How do I prove that 4 is even using Isabelle - isabelle

Question

Assuming that I have this inductive definition for Even
inductive_set Even :: "Int.int set"
where null: "0 ∈ Even"
| plus: "x ∈ Even ⟹ x+2 ∈ Even"
| min: "x ∈ Even ⟹ x-2 ∈ Even"
How do I prove this lemma
lemma four_is_even: "4 ∈ Even" ?
I am relatively new to proof assistant.

Answer 1

As #Mathias Fleury informally explained above, there are several ways in which you can prove your lemma:
Using forward-style reasoning
You can first prove the following auxiliary lemma using the OF fact combinator:
lemma four_is_even_fwd': "0+2+2 ∈ Even"
by (fact plus [OF plus [OF null]])
and then let the simplifier take care of proving 0+2+2 = 4:
lemma four_is_even_fwd: "4 ∈ Even"
using four_is_even_fwd' by simp
Also, you can use the Isabelle/Isar proof language to produce a more structured and pedagogical proof:
lemma four_is_even_fwd_str: "4 ∈ Even"
proof -
have "0 ∈ Even"
by (fact null)
then have "0+2 ∈ Even"
by (fact plus)
then have "0+2+2 ∈ Even"
by (fact plus)
then show ?thesis
by simp
qed
Using backward-style reasoning
Similarly to the first example of the forward-style reasoning above, you can prove the following auxiliary lemma using apply-scripts:
lemma four_is_even_bwd': "0+2+2 ∈ Even"
apply (rule plus) (* subgoal: 0+2 ∈ Even *)
apply (rule plus) (* subgoal: 0 ∈ Even *)
apply (rule null) (* no subgoals! *)
done
and then again let the simplifier take care of proving 0+2+2 = 4:
lemma four_is_even_bwd: "4 ∈ Even"
using four_is_even_bwd' by simp
Of course, trying to prove your lemma using Even.min is much more involved.