc, ok := children[t.parent]
+			if !ok {
+				cc = make(setOid, 1)
+				children[t.parent] = cc
+			}
+			cc.Add(oid)
+		}
+	}
+
+	for oid, t := range S {
+		cc := children[oid]
+		if t.nchild != len(cc) {
+			badf("[%s].nchild=%d  children: %s", oid, t.nchild, cc)
+		}
+	}
+
+	// verify that all pointed-to parents are present in the set (= PP-connected)
+	for oid := range children {
+		_, ok := S[oid]
+		if !ok {
+			badf("oid %s is pointed to via some .parent, but is not present in the set", oid)
+		}
+	}
+}
+
+
+// ---- misc ----
+
+// Clone returns copy of the set.
+func (orig PPTreeSubSet) Clone() PPTreeSubSet {
+	klon := make(PPTreeSubSet, len(orig))
+	for oid, t := range orig {
+		klon[oid] = &nodeInTree{parent: t.parent, nchild: t.nchild}
+	}
+	return klon
+}
+
+// Equal returns whether A == B.
+func (A PPTreeSubSet) Equal(B PPTreeSubSet) bool {
+	if len(A) != len(B) {
+		return false
+	}
+
+	for oid, ta := range A {
+		tb, ok := B[oid]
+		if !ok {
+			return false
+		}
+
+		if !(ta.parent == tb.parent && ta.nchild == tb.nchild) {
+			return false
+		}
+	}
+
+	return true
+}
+
+// Empty returns whether set is empty.
+func (S PPTreeSubSet) Empty() bool {
+	return len(S) == 0
+}
+
+func (t nodeInTree) String() string {
+	return fmt.Sprintf("{p%s c%d}", t.parent, t.nchild)
+}
+
+
+// ---- diff/patch ----
+
+// ΔPPTreeSubSet represents a change to PPTreeSubSet.
+//
+// It can be applied via PPTreeSubSet.ApplyΔ .
+//
+// The result B of applying δ to A is:
+//
+//	B = A.xDifference(δ.Del).xUnion(δ.Add)		(*)
+//
+// (*) NOTE δ.Del and δ.Add might have their leafs starting from non-leaf nodes in A/B.
+//     This situation arises when δ represents a change in path to particular
+//     node, but that node itself does not change, for example:
+//
+//            c*             c
+//           / \            /
+//         41*  42         41
+//          |    |         | \
+//         22   43        46  43
+//               |         |   |
+//              44        22  44
+//
+//     Here nodes {c, 41} are changed, node 42 is unlinked, and node 46 is added.
+//     Nodes 43 and 44 stay unchanged.
+//
+//         δ.Del = c-42-43   | c-41-22
+//         δ.Add = c-41-43   | c-41-46-22
+//
+//     The second component with "-22" builds from leaf, but the first
+//     component with "-43" builds from non-leaf node.
+//
+//         ΔnchildNonLeafs = {43: +1}
+//
+//     Only complete result of applying all
+//
+//         - xfixup(-1, ΔnchildNonLeafs)
+//         - δ.Del,
+//         - δ.Add, and
+//         - xfixup(+1, ΔnchildNonLeafs)
+//
+//     produces correctly PP-connected set.
+type ΔPPTreeSubSet struct {
+	Del PPTreeSubSet
+	Add PPTreeSubSet
+	ΔnchildNonLeafs map[zodb.Oid]int
+}
+
+// NewΔPPTreeSubSet creates new empty ΔPPTreeSubSet.
+func NewΔPPTreeSubSet() *ΔPPTreeSubSet {
+	return &ΔPPTreeSubSet{
+		Del: PPTreeSubSet{},
+		Add: PPTreeSubSet{},
+		ΔnchildNonLeafs: map[zodb.Oid]int{},
+	}
+}
+
+// Update updates δ to be combination of δ+δ2.
+func (δ *ΔPPTreeSubSet) Update(δ2 *ΔPPTreeSubSet) {
+	δ.Del.UnionInplace(δ2.Del)
+	δ.Add.UnionInplace(δ2.Add)
+	for oid, δnc := range δ2.ΔnchildNonLeafs {
+		δ.ΔnchildNonLeafs[oid] += δnc
+	}
+}
+
+// Reverse changes δ=diff(A->B) to δ'=diff(A<-B).
+func (δ *ΔPPTreeSubSet) Reverse() {
+	δ.Del, δ.Add = δ.Add, δ.Del
+	// ΔnchildNonLeafs stays the same
+}
+
+
+// ApplyΔ applies δ to S.
+//
+// See ΔPPTreeSubSet documentation for details.
+func (S PPTreeSubSet) ApplyΔ(δ *ΔPPTreeSubSet) {
+	if tracePPSet {
+		fmt.Printf("\n\nApplyΔ\n")
+		fmt.Printf("  A: %s\n", S)
+		fmt.Printf("  -: %s\n", δ.Del)
+		fmt.Printf("  +: %s\n", δ.Add)
+		fmt.Printf("  x: %v\n", δ.ΔnchildNonLeafs)
+		defer fmt.Printf("\n->B: %s\n", S)
+	}
+
+	S.verify()
+	δ.Del.verify()
+	δ.Add.verify()
+	defer S.verify()
+
+	S.xfixup(-1, δ.ΔnchildNonLeafs)
+	S.xDifferenceInplace(δ.Del)
+	S.xUnionInplace(δ.Add)
+	S.xfixup(+1, δ.ΔnchildNonLeafs)
+}

wcfs/internal/xbtree/blib/pptreesubset_test.go

+// Copyright (C) 2021  Nexedi SA and Contributors.
+//                     Kirill Smelkov <kirr@nexedi.com>
+//
+// This program is free software: you can Use, Study, Modify and Redistribute
+// it under the terms of the GNU General Public License version 3, or (at your
+// option) any later version, as published by the Free Software Foundation.
+//
+// You can also Link and Combine this program with other software covered by
+// the terms of any of the Free Software licenses or any of the Open Source
+// Initiative approved licenses and Convey the resulting work. Corresponding
+// source of such a combination shall include the source code for all other
+// software used.
+//
+// This program is distributed WITHOUT ANY WARRANTY; without even the implied
+// warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
+//
+// See COPYING file for full licensing terms.
+// See https://www.nexedi.com/licensing for rationale and options.
+
+package blib
+
+import (
+	"strings"
+	"testing"
+
+	"lab.nexedi.com/kirr/neo/go/zodb"
+)
+
+func TestPPTreeSubSetOps(t *testing.T) {
+	const (
+		a zodb.Oid = 0xa + iota
+		b
+		c
+		d
+		ø = zodb.InvalidOid
+	)
+	type S = PPTreeSubSet
+	type testEntry struct {
+		A, B       S
+		Union      S
+		Difference S
+	}
+	E := func(A, B, U, D S) testEntry {
+		return testEntry{A, B, U, D}
+	}
+
+	testv := []testEntry{
+		E(
+			S{},	// A
+			S{},	// B
+				S{},	// U
+				S{}),	// D
+
+		E(
+			S{a:{ø,0}},	// A
+			S{a:{ø,0}},	// B
+				S{a:{ø,0}},	// U
+				S{}),		// D
+
+		E(
+			S{a:{ø,0}},	// A
+			S{b:{ø,0}},	// B
+				S{a:{ø,0}, b:{ø,0}},	// U
+				S{a:{ø,0}}),		// D
+
+		E(
+			S{a:{ø,1}, b:{a,0}},	// A
+			S{a:{ø,1}, c:{a,0}},	// B
+				S{a:{ø,2}, b:{a,0}, c:{a,0}},	// U
+				S{a:{ø,1}, b:{a,0}}),		// D
+
+		E(
+			S{a:{ø,1}, b:{a,1}, c:{b,0}},	// A
+			S{a:{ø,1}, b:{a,1}, d:{b,0}},	// B
+				S{a:{ø,1}, b:{a,2}, c:{b,0}, d:{b,0}},	// U
+				S{a:{ø,1}, b:{a,1}, c:{b,0}}),		// D
+
+		E(
+			S{a:{ø,1}, b:{a,0}},	// A
+			S{a:{ø,1}, b:{a,0}},	// B
+				S{a:{ø,1}, b:{a,0}},	// U
+				S{}),			// D
+
+		E(
+			S{a:{ø,1}, b:{a,1}, c:{b,0}},	// A
+			S{a:{ø,1}, b:{a,1}, c:{b,0}},	// B (=A)
+				S{a:{ø,1}, b:{a,1}, c:{b,0}},	// U (=A)
+				S{}),				// D
+	}
+
+	// assert1 asserts that result of op(A,B) == resOK.
+	assert1 := func(op string, A, B, res, resOK S) {
+		t.Helper()
+		if res.Equal(resOK) {
+			return
+		}
+		op1 := op[0:1]
+		t.Errorf("%s:\n  A:   %s\n  B:   %s\n  ->%s: %s\n  ok%s: %s\n",
+			strings.Title(op), A, B, op1, res, strings.ToUpper(op1), resOK)
+	}
+
+	for _, tt := range testv {
+		Uab := tt.A.Union(tt.B)
+		Uba := tt.B.Union(tt.A)
+		Dab := tt.A.Difference(tt.B)
+
+		assert1("union",      tt.A, tt.B, Uab, tt.Union)
+		assert1("union",      tt.B, tt.A, Uba, tt.Union)
+		assert1("difference", tt.A, tt.B, Dab, tt.Difference)
+
+		Uaa := tt.A.Union(tt.A)
+		Ubb := tt.B.Union(tt.B)
+		Daa := tt.A.Difference(tt.A)
+		Dbb := tt.B.Difference(tt.B)
+
+		assert1("union",      tt.A, tt.A, Uaa, tt.A)
+		assert1("union",      tt.B, tt.B, Ubb, tt.B)
+		assert1("difference", tt.A, tt.A, Daa, S{})
+		assert1("difference", tt.B, tt.B, Dbb, S{})
+
+		// TODO also verify U/D properties like (A+B)\B + (A+B)\A + (A^B) == (A+B) ?
+	}
+}