Here is an exercise for ZDD programming. Let X, Y be a family of sets. For X, Y, compute a family Z of sets defined

```
Z = { A in X | forall B in Y B is a subset of A }.
```

Let the symbol X / Y be for Z for simplicity in the following. For example.

```
{{a, b}} / {{a}} = {{a, b}}
{{a, b}, {c, d}, {a, d}} / {{a}, {d}} = {{a, d}}
```

This should be an easy exercise, but I took time to point out bugs on my initial easy codes. The codes listed below is now correct, I hope. The first one of the following equations was most difficult to find, which was against my intuition.

```
X / {} = X
X / {{}} = X
```

However, I noticed, if one stands on logical reading of forall/exists used in the given problem specification, both of them sounds quite natural.

## Sample Query, and codes.

```
% ?- #S, X<< +[*[a,b], *[c,d], *[a, d]], Y<< +[*[a], *[d]],
% zdd_divisible_by_all(X, Y, Z, S), psa(Z, S).
%@ zid = 8
%@ [a,d]
%@ -------------------
%@ S = ..,
%@ X = 11,
%@ Y = 12,
%@ Z = 8.
zdd_divisible_by_all(X, Y, X, _):-Y<2, !. % was a bug.
zdd_divisible_by_all(X, _, 0, _):-X<2, !. % was a bug.
zdd_divisible_by_all(X, Y, Z, S):- memo(div_by_all(X, Y)-Z, S),
( nonvar(Z) -> true
; cofact(X, t(A, L, R), S),
zdd_divisible_by_all(L, Y, L0, S),
zdd_divisible_by_all(A, R, Y, R0, S),
zdd_join(L0, R0, Z, S)
).
%
zdd_divisible_by_all(A, X, Y, Z, S):- Y<2, !,
cofact(Z, t(A, 0, X), S).
zdd_divisible_by_all(A, X, Y, Z, S):- cofact(Y, t(B, L, R), S),
zdd_compare(C, A, B, S),
( C = (<) ->
zdd_divisible_by_all(X, Y, Z0, S),
zdd_insert(A, Z0, Z, S)
; C = (=) ->
zdd_divisible_by_all(X, R, R1, S),
zdd_insert(A, R1, R0, S),
zdd_divisible_by_all(R0, L, Z, S)
; Z = 0 % Was a bug.
).
```

## `zdd_divisible_by_all/4` seems efficient.

```
% ?- N = 300, #S, numlist(1, N, Ns), X<<pow(Ns), Y << +[*Ns],
% time(zdd_divisible_by_all(X, Y, Z, S)), psa(Z, S).
%@ % 1,772,696 inferences, 0.127 CPU in 0.133 seconds (95% CPU, 14007317 Lips)
%@ zid = 900
%@ [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300]
%@ -------------------
%@ N = 300,
%@ S = ..,
%@ Ns = [1, 2, 3, 4, 5, 6, 7, 8, 9|...],
%@ X = 301,
%@ Y = Z, Z = 900.
```