Still on negative translation for substructural logics

On 01/12/2024 15:32, Mild Shock wrote:
 > So although it was very temping to download
 > you software, and then replace these line:
 > <https://gist.github.com/jp-diegidio/b6a7a071f9f81a469c493c4afcc5cb11>
 > By these line:
 > notation(gliv(X), (~(~X)))
 > solve_t__sel(neg, C=>X) :-
 >     solve(C=>gliv(X)).
Or just change the definition of `dnt`, or create another one.  The code is functional and pretty flexible actually, `notation/5` is `multifile`.
 > I am afraid I have no time for that. You
 > could do it by yourself. Or what
I have already done it, and I have already told you the results!  You have no time for anything apparently, except for posting random shit and fucking with thread titles: I have messages all over the place, and it's just me, you, and Ross once a month...
---
Here is an interesting new case, which I had thought should be *unsolvable*:
```
?- solve_w([]=>dnt((~(~p)->p)<->p\/(~p))), !.  % DNE<->LEM
```
Unsolvable not just because my logic is *affine*, but because the actual statement, provable intuitionistically, is intrinsically higher-order:
```
Theorem DNE_is_LEM : (forall p, ~ ~ p -> p) <-> (forall p, p \/ ~ p).
```
while the statement I am proving above is this one, and it is not intuitionistically provable (AFAICT):
```
Theorem DNE_is_LEM_not_quite : forall p, (~ ~ p -> p) <-> (p \/ ~ p)
```
Yet with my `dnt`, my solver proves even that one (but I still have to inspect the proof tree, what I actually get).  Indeed, I am rather worried that it just solves everything I throw at it, though not the falsities... which is why I am also developing a meta-theory for it, in Coq:
<https://gist.github.com/jp-diegidio/b6a7a071f9f81a469c493c4afcc5cb11#file-gentzen-v>
-Julio