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[mukesh tiwari <mukeshtiwari.iiitm AT gmail.com>]

Hi Marko,

Thanks for the answer! Now my question is: how did you figure out ‘ go_app_stated_nicely’? Is there any theory behind it or just experience? (I generally,m avoid let in or match with syntax in proofs, but it seems they are very helpful)

Best,

Mukesh 

On Thu, 22 Dec 2022 at 23:01, Marko Doko <m.doko AT hw.ac.uk> wrote:

Hi Mukesh,

take a look at this:

Theorem go_app_stated_nicely:
   ∀ (xs : list A) (z : A),
   let (ys, zs) := go z xs in Permutation (ys ++ zs) xs.
Proof.
   intros xs z; induction xs; intros; simpl; [trivial | ].
   destruct (go z xs) as [ys zs].
   destruct (R a z); simpl; eauto using Permutation_cons, Permutation_sym, Permutation_cons_app.
Qed.

Theorem go_app :
     ∀ (xs ys zs : list A) (z : A),
     In z xs ->
     (ys, zs) = go z xs -> Permutation (ys ++ zs) xs.
Proof.
  intros xs ys zs z IN EQ.
  specialize (go_app_stated_nicely xs z).
  rewrite <- EQ.
  trivial.
Qed.




Best,
--
Marko
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