Isabelle: prove that this set notation of a matrice results in a finite set

Question

I am using the development version of Isabelle from the repository. I hope it is a good decicion. Sledgehammer is again greatly improved! The isabelle developpers are really great! The reason for the dev version is that Isabelle2013-2 was crashing too often.

How can I prove this lemma:

lemma finite_f_A:
  fixes A :: "('a::comm_ring_1 poly)^'n∷finite^'n∷finite"
    and f :: "('a::comm_ring_1 poly) ⇒ nat"
  shows "finite {f (A $ i $ j) | i j. True }"
sorry

My old proof that is no longer working with the development version of Isabelle is this:

proof-
  have "⋀ K. finite ((λ (i, j). f (A $ i $ j)) ` K)" using finite_imageI by simp
  from this show ?thesis by simp 
qed

Answer 1

The simplification procedure finite_Collect has been deactivated after the Isabelle2013-2 release in changeset 31afce809794, because it sometimes behaves surprisingly or crashes. You can reactivate it with using [[simproc add: finite_Collect]] between the lemma statement and the start of your proof. Then, your former proof works again.

By the way, it is in general not a good idea to use the repository version for your work unless you desparately need a new feature or want to participate in the Isabelle development process.