# [isabelle] evaluation of expressions

• To: "cl-isabelle-users at lists.cam.ac.uk" <cl-isabelle-users at lists.cam.ac.uk>
• Subject: [isabelle] evaluation of expressions
• From: Viorel Preoteasa <viorel.preoteasa at gmail.com>
• Date: Fri, 1 Mar 2019 10:54:48 +0200
• References: <875B1B8C-280D-413D-A1F1-2C98D2FC785D@gmail.com> <0F563D2E-1CDD-411A-BED7-1614D1DE0882@uibk.ac.at> <16c07875-31b3-adf4-e098-b27a537d71e3@gmail.com> <d0e84b6ff7e1485cae265938dcaf5ac6@chalmers.se>
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```Hello,

```
I have a lemma of the following form: "g (S f) = f" and I would like to use it evaluate for example "g (S Suc) 3" using
```

value "g (S Suc) 3"

```
Is it possible to achieve this behavior? Simplification works OK, but I need also a quick way to calculate this kind of expressions.
```

Next is the complete example

definition "S f x = f x"

lemma S_simp: "S f = f"

definition "g A = (SOME f . A = S f)"

lemma g_S_simp[code,simp]: "g (S f) = f"

lemma "g (S Suc) 3 = 4"
by (simp) (*this works OK*)

value "g (S Suc) 3" (*this gives Wellsortedness error*)

Best regards,

Viorel Preoteasa

On 19/02/2019 15.14, Thomas Sewell wrote:
```
```
```
I recall the addition of the signed words, and maybe I can add some information.
```

```
I think that "'a signed word" and "'a word" are equivalent for all key operations. The signed variant was added as part of an adjustment of the C-to-Isabelle parser, so that in the resulting code it was possible to tell whether the variables were designated signed or unsigned in C. Note that most arithmetic is exactly the same.
```

```
Also note that the outer type constructor is word, that is, "32 signed word" is an instance of "'a word" where the length parameter is "32 signed". To make use of this, I think you want to define some kind of is_signed class with a class constant of type bool, and ensure that "32 signed" is signed, and that each numeral type counts as unsigned. Finally, you can lift that to a query on the outer word type.
```

I think that all might work, but I haven't tried any of it.

Cheers,

Thomas.

------------------------------------------------------------------------
```
*From:* cl-isabelle-users-bounces at lists.cam.ac.uk <cl-isabelle-users-bounces at lists.cam.ac.uk> on behalf of Viorel Preoteasa <viorel.preoteasa at gmail.com>
```*Sent:* Tuesday, February 19, 2019 10:40:26 AM
*To:* Thiemann, René
*Cc:* cl-isabelle-users at lists.cam.ac.uk
*Subject:* Re: [isabelle] signed and unsigned words
Dear René,

On 18/02/2019 15.18, Thiemann, René wrote:
> Dear Viorel,
>
```
> it is difficult to test your problem, since the “signed” and “usigned” types are something that I do neither find in the distribution nor in the AFP.
```
The definition of the signed type is here:

https://www.isa-afp.org/browser_info/current/AFP/Word_Lib/Signed_Words.html

and the definition of unsigned is identical.

>
> Hence, I can only blindly guess:
>
```
> There might be a problem in your definition of “overflow_add” which uses the generic “to_int”, not a specific implementation. > This will work if “to_int” is defined as a class-constant (class some_name = fixes to_int :: …), but I’m not sure whether your approach via “consts to_int” is supported with code-generation.
```>
```
> So, you might try to reformulate the definition of to_int via class and instantiation.
```
This was my first attempt, but I did not manage the instantiation  of
class overflow as 'a signed word .

Now I have implemented a different solution. I created a new copy of the
type 'a word:

typedef (overloaded) 'a::len sword = "UNIV::'a word set"
morphisms sword_to_word word_to_sword by auto

and I lifted all operations from word to sword. This seems to work, but
I am interested to find the best possible solution.

Best regards,

Viorel

>
> Best regards,
> René
>
```
>> I need new types of signed and unsigned words with operations (+, -, ...) identical with the operations on words, but with a new operation:
```>>
```
>> overflow_add a b which is defined differently for signed and unsigned words.
```>>
```
>> Basically I need the two types to be instantiations of the following class:
```>>
>> class overflow = plus +
>>   fixes overflow_add::  "'a ⇒ 'a ⇒ bool"
>>
```
>> I tried something inspired from the AFP entry "Finite Machine Word Library":
```>>
>> type_synonym 'a sword = "'a signed word"
>> type_synonym 'a uword = "'a usigned word"
>>
>> consts to_int :: "'a word ⇒ int"
>>
>>   to_int_sword ≡ "to_int:: 'a sword ⇒ int"
>>   to_int_uword ≡ "to_int::'a uword ⇒ int"
>> begin
>>   definition "to_int_sword (a:: 'a::len sword) = sint a"
>>   definition "to_int_uword (a:: 'a::len uword) = uint a"
>> end
>>
>> instantiation  word :: (len0)  overflow
>> begin
```
>>   definition "overflow_add a b = (to_int (a::'a word) + to_int b = to_int (a + b))"
```>> instance ..
>> end
>>
```
>> this seems to work, but I don't know how to get code generation for overflow_add:
```>>
>> value "overflow_add (-2::4 sword) (-3)"
>>
>> gives the error:
>>
>> No code equations for to_int

```
```
```

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