multi-relation-composition.page

---
title: Example of composing of a multi-set with a multi-relation
...

A span from Bundle to Item. 

```clafer
abstract Bundle
    contains ->> Item +
```

Two bundles.

```clafer
B1 : Bundle
    [ contains in Bread ]
    [ #contains = 2 ]

B2 : Bundle
    [ contains = Milk, Butter ] 
```

And three items.

```clafer
abstract Item
Bread : Item
Butter : Item 
Milk : Item
```

A family of bundles

```clafer
abstract bundles ->> Bundle
bundlesIB1 : bundles ->> B1 2
bundlesIB2 : bundles ->> B2
``` 

A family to represent the result of the composition `bundles ; contains`.

```clafer
allItem ->> Item 6..*

[alloy|
fact {
  compose 
  [ c0_bundles,  c0_bundles_ref
  , c0_contains, ~r_c0_contains, c0_contains_ref
  , c0_allItem,  c0_allItem_ref
  ]
}

pred compose[
  idx: univ, i_elem: idx->univ,
  hd:  univ, sleg:   hd->univ,  tleg: hd->univ,
  res: univ, r_elem: res->univ]               
{
  some r1: res->idx | some r2: res->hd |
   all a: idx, b: hd |
    a.i_elem = b.sleg implies { one r: res | r.r1 = a && r.r2 = b } &&
    #res = #(idx<:i_elem).~(hd<:sleg) &&
    res<:r_elem = r2.(hd<:tleg)
}
|]
```

Escape to Alloy to compose `allItem = bundles ; contains`:

```
[alloy|
fact {
  compose 
  [ c0_bundles,  c0_bundles_ref                   // family
  , c0_contains, ~r_c0_contains, c0_contains_ref  // span
  , c0_allItem,  c0_allItem_ref                   // family
  ]
}

pred compose[
  idx: univ, i_elem: idx->univ,                 // family
  hd:  univ, sleg:   hd->univ,  tleg: hd->univ, // span
  res: univ, r_elem: res->univ]                 // family
{
  some idx && some hd    // why is it needed? we should be able to compose empty families and spans as well
  some r1: res->idx | some r2: res->hd |
   all a: idx, b: hd |
    a.i_elem = b.sleg implies { one r: res | r.r1 = a && r.r2 = b } &&
    #res = #(idx<:i_elem).~(hd<:sleg) &&
    res<:r_elem = r2.(hd<:tleg)
}
|]
```

```{.clafer .ide}
```

Next, we elaborate the model to illustrate the [composition of two multi-relations](multi-relation-composition2).