range restriction/domain restriction in Isabelle

Question

I am trying to input a schema into Isabelle however when I add range restriction or domain restriction into the theorem prover it doesn't want to parse. I have the following schema in LaTeX:

\begin{schema}{VideoShop}
members: \power PERSON \\
rented: PERSON \rel TITLE \\
stockLevel: TITLE \pfun \nat
\where
\dom rented \subseteq members \\
\ran rented \subseteq \dom stockLevel \\
\forall t: \ran rented @ \# (rented \rres \{t\}) \leq stockLevel~t
\end{schema}

When inputting this into Isabelle I get the following:

locale videoshop = 
fixes members :: "PERSON set"
and rented :: "(PERSON * TITLE) set"
and stockLevel :: "(TITLE * nat) set"
assumes "Domain rented \<subseteq> members" 
 and "Range rented \<subseteq> Domain stockLevel" 
 and "(\<forall> t. (t \<in> Range rented) \<and> (card (rented \<rhd> {t}) \<le> stockLevel t))"
begin
.....

It all parses except for the last expression \<forall> t.....

I just don't understand how to add range restriction into Isabelle.

score 1 · Accepted Answer · answered Feb 09 '16 at 08:16

1

There are multiple problems with your input.

The ⊳ symbol you are using in the expression
```
(rented ⊳ {t})
```
is not associated with any operator, so it can't be parsed. I'm not quite sure what it's supposed to mean. From the high-level idea of the specification I'm guessing something along the lines of "all persons who rented a specific title". This can be expressed most easily with a set comprehension:
```
{p. (p, t) ∈ rented}
```
You translated the bounded universal quantifier into a quantifier containing a conjunction. This is likely not what you want, because it says "for all t, t is in the range of rented and something else". Isabelle has notation for bounded quantifiers.
```
∀t ∈ Range rented. ...
```
You are trying to use stockLevel as a function, which it isn't. From your LaTeX input I gather that it's supposed to be a partial function. Isabelle calls these maps. The appropriate type is:
```
TITLE ⇀ nat
```
Note the "harpoon" symbol instead of a function arrow. The domain function for maps is called dom. The second locale assumption can be expressed as:
```
Range rented ⊆ dom stockLevel
```
Given that, you can use stockLevel as a function from TITLE to nat option.

answered Feb 09 '16 at 08:16

larsrh

2,579
8
30

Thanks, that helped a lot. What is the syntax to get the nat option for stockLevel? I've changed the \ expression to "(\ t \ Range rented. card ({p. (p, t) \ rented}) < (stockLevel t))" but theres a clash of types with the (stockLevel t) – lburski Feb 10 '16 at 16:51
1

`stockLevel t` gives you a `nat option`. To convert that to `nat`, you can use the `the` function: `the (stockLevel t)`. – larsrh Feb 10 '16 at 18:14
Another quick question, say i wanted to update the stockLevel so I had the expression stockLevel' = stockLevel \cup (t \mapsto n). How would i write that in Isabelle syntax, I've looked at the maps_add function but i don't quite get it and i don't know if thats what i need. @larsrh – lburski May 18 '16 at 11:14
Is `stockLevel` a function now, or still a set? – larsrh May 18 '16 at 14:03
It is a function using the ⇀ sign @larsrh – lburski May 18 '16 at 14:56
1

Try this: `stockLevel(t ↦ n)` – larsrh May 19 '16 at 07:52

range restriction/domain restriction in Isabelle

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