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# ?SetExp1 `= ?SetExp2

Set equality constraint
SetExp1
A set expression.
SetExp2
A set expression.

## Description

Constrain sets in both hand sides of equation so that SetExp1 and SetExp2 represent the same set.

A set expression is a set (variable or ground) or a set operation between two set expressions. Possible set operations are set union, set intersection and set difference, and the respective operators are `\/, `/\ and `\. These operators are defined as:

```:- op(500, yfx, `\/).   %set union
:- op(400, yfx, `/\).   %set intersection
:- op(300, yfx, `\ ).   %set difference
```

### Fail Conditions

Fails if SetExp1 can not be the same set as SetExp2.

No.

## Examples

```/*
Examples of equalities between set expressions, being S,T,U,V,W,X,Y,Z set variables:

X `= Y
S `\/ Y `\ Z `= T `/\ S `\/ U `/\ W
V `\ ([a,b,d] `\/ W) `\/ (S `\ W) `= ((T `/\ [3,9]) `\/ W) `\ U
*/

% just union:
?- [8,a,9] `\/ [i,8,o] `\/ [] `\/ [a,8,5] `= U.
U = [5,8,9,a,i,o]

?- X `\/ Y `= [8,9], glb_poss(X,GX,PX), glb_poss(Y,GY,PY).
GX = [], PX = [8,9], GY = [], PY = [8,9]

?- X `:: [a]+[b,c], X `\/ [b,n] `= U, glb_poss(X,GX,PX), glb_poss(U,GU,PU).
GX = [a], PX = [b,c], GU = [a,b,n], PU = [c]

?- sets([A,B],[a,b],[d,g,h,j],[cardinality:4]), A`\/B`=U, #(U,C), fd:dom(C,DomC).
DomC = [4,5,6]

?- sets([X,Y],[a,b],[d,g,h,j],[]), X`\/Y`=U, #(U,4), #(Y,C), fd:dom(C,DomC).
DomC = [2,3,4]

?- sets([S,X], [],[a,b], [cardinality:[0,2]]), U `= X `\/ S, #(U,C), fd:dom(C,DomC).
DomC = [0,2]

?- S `:: [a,c]+[b,g,h,j,l]:3, X`::[a]+[b,h,t,u,y]:2, U `= X `\/ S, #(U,C), fd:dom(C,DomC).
DomC = [3,4]

?- S `:: [a,c]+[b,g,h,j,l], X`::[a]+[b,h,t,u,y], U `= X `\/ S, #(U,C), fd:(C::0..3), #(X,CX), fd:dom(CX,DomCX).
DomCX = [1,2]

% just intersection:
?- I `= [4,6] `/\ [3,6,8].
I = [6]

?- [a,b,c,d,e] `/\ [a,b,c,e,f,g] `/\ [b,d,e,f,x,y] `= I.
I = [b,e]

?- X `:: [a]+[b,c], X `/\ [b,n] `= I, glb_poss(X,GX,PX), glb_poss(I,GI,PI).
GX = [a], PX = [b,c], GI = [], PI = [b]

?- S `:: [a]+[b,c], X`::[]+[7,8,9], I`::[]+[a,b,c,7,z,99], I `= X `/\ S.
I = []

?- sets([A,B],[a,b],[d,g,h,j],[cardinality:5]), A`/\B`=I, #(I,C), fd:dom(C,DomC).
DomC = [4,5]

?- S `:: [a,c]+[g,h,j,l], X`::[a]+[b,h,t,u,y], I `= X `/\ S, #(I,C), fd:dom(C,DomC).
DomC = [1,2]

% just difference:
?- D `= [4,6] `\ [3,6,8].
D = [4]

?- [a,b,c,d,e] `\ [a,e,f,g] `\ [b,d,e,f,x,y] `= D.
D = [c]

?- [a,b,c,d,e] `\ ([a,e,f,g] `\ [b,d,e,f,x,y]) `= D.
D = [b,c,d,e]

?- X `:: [a]+[b,c], X `\ [b,n] `= D, glb_poss(X,GX,PX), glb_poss(D,GD,PD).
GX = [a], PX = [b,c], GD = [a], PD = [c]

?- S `:: [a]+[b,c], X`::[]+[7,8,9], D`::[]+[a,b,c,7,z,99], D `= X `\ S, glb_poss(D,GD,PD).
GD = [], PD = [7]

?- sets([A,B],[a,b],[d,g,h,j],[cardinality:5]), A`\B`=D, #(D,C), fd:dom(C,DomC).
DomC = [0,1]

?- sets([X,Y],[a,b],[d,g,h,j],[]), X`\Y`=D, #(D,4).
X = [a,b,d,g,h,j], Y = [a,b]

?- S `:: [a,c,z]+[g,h], X`::[a]+[b,c,h,t], D `= S `\ X, #(D,C), fd:dom(C,DomC).
DomC = [1,2,3,4].

?- S `:: []+[a,b], X=S, #(S `\ X, C).
C = 0```