Re: [isabelle] Instantiating to a lambda expression
On Sat, 2011-01-15 at 15:52 +0000, John Munroe wrote:
> If i have
> foo :: "real => real"
> bar :: "real => real"
> and an axiom
> same_ax: "ALL (g1::real=>real) g2 x y. (y > x & g1 y - g1 x = g2 y -
> g2 x) --> g1 = g2"
Axioms are evil. It's all too easy to introduce inconsistencies.
Here's a proof of False from same_ax:
apply (cut_tac same_ax)
apply (drule_tac x="%x. 0" in spec)
apply (drule_tac x="%x. 1" in spec)
apply (auto dest: fun_cong)
(The problem is that the scope of x and y in same_ax extends all the way
to the end of the formula.)
> However, if I change the type of bar to:
> bar :: "real => real => real", the following won't go through:
> lemma lem1: "ALL x. foo x = bar x 3"
> proof -
> have "ALL x y. y > x --> foo y - foo x = bar y 3 - bar x 3"
> then obtain r1 r2 where #: "r2 > r1" and ##: "foo r2 - foo r1 = bar
> r2 3 - bar r1 3"
> by auto
> then show ?thesis
> using same_ax
> by auto
Here's a proof of lem1 that's similar to the one you provided:
lemma lem1: "ALL x. foo x = bar x 3"
have "ALL x y. y > x --> foo y - foo x = bar y 3 - bar x 3"
then obtain r1 r2 where #: "r2 > r1" and ##: "foo r2 - foo r1 = bar
r2 3 - bar r1 3"
then show ?thesis
using same_ax[rule_format, of _ _ _ "%x. bar x 3"]
by (auto dest: fun_cong)
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