How to generate code for the existential quantifier

Question

Here is a sample theory:

datatype ty = A | B | C

inductive test where
  "test A B"
| "test B C"

inductive test2 where
  "¬(∃z. test x z) ⟹ test2 x"

code_pred [show_modes] test .
code_pred [show_modes] test2 .

values "{x. test2 A}"

The generated code tries to enumerate over ty. And so it fails.

I'm tring to define an executable version of test predicate:

definition "test_ex x ≡ ∃y. test x y"

definition "test_ex_fun x ≡
  Predicate.singleton (λ_. False)
    (Predicate.map (λ_. True) (test_i_o x))"

lemma test_ex_code [code_abbrev, simp]:
  "test_ex_fun = test_ex"
  apply (intro ext)
  unfolding test_ex_def test_ex_fun_def Predicate.singleton_def
  apply (simp split: if_split)

But I can't prove the lemma. Could you suggest a better approach?

Answer 1

Existential quantifiers over an argument to an inductive predicate can be made executable by introducing another inductive predicate. For example:

inductive test2_aux where "test x z ==> test2_aux x"
inductive test2 where "~ test2_aux x ==> test2 x"

with appropriate code_pred statements. The free variable z in the premise of test2_aux acts like an existential. Since this transformation is canonical, code_pred has a preprocessor to do so:

code_pred [inductify] test2 .

does the job.