datatype 'a list = Cons 'a "'a list" | Nil
instantiation list :: (order) order
begin
fun less_eq_list :: "'a list ⇒ 'a list ⇒ bool" where
"less_eq_list Nil Nil = True" |
"less_eq_list (Cons _ _) Nil = True" |
"less_eq_list Nil (Cons _ _) = False" |
"less_eq_list (Cons _ a) (Cons _ b) = less_eq_list a b"
instance
proof
fix x y:: "'a list"
show "x ≤ x"
apply(induct_tac x)
apply(auto)
done
(* at this point the state is
show x ≤ x
Successful attempt to solve goal by exported rule:
?x2 ≤ ?x2
proof (state)
this:
x ≤ x
goal (3 subgoals):
1. ⋀x y. (x < y) = (x ≤ y ∧ ¬ y ≤ x)
2. ⋀x y z. x ≤ y ⟹ y ≤ z ⟹ x ≤ z
3. ⋀x y. x ≤ y ⟹ y ≤ x ⟹ x = y
*)
show "(x < y) = (x ≤ y ∧ ¬ y ≤ x)"
(* I get an error here
Failed to refine any pending goal
Local statement fails to refine any pending goal
Failed attempt to solve goal by exported rule:
(?x2 < ?y2) = (?x2 ≤ ?y2 ∧ ¬ ?y2 ≤ ?x2)
*)
qed
end
What is wrong with this? The proof of "x ≤ x" worked like a charm. Somehow "(x < y) = (x ≤ y ∧ ¬ y ≤ x)" doesn't match any subgoal.
Class order is a subclass of preorder, which in turn is a subclass of ord. Class ord requires you to define both less_eq (≤) and less (<). In your code, you have correctly defined less_eq_list but forgot to define less_list, and that's why you got an error when trying to prove (x < y) = (x ≤ y ∧ ¬ y ≤ x).