Re: [isabelle] Trying to use "datatype" to restrict types of formulas used, getting error



On 1/18/2013 1:28 AM, Andreas Lochbihler wrote:
you need to embed your formulas deeply in HOL, i.e., you cannot use the standard HOL connectives &, |, !, etc. for your syntax. So you must use your formula datatype also for the subexpressions of your connectives:
Hope this helps,
Andreas,

It helped a lot. I wouldn't have gotten to this next step without a fix.

I'll answer a few questions and then show you the failure I'm at now.

I am not sure what sT is supposed to stand for. Is it the type for variable names or the type of values that variables can take. In the former case: what is the type of values?

Type "sT" is a primitive type that represents a set. By "can take", I guess you mean "mapped to" as shown by your "env" below.

Semantically, I don't think a variable of type sT is mapped to another value in the sense of a function.

I have a function:

consts seS :: "sT => (sT => bool) => sT"

I then use a function of that type, (seS q P), in an axiom describing a property for every (q::sT) and (P::(sT => bool)), where P is a property that holds for q. (Actually the (sT => bool) type function is what I'm trying to tighten up with this sFOLf function.)

Variables of type sT are only used with the predicate \<in>, such as (x::sT \<in> y::sT), or as a binder variable in \<exists> or \<forall>, such as (!x. phi) or (? x. phi), where phi is a FOL formula built up starting with the atomic formulas (x \<in> y) and (x = y).

Variables of type sT are used in HOL functions, but that's only because that's how Isabelle makes me do it. The constant functions I define represent sets, and axioms are used to state what is true about those functions.

I could get more detailed, but I now get wellsortedness errors when trying to use the function sFOLf in a "value" statement.

The command

value "sFOLf1 sID (In x y)"

gives the error

"Wellsortedness error... Type sT not of sort equal. No type arity sT :: equal"

A similar error with "enum" in place of "equal" is after the third "value" command. The "Eq" and the "Forall" mess things up.

I'm trying to keep this short, but I'm not all that clear on the "environment" requirement. However, like I said, variables of type sT aren't really mapped anywhere.

To use the "value" command, I just made my "env" function the identity function.

The code is below, and I attached it as a THY.

Thanks for the help,
GB

theory sts__sFOLdt
imports Complex_Main
begin

typedecl sT

consts inS :: "sT => sT => bool"

datatype sFOLdt =
  In sT sT
| Eq sT sT
| Forall sT sFOLdt

type_synonym env = "(sT => sT)"

definition sID :: "sT => sT" where
  "sID s = s"

fun sFOLf :: "env => sFOLdt => bool" where
  "sFOLf E (In x y) = inS x y"

value "sFOLf sID (In x y)"

fun sFOLf1 :: "env => sFOLdt => bool" where
  "sFOLf1 E (In x y) = inS x y"
| "sFOLf1 E (Eq x y) = (E x = E y)"

value "sFOLf1 sID (In x y)"

fun sFOLf2 :: "env => sFOLdt => bool" where
  "sFOLf2 E (In x y) = inS x y"
| "sFOLf2 E (Forall x f) = (!v. sFOLf2 (E(x := v)) f)"

value "sFOLf2 sID (In x y)"

end





theory sTs__sFOLdt
imports Complex_Main
begin

typedecl sT

consts inS :: "sT => sT => bool"

datatype sFOLdt =
  In sT sT
| Eq sT sT
| Forall sT sFOLdt 

type_synonym env = "(sT => sT)"

definition sID :: "sT => sT" where
  "sID s = s"
  
fun sFOLf :: "env => sFOLdt => bool" where
  "sFOLf E (In x y) = inS x y"

value "sFOLf sID (In x y)"

fun sFOLf1 :: "env => sFOLdt => bool" where
  "sFOLf1 E (In x y) = inS x y"
| "sFOLf1 E (Eq x y) = (E x = E y)"

value "sFOLf1 sID (In x y)"

fun sFOLf2 :: "env => sFOLdt => bool" where
  "sFOLf2 E (In x y) = inS x y"
| "sFOLf2 E (Forall x f) = (!v. sFOLf2 (E(x := v)) f)"

value "sFOLf2 sID (In x y)"

end


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